Search Results for Topology

Biographies

1. Aleksandrov biography
   
   • After taking his examinations in 1921, Aleksandrov was appointed as a lecturer at Moscow university and lectured on a variety of topics including functions of a real variable, topology and Galois theory.
     
   • In July 1922 Aleksandrov and Urysohn went to spend the summer at Bolshev, near to Moscow, where they began to study concepts in topology.
     
   • In the summer of 1924 they also visited Hausdorff in Bonn and he was fascinated to hear the major new directions that the two were taking in topology.
     
   • Of course Aleksandrov also taught in Moscow University and from 1924 he organised a topology seminar there.
     
   • This was an important year in the development of topology with Aleksandrov and Hopf in Princeton and able to collaborate with Lefschetz, Veblen and Alexander.
     
   • During their year in Princeton, Aleksandrov and Hopf planned a joint multi-volume work on Topology the first volume of which did not appear until 1935.
     
   • In 1935 Aleksandrov went to Yalta with Kolmogorov, then finished the work on his Topology book in the nearby Crimea and the book was published in that year.
     
   • His methods allowed arguments of combinatorial and algebraic topology to be applied to point set topology and brought together these areas.
     
   • His influence on the class of young men studying topology under him was never purely mathematical, however real and significant that was.
     
   • Today the Department of General Topology and Geometry of Moscow State University is Russia's leading centre of research in set-theoretic topology.
     
   • After Aleksandrov's death in November 1982, his colleagues from the Department of Higher Geometry and Topology, in which he had held the chair, sent a letter to Moscow University's rector A A Logunov proposing that one of Aleksandrov's former students should become Head of the Department, to preserve Aleksandrov's scientific school.
     
   • On 28 December 1982 the rector issued a circular creating the Department of general topology and Geometry.
     
   • Also in memory of Aleksandrov's contributions to topology at Moscow University and his work with the Moscow Mathematical Society, there is an annual topological symposium Aleksandrov Proceedings held every May.
     

2. Brouwer biography
   
   • Other topics which interested Brouwer were topology and the foundations of mathematics.
     
   • His doctoral thesis [I M James (ed.), History of topology (Amsterdam, 1999), 1-24.',13)">13]:- .
     
   • revealed the twin interests in mathematics that dominated his entire career; his fundamental concern with critically assessing the foundations of mathematics, which led to his creation of intuitionism, and his deep interest in geometry, which led to his seminal work in topology ..
     
   • He quickly discovered that his ideas on the foundations of mathematics would not be readily accepted [I M James (ed.), History of topology (Amsterdam, 1999), 1-24.',13)">13]:- .
     
   • I discovered all of a sudden that the Schoenfliesian investigations concerning topology of the plane, on which I had relied in the fullest way, could not be taken as correct in all parts, so that my group-theoretic results also became doubt.
     
   • Despite the substantial contributions he had made to topology by this time, Brouwer chose to give his inaugural professorial lecture on intuitionism and formalism.
     
   • Even though his most important research contributions were in topology, Brouwer never gave courses on topology, but always on -- and only on -- the foundations of intuitionism.
     
   • It seemed that he was no longer convinced of his results in topology because they were not correct from the point of view of intuitionism, and he judged everything he had done before, his greatest output, false according to his philosophy.
     
   • As is mentioned in this quotation, Brouwer was a major contributor to the theory of topology and he is considered by many to be its founder.
     
   • The status of the subject when he began his research is well described in [I M James (ed.), History of topology (Amsterdam, 1999), 1-24.',13)">13]:- .
     
   • When Brouwer was beginning his career as a mathematician, set-theoretic topology was in a primitive state.
     
   • He did almost all his work in topology early in his career between 1909 and 1913.
     
   • As well as proving theorems of major importance in topology, Brouwer also developed methods which have become standard tools in the subject.
     
   • History Topics: Topology enters mathematics .
     

3. Listing biography
   
   • It is in this letter that the word "topology" appears for the first time.
     
   • He disliked the term "geometria situs", then used for topological ideas, and [I M James (ed.), History of Topology (Amsterdam, 1999), 909-924.',4)">4]:- .
     
   • The entire doctrine being rather new, he felt justified to give it a new name and therefore called it "topology", which he though more appropriate.
     
   • It was the first published use of the word topology although, as we mentioned above, it was first used in Listing's letter of 1836.
     
   • The subject was known as analysis situs for many years and only in the late 1920s was the English word topology used by Lefschetz.
     
   • By topology we mean the doctrine of the modal features of objects, or of the laws of connection, of relative position and of succession of points, lines, surfaces, bodies and their parts, or aggregates in space, always without regard to matters of measure or quantity.
     
   • However Pauline Listing's [I M James (ed.), History of Topology (Amsterdam, 1999), 909-924.',4)">4]:- .
     
   • Neither Listing nor his wife seemed capable of managing the family finances [I M James (ed.), History of Topology (Amsterdam, 1999), 909-924.',4)">4]:- .
     
   • The near bankruptcy came around the time that Listing was publishing another remarkable contribution to topology.
     
   • This second work on topology by Listing is discussed by detail in [Accad.
     
   • Yet he was [I M James (ed.), History of Topology (Amsterdam, 1999), 909-924.',4)">4]:- .
     
   • Breitenberger in [I M James (ed.), History of Topology (Amsterdam, 1999), 909-924.',4)">4] writes:- .
     
   • he invented a good many terms [other than topology], some of which have became current: "entropic phenomenona", "nodal points", "homocentric light", "telescopic system", " geoid" ..
     
   • History Topics: Topology enters mathematics .
     

4. Kerekjarto biography
   
   • In the summer semester of 1922, at the invitation of the University of Gottingen, he gave a course on topology and the following semester a course titled Mathematische Betrachtungen zur Kosmologie (Mathematical considerations in Cosmology).
     
   • The first of these courses was enlarged into a book Vorlesungen uber Topologie (Lectures on Topology) which appeared in the series Grundlehren der Mathematischen Wissenschaften in 1923.
     
   • Solomon Lefschetz was at that time writing his own famous monograph on topology in 1924 entitled L'analysis situs et la geometrie algebrique and he wrote a review (published in 1925) of Kerekjarto's book for the Bulletin of the American Mathematical Society [Bull.
     
   • It also gives under one cover a fairly complete treatment of the results obtained by Brouwer and his school on two dimensional topology, a useful thing indeed.
     
   • The material in the book may be essentially classified into three groups: (a) Topology of the plane and its curves, centering around the Jordan curve theorems and including such questions as invariance of dimensionality and regionality, structure of regions and their boundaries, the general closed curve, etc.
     
   • The level of the book is far below that of topology at that time, and the organisation is chaotic.
     
   • First, Hermann Weyl wrote that Kerekjarto's book completely changed his views on topology.
     
   • He spent the years 1923-24 and 1924-25 as a visiting lecturer at Princeton where he gave courses on topology and on continuous groups.
     
   • With his Lessons on topology and its applications he gained the respect of the most distinguished French mathematicians.
     
   • His most important scientific results were in the area of "classical" topology founded by Poincare and Brouwer and in the theory of continuous groups.
     
   • The famous speech by Hilbert at the International Congress in Paris had been of great importance to the development of topology.
     
   • It was these methods which also led to fundamental results on topology and Euclidean and hyperbolic geometry in 3 dimensions.
     
   • His expertise in this new branch of geometry, topology, was recognised in, among other ways, his being asked to write the chapter on 'Topology' in the Encyclopedie Francaise.
     

5. Steenrod biography
   
   • He took just one mathematics course but it was an important one for the future direction of his research, for he took a topology course given by Raymond Wilder who had been a student of Robert Moore.
     
   • He worked hard on topology problems which Wilder had given him and he made sufficient progress, despite working on his own, that by the end of the year he had written his first paper.
     
   • After his first research work on point-set topology Steenrod then worked on algebraic topology.
     
   • The book The Topology of Fibre Bundles was published in 1951.
     
   • The book presupposes little knowledge of algebraic topology from the reader.
     
   • The lectures gave an excellent introduction to a central problem of algebraic topology and its applications, namely the problem of extending continuous functions.
     
   • Finally we mention the important work which Steenrod did on homology theories which appeared in the famous book Foundations of algebraic topology which he wrote with Samuel Eilenberg and was published in 1952.
     
   • Steenrod received many honours for his major contribution to topology, the most important of which was his election to the National Academy of Sciences.
     
   • James writes in [I M James (ed.), History of Topology (Amsterdam, 1999), 883-908.',2)">2]:- .
     
   • Algebraic topology underwent a spectacular development in the years following the second world war.
     

6. Lefschetz biography
   
   • Solomon Lefschetz was a Russian born, Jewish mathematician who was the main source of the algebraic aspects of topology.
     
   • The result was a deep depression, but the tragedy eventually pushed him towards mathematics, which was the right subject for him and, of course, it was fortunate for topology that he found his true love of mathematics.
     
   • During these years he wrote a series of important papers on topology despite being out the mainstream of mathematical research.
     
   • It was my lot to plant the harpoon of algebraic topology into the body of the whale of algebraic geometry.
     
   • Our greatest debt to Lefschetz lies in the fact that he showed us that a study of topology was essential for all algebraic geometers.
     
   • In doing this he developed a theory of algebraic topology of algebraic varieties of higher dimension.
     
   • The word 'topology' comes from the title of a monograph written by Lefschetz in 1930.
     
   • Another text which would have a huge influence on the development of the field was Algebraic topology which was published in 1942.
     
   • In the course of his work he introduced many of what would be considered today the basic tools of algebraic topology [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
     
   • In his last book, he wrote on recent ideas outside his own area, namely the topology of Feynman integrals.
     
   • for indomitable leadership in developing mathematics and training mathematicians, for fundamental publications in algebraic geometry and topology, and for stimulating needed research in nonlinear control processes.
     

7. James biography
   
   • From there he won an Open Scholarship to the University of Oxford where he studied mathematics at Queen's College in the topology school of Henry Whitehead.
     
   • James has done wide ranging work in topology, particularly in homotopy theory.
     
   • These volumes covered Henry Whitehead's work in differential geometry, complexes and manifolds, homotopy theory, and algebraic and classical topology.
     
   • In 1976 James published The topology of Stiefel manifolds which was based on his lecture notes.
     
   • general topology and homotopy theory (1984), another book by James which was based on his lectures, is described as follows:- .
     
   • In this monograph, based on a set of sixteen lectures to students, the author expounds certain parts of general topology which are particulary relevant to homotopy theory.
     
   • Fibrewise topology (1988) is a treatise on general topology, uniform spaces, and homotopy theory from the point of view of fibres.
     
   • James published Handbook of algebraic topology in 1995.
     
   • In the late 1950s Henry Whitehead approached Robert Maxwell, the chairman of Pergamon Press, to start a new journal Topology although Whitehead never lived to see the first part appear.
     
   • James became an editor of Topology in 1962 and he has continued in that role ever since.
     
   • In 1999 he edited History of topology and contributed three chapters to the book: From combinatorial topology to algebraic topology; Topologists at conferences; and Some topologists.
     
   • This was especially true of topology, and the prince of Princeton topologists for most of that time was James Waddell Alexander.
     

8. Novikov Sergi biography
   
   • Sergei's mother, Ludmila Vsevolodovna Keldysh, was also an outstanding mathematician who became a professor of mathematics and made important contributions to set theory and geometric topology.
     
   • He decided to work on algebraic topology and his work was supervised by M M Postnikov.
     
   • Topology, on the other hand, was not such an important topic at that time as far as Moscow University was concerned, so when Postnikov went to China for the academic year 1958-59, Novikov was left without a supervisor.
     
   • Another important paper Some problems in the topology of manifolds connected with the theory of Thom spaces was published by Novikov in 1960.
     
   • From personal meeting with them Novikov learnt about the major directions and problems which were being studied in topology [European Mathematical Society Newsletter 42 (December 2001), 17-20.',5)">5]:- .
     
   • Novikov also became head of the Department of Higher Geometry and Topology of Moscow University in 1983 and, the following year he became head of the Department of Geometry and Topology of the Mathematical Institute of the USSR Academy of Sciences.
     
   • Since 1996 he has been working at the University of Maryland in the United States but retains close links with Russia with a research appointments in Moscow University, in the Landau Institute for Theoretical Physics, and as Head of the Geometry and Topology research groups at the Steklov Institute.
     
   • Novikov's work up to 1971 was on algebraic and differential topology; in particular he studied calculating stable homotopy groups and classifying smooth simply-connected manifolds of dimension greater than 4.
     
   • It is one of the most fundamental problems in topology.
     
   • for his fundamental and pioneering contributions to topology and to mathematical physics.
     
   • His early work in algebraic and differential topology includes such milestones as the calculation of cobordism rings and stable homotopy groups, proof of the topological invariance of rational Pontrjagin classes, formulation of the "Novikov Conjecture" on higher signature invariants, and proof of the existence of closed leaves in two-dimensional foliations of the 3-sphere.
     

9. Jones Burton biography
   
   • In [Proceedings of the 1999 Topology and Dynamics Conference, Salt Lake City, UT, Topology Proc.
     
   • results that had an enormous impact on the development of point set topology and continuum theory ..
     
   • Nyikos, in [Topology Conference,1979, Greensboro, N.C., 1979 (Greensboro, N.C., 1980), 27-38.',2)">2], writes of Jones's contributions to the normal Moore space problem:- .
     
   • When Burton Jones posed the normal Moore space problem in 1933, he probably had little inkling of the role this problem would play in the history of point-set topology.
     
   • It has given rise to hundreds of papers in topology and quite a few (and their number is growing!) in set theory as well.
     
   • But more importantly, it has helped to generate interest in topology by set theorists, and it has kept us point-set topologists happy with our subject by giving us a tantalizing set of problems and partial results for us to think and talk about.
     
   • However, Jones's contributions were not only in research; perhaps, as Rogers' suggests, the most important came though his teaching abilities [Proceedings of the 1999 Topology and Dynamics Conference, Salt Lake City, UT, Topology Proc.
     
   • In particular one's intuition is aided by examples of spaces that do not satisfy the axioms as much as by examples that do; for instance, (in my approach to general topology) topological spaces (even compact and Hausdorff) that are not semi-metric and semi-metric spaces which are not metric.
     

10. Hopf biography
    
    • He attended several courses by Schur in Berlin and he received his doctorate in 1925 with a thesis, supervised by Schmidt, studying the topology of manifolds.
      
    • It was an impressive piece of work which received the following praise from Schmidt in his report (see for example [History of topology (Amsterdam, 1999), 991-1008.',11)">11]):- .
      
    • The meeting soon became friendship; not only topology, not only mathematics was discussed; it was a fortunate and also a very happy time, not restricted to Gottingen but continued on many joint journeys.
      
    • This was an important year in the development of topology with Aleksandrov and Hopf in Princeton and able to collaborate with Lefschetz, Veblen and Alexander.
      
    • During their year in Princeton, Aleksandrov and Hopf planned a joint multi-volume work on Topology the first volume of which did not appear until 1935.
      
    • Most of Hopf's work was in algebraic topology where he can be thought of as continuing Brouwer's work.
      
    • In the early 1940s Hopf published [History of topology (Amsterdam, 1999), 991-1008.',11)">11]:- .
      
    • Frei and Stammbach in [History of topology (Amsterdam, 1999), 991-1008.',11)">11] pay this tribute to Hopf:- .
      
    • His work is closely linked with the emergence of algebraic topology; it is most decisively thanks to his early works that this area established itself as a new and important branch of mathematics.
      
    • his work has influenced profoundly the evolution not only of topology but of a large part of mathematics.
      

11. Cech biography
    
    • Through his widening mathematical interests, to some extent forced on him by his teaching duties, Cech became interested in topology, in particular he became one of the foremost experts on combinatorial topology.
      
    • The topology papers written by Polish mathematicians in the new journal Fundamenta mathematicae greatly excited him.
      
    • His early interests in topology were in homology theory, a topic on which he published in 1932, and he proved duality theorems for manifolds.
      
    • His aim was to bring together point-set topology and algebraic topology with his 1932 paper.
      
    • On hearing Cech talk about his results at a combinatorial topology conference in Moscow, Lefschetz invited him to visit Princeton and Cech made the visit during session 1935-36.
      
    • Cech was influenced by the work of Aleksandrov and Urysohn and he set up a topology seminar at Brno in 1936 which went on to produce 26 papers in 3 years.
      
    • Cech's interpretation became a very important tool of general topology and also of some branches of functional analysis.
      
    • The conference took place in 1961 under the name Symposium on general topology and its relations to Modern Analysis and Algebra.
      

12. Urysohn biography
    
    • The authors of [I M James (ed.), History of topology (Amsterdam, 1999), 1-24.
      
    • Urysohn soon turned to topology.
      
    • During the following year Urysohn worked through the consequences building a whole new area of dimension theory in topology.
      
    • It was an exciting time for the topologists in Moscow for Urysohn lectured on the topology of continua and often his latest results were presented in the course shortly after he had proved them.
      
    • As Crilly and Johnson write [I M James (ed.), History of topology (Amsterdam, 1999), 1-24.
      
    • It was an occasion which made Brouwer begin to think about topology again, for his interests had turned to intuitionism, the subject of his talk at Marburg.
      
    • Van Dalen writes in [I M James (ed.), History of topology (Amsterdam, 1999), 947-964.',13)">13] about their final mathematical visit which was to Brouwer:- .
      
    • As van Dalen writes [I M James (ed.), History of topology (Amsterdam, 1999), 947-964.',13)">13]:- .
      
    • Crilly and Johnson write [I M James (ed.), History of topology (Amsterdam, 1999), 1-24.
      
    • Considering that he only had three years to devote to topology, he made his mark in his chosen field with brilliance and passion.
      

13. Leray biography
    
    • This 1934 paper on topology and partial differential equations is of major importance:- .
      
    • After his 1934 paper with Schauder, Leray published a paper on algebraic topology in the following year on the topology of Banach spaces.
      
    • After his release in 1945 Leray published a three part work Algebraic topology taught in captivity.
      
    • algebraic topology should not only study the topology of a space, i.e.
      
    • algebraic objects attached to a space, invariant under homomorphisms, but also the topology of a representation (continuous map), i.e.
      
    • In his hands, energy estimates for partial differential equations became combined with ideas from algebraic topology (such as fixed point theorems) in a highly original combination which cracked open the toughest problems.
      
    • Mathematician of penetration and originality, whose inventions revolutionized partial differential equations and algebraic topology.
      

14. Adams Frank biography
    
    • He changed supervisors and began working on algebraic topology with Shaun Wylie.
      
    • However, he was most strongly influenced by Henry Whitehead, who led the foremost British school of algebraic topology.
      
    • These books are of major importance, and include Stable homotopy theory (1964), Lectures on Lie groups (1969), Algebraic topology: a student's guide (1972), Stable homotopy theory and generalized homology (1974), Localisation and completion (1975), and Infinite loop spaces (1978).
      
    • The exposition of the book is aimed at the reader who has some understanding of algebraic topology and would like to understand the aspects of the theory of compact Lie groups that are relevant to algebraic topology.
      
    • Algebraic topology: a student's guide (1972) is rather unusual.
      
    • It is in two parts, the first contains a description of the topics that Adams thought essential for any young mathematician interested in algebraic topology.
      
    • The second part contains excerpts from some famous papers on algebraic topology together with surveys of generalized cohomology theories and complex cobordism written by Adams.
      
    • in recognition of his solution of several outstanding problems of algebraic topology and of the methods he invented for this purpose which have proved of prime importance in the theory of that subject.
      

15. Eilenberg biography
    
    • It is not surprising that Eilenberg's interests quickly turned towards point set topology which, of course, was an area which flourished at the University of Warsaw at that time.
      
    • His thesis, concerned with the topology of the plane, was published in Fundamenta Mathematicae in 1936.
      
    • Most of Eilenberg's publications from this period were on point-set topology but there were signs, even at this early stage of his career, that he was moving towards more algebraic topics.
      
    • Algebra was not foreign to his topology! .
      
    • In 1940 there was an important topology conference organised at Michigan.
      
    • In 1945 they set out the axioms for homology and cohomology theory but they did not give proofs in their paper, leaving these to appear in their famous text Foundations of algebraic topology in 1952.
      
    • Thanks to Sammy's insight and his enthusiasm, this text drastically changed the teaching of topology.
      
    • The title "Homological Algebra" is intended to designate a part of pure algebra which is the result of making algebraic homology theory independent of its original habitat in topology and building it up to a general theory of modules over associative rings.
      
    • The conceptual flavour of homological algebra derives less specifically from topology than from the general "naturalistic" trend of mathematics as a whole to supplement the study of the anatomy of any mathematical entity with an analysis of its behaviour under the maps belonging to the larger mathematical system with which it is associated.
      

16. Seifert biography
    
    • it was in 1927 that his whole life took a new turn when he attended a topology course by William Threlfall who was a Privatdozent at the technical university.
      
    • Not only did Threlfall turn Seifert into an enthusiastic student of topology, but far more than that, they became firm friends and mathematical collaborators.
      
    • During Seifert's time at Gottingen, Aleksandrov was again a visitor and Seifert's visit only heightened his knowledge of, and passion for, topology.
      
    • Seifert and Threlfall also spent vacations together working on mathematics but of course it was to Leipzig that Seifert submitted his dissertation Topology of 3-dimensional fibred spaces on 1 February 1932 and he was awarded his doctorate of philosophy after his oral examination on 3 March.
      
    • Much of Seifert and Threlfall's collaboration at this time was working on making a textbook out of Threlfall's lecture notes on topology.
      
    • The book Lectures on topology was published in 1934.
      
    • Threlfall wrote a preface which reads (see for example [I M James (ed.), History of Topology (Amsterdam, 1999), 1021-1028.',1)">1]):- .
      
    • Puppe describes the merits of the text in [I M James (ed.), History of Topology (Amsterdam, 1999), 1021-1028.',1)">1]:- .
      
    • The book gives an excellent account of what was known in topology at that time.
      

17. Spanier biography
    
    • Spanier's doctoral supervisor was Norman Steenrod and under his supervision Spanier wrote a thesis on algebraic topology for which he was awarded his doctorate in 1947.
      
    • At Berkeley, Spanier built up a strong group working in geometry and topology by several appointments of topologists to the faculty of Berkeley and also by attracting many top topologists to spend periods as visitors at Berkeley.
      
    • From the time of his doctoral work until around the time of the publication of his classic teext algebraic topology in 1966, Spanier work almost exclusively on algebraic topology.
      
    • In all, Spanier published moer than forty papers in algebraic topology, contributing to most to most of the major research areas in the field, including cohomology operations, obstruction theory, homotopy theory, imbeddability of polyhedra in Euclidean spaces, and topology of function spaces.
      
    • Interestingly, one of Spanier's theories, now called Alexander-Spanier homology, is currently being applied to analyse differential equations - a return to Poincare's original use of algebraic topology.
      
    • We have suggested that his work on algebraic topology went on until around the time that his famous book was published in 1966.
      
    • Spanier returned to algebraic topology for the publications in the last years of his life.
      

18. Kuratowski biography
    
    • As early as 1917 [Janiszewski and Mazurkiewicz] were conducting a topology seminar, presumably the first in that new, exuberantly developing field.
      
    • During his time in the United States he also made contact with Robert Moore's topology group, meeting mathematicians who he would keep in contact with for many years.
      
    • He contributed the third volume in this series with his monograph on topology which we will mention again below.
      
    • Kuratowski's main work was in the area of topology and set theory.
      
    • he used Boolean algebra to characterise the topology of an abstract space independently of the notion of points.
      
    • He was the author of Topologie, referred to above, which was the crowning achievement of the Warsaw School in point set topology.
      
    • He also considered the topology of the continuum, the theory of connectivity, dimension theory, and answered measure theory questions.
      
    • Kuratowski's Foreword to Set Theory and Topology .
      
    • Kuratowski's Introduction to Topology .
      

19. Whyburn biography
    
    • Of course with Moore having a deep interest in topology, that was the direction that Whyburn took and it was to become the topic of his research throughout his life.
      
    • 79 (1973), 1174-1182.',4)">4] and [Topology Proc.
      
    • Whyburn presented his first paper, which was on cyclic elements for locally connected plane continua, at the Western Christmas Meeting of the American Mathematical Society in Chicago on 31 December 1926 (see [Topology Proc.
      
    • In 1942 he published his famous text Analytic Topology in which he says [1]:- .
      
    • Analytic topology is meant to cover those phases of topology which are being developed advantageously by methods in which continuous transformations play the essential role.
      
    • Later major texts by Whyburn were Topological analysis (1958) and Dynamic topology which was jointly authored by Edwin Duda and was published 10 years after Whyburn's death.
      
    • The term "dynamic topology", writes Whyburn [Topology Proc.
      

20. Bott biography
    
    • Topology and Lie groups (Birkhauser Boston, Inc., Boston, MA, 1994), 3-9.',9)">9]:- .
      
    • Topology and Lie groups (Birkhauser Boston, Inc., Boston, MA, 1994), 3-9.',9)">9]:- .
      
    • The aim of the invitation was so that Bott could write a book on network theory but in fact the experience took his interests away from network theory and towards topology [17:- .
      
    • In particular Steenrod was giving a course on fibre bundles which he was running as part of his programme of writing the now classic text The Topology of Fibre Bundles which was published in 1951.
      
    • Volume 1 contains papers on topology and Lie groups most of which were written between 1949 and 1962.
      
    • The third volume of collected papers by Raoul Bott represents his works on the algebraic topology aspects of foliations and Gelfand-Fuchs cohomology.
      
    • The main themes of the papers included in [Volume 4] are the geometry and topology of the Yang-Mills equations and the rigidity phenomena of vector bundles.
      
    • his many fundamental contributions in topology and differential geometry and their application to Lie groups, differential operators and mathematical physics.
      

21. Moore Robert biography
    
    • By the time he was appointed in 1920 he had published 17 papers on point-set topology (a term which he coined).
      
    • For his doctoral thesis Moore had worked on the foundations of topology.
      
    • Moore wrote up his work on point-set topology in the important book Foundations of point set topology published in 1932.
      
    • Moore would begin his graduate course in topology by carefully selecting the members of the class.
      
    • If a student had already studied topology elsewhere or had read too much, he would exclude him (in some cases, he would run a separate class for such students).
      
    • He would usually caution the group not to read topology but simply to use their own ability.
      

22. Whitney biography
    
    • As impressive as each individual paper is, it is even more impressive to see them grouped together in two volumes, the first including papers in graphs and combinatorics, differentiable functions and singularities, and analytic spaces, and the second containing contributions to manifolds, bundles and characteristic classes, topology and algebraic topology, and geometric integration theory.
      
    • His main work, however, was in topology, particularly in the theory of manifolds.
      
    • Continuing work started by Veblen and Henry Whitehead, Whitney produced fundamental work in differential topology in 1935.
      
    • He published the book Geometric integration theory In 1957 which describes his work on the interactions between algebraic topology and the theory of integration.
      
    • for his fundamental work in algebraic topology, differential geometry and differential topology.
      

23. Veblen biography
    
    • Under their direction he laid the basis for the important work he was later to achieve in the fields of foundations of geometry, projective geometry, topology, differential invariants and spinors.
      
    • Already at this time he had begun to undertake research in topology (or analysis situs as it was then called) and he published Theory on plane curves in non-metrical analysis situs in 1905.
      
    • We mentioned above that Veblen's first work on topology appeared just before he arrived in Princeton.
      
    • He went on to establish Princeton as one of the leading centres in the world for topology research.
      
    • Analysis Situs (1922) provided the first systematic coverage of the basic ideas of topology and contributed to the development of modern topology.
      
    • He was the Colloquium Lecturer for the Society in 1916 when he gave a series of lectures on topology.
      

24. Frechet biography
    
    • It was after going to Strasbourg that he began to become interested in statistics but he only published a small number of articles on probability at this stage, most of his papers being on general analysis and topology.
      
    • As we have indicated, Frechet made major contributions to the topology of point sets, and defined and founded the theory of abstract spaces.
      
    • His introduction of general topology has been somewhat less appreciated than would otherwise have been the case since the publication of Hausdorff's major text in 1914 provided a more popular view.
      
    • made a major contribution toward laying the foundations of general topology and abstract analysis.
      
    • Entertaining reading about combinatorial topology accessible to a reader with very little mathematical preparation.
      
    • Of particular interest in connection with the development of topology are the letters written between 1920 and the 1930s.
      
    • History Topics: Topology enters mathematics .
      

25. Borsuk biography
    
    • Rapidly a strong school of mathematics grew up in the University of Warsaw, with topology being one of the main topics.
      
    • When Borsuk entered the University of Warsaw to study mathematics it was an exciting centre for research in topology; Saks and Mazurkiewicz were both teaching at the University and making major advances in the topic.
      
    • From him I learned about the truly geometric, more visual, almost "palpable" tricks and methods of topology.
      
    • He influenced strongly the development of the whole area of infinite-dimensional topology with his theory of retracts and his theory of shape which became major topics for discussion at his seminar.
      
    • Shape theory grew up at the same time as infinite-dimensional topology and the interaction between the two fields was of great mutual benefit.
      
    • The year 1978 is also that in which Borsuk organised the International Conference on Geometric Topology, held in Warsaw, which [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
      
    • demonstrated his widespread and profound influence on topology and the high regard in which he was held.
      

26. Rennie biography
    
    • The author defines a new topology (the "L-topology") in a lattice L, by taking as a basis of open sets, those convex sets S, whose intersection with any chain is an open set of the chain.
      
    • In a metric lattice, vx is continuous in the L-topology if and only if it is continuous on all chains; in this case ("continuous metric lattice"), metric convergence is equivalent to L-convergence.
      
    • In the lattice L(H) of closed subsets of a Hausdorff space, the empty set 0 is isolated if and only if H is compact; moreover, if H is locally compact, then L is a Hausdorff space in the L-topology.
      
    • In any Banach lattice Lp , p > 1, the L-topology is the metric topology; in any Banach lattice B, it is the star topology.
      

27. Tikhonov biography
    
    • However he did not stop there and continued his investigations in topology.
      
    • By 1926 he had discovered the topological construction which is today named after him, the Tikhonov topology defined on the product of topological spaces.
      
    • How was it possible that a topology introduced by means of such enormous neighbourhoods, which are only distinguished from the whole space by a finite number of the coordinates, could catch any of the essntial characteristics of a topological product? .
      
    • His results on the Tikhonov topology of products were achieved before he graduated in 1927.
      
    • His first-class achievements in topology and functional analysis, in the theory of ordinary and partial differential equations, in the mathematical problems of geophysics and electrodynamics, in computational mathematics and in mathematical physics are all widely known.
      
    • In fact Tikhonov's work led from topology to functional analysis with his famous fixed point theorem for continuous maps from convex compact subsets of locally convex topological spaces in 1935.
      
    • These results are of importance in both topology and functional analysis and were applied by Tikhonov to solve problems in mathematical physics.
      

28. Riemann biography
    
    • Through Weber and Listing, Riemann gained a strong background in theoretical physics and, from Listing, important ideas in topology which were to influence his ground breaking research.
      
    • Monastyrsky writes in [Rieman, Topology and Physics (Boston-Basel, 1987).',6)">6]:- .
      
    • However [Rieman, Topology and Physics (Boston-Basel, 1987).',6)">6]:- .
      
    • In 1858 Betti, Casorati and Brioschi visited Gottingen and Riemann discussed with them his ideas in topology.
      
    • Their proposal read [Rieman, Topology and Physics (Boston-Basel, 1987).',6)">6]:- .
      
    • Monastyrsky writes in [Rieman, Topology and Physics (Boston-Basel, 1987).',6)">6]:- .
      
    • History Topics: Topology enters mathematics .
      

29. Bing biography
    
    • In fact he had been invited by Lefschetz to join the Faculty at Princeton but only on the condition that he gave up topology which Lefschetz at this time seemed to consider as having little future.
      
    • Bing's work on topology ranged across many different areas of the subject.
      
    • He wrote papers on general topology, particularly on metrization; planar sets where he examined in particular planar webs, cuttings and planar embeddings.
      
    • Bing published The geometric topology of 3-manifolds in 1983.
      
    • This book is a classic in the study of the geometric topology of 3-manifolds.
      
    • Virtually everything that is known about 3-manifolds from the standpoint of geometric topology is included here.
      

30. Montgomery biography
    
    • His thesis advisor at Iowa was Edward W Chittenden and his thesis was on point-set topology.
      
    • His research interests included algebraic and geometric topology and he made major advances to transformation groups.
      
    • For many years he was at the centre of activity in topology at the Institute for Advanced Study.
      
    • His interests turned from point-set topology to transformation groups quite early in his career and he published a series of papers on the topic in collaboration with Leo Zippin.
      
    • In a long series of papers written in the late 1960s and early 1970s, [Montgomery and C T Yang] used the study of group actions on homotopy 7-spheres to showcase and test the growing new techniques of differential topology, especially index theory and surgery theory.
      
    • At a time when much work in topology consisted in building these machines, their papers demonstrated the beauty of applying this theory to unfurl complexities of symmetry and structure.
      

31. Wilder biography
    
    • When he asked permission from Moore to take his topology course, Moore replied (see for example [Biographical Memoirs National Academy of Sciences 82 (2003), 336-351.',3)">3]):- .
      
    • No, there is no way a person interested in actuarial mathematics could do, let alone be really interested in, topology.
      
    • The initial phase of Wilder's research on the Schonflies programme, which we described above, was in in set-theoretic topology and lasted until around 1930.
      
    • After this he worked in algebraic topology, and in 1932 he called for the unification of the two areas.
      
    • This work was presented in a unified form in Topology of Manifolds (1949); this was reprinted in 1963 and again in 1979 with a few notes on the current status of the problems.
      
    • It was around the time that Wilder published the first edition of Topology of Manifolds that his research interests underwent a major change.
      

32. Dieudonne biography
    
    • He worked in a wide variety of mathematical areas including general topology, topological vector spaces, algebraic geometry, invariant theory and the classical groups.
      
    • Here the term "classical group" is used as in the author's monograph, Sur les groupes classiques (1948) and the "elementary theory" refers roughly to results which involve subgroup and homomorphisms as opposed to results concerned for example with topology, differential geometry, etc.
      
    • He published texts such as History of functional analysis (1981), History of algebraic geometry (1985), Pour l'honneur de l'esprit humain (1987), A history of algebraic and differential topology 1900-1960 (1989), and L'ecole mathematique francaise du XXe siecle (2000).
      
    • Mac Lane, in a review of A history of algebraic and differential topology, writes:- .
      
    • is a well-informed and detailed analysis of the problems and development of algebraic topology, from Poincare and Brouwer to Serre, Adams, and Thom.
      
    • This book demonstrates in the case of topology how these obstacles can be overcome, with enlightening results.
      

33. Newman biography
    
    • in recognition of his distinguished contributions to combinatory topology, Boolean algebras and mathematical logic.
      
    • His mathematical work was in the field of combinatorial topology where he greatly influenced his friend Henry Whitehead.
      
    • He only wrote one book Elements of the topology of plane sets of points (1939).
      
    • this is the only text in general topology which can be wholeheartedly recommended without qualification.
      
    • Newman saw, and presented, topology as part of the whole of mathematics, not as an isolated discipline: and many must wish he had written more.
      
    • His early work on Combinatory Topology has exercised a decisive influence on the development of that subject.
      

34. Poincare biography
    
    • Poincare's Analysis situs , published in 1895, is an early systematic treatment of topology.
      
    • He can be said to have been the originator of algebraic topology and, in 1901, he claimed that his researches in many different areas such as differential equations and multiple integrals had all led him to topology.
      
    • For 40 years after Poincare published the first of his six papers on algebraic topology in 1894, essentially all of the ideas and techniques in the subject were based on his work.
      
    • Even today the Poincare conjecture remains as one of the most baffling and challenging unsolved problems in algebraic topology.
      
    • History Topics: Topology enters mathematics .
      

35. Alexander biography
    
    • In a collaboration with Veblen, he showed that the topology of manifolds could be extended to polyhedra.
      
    • Soon after arriving in Princeton, Alexander generalised the Jordan curve theorem and continued his work, now exclusively on topology, with an important paper on the Jordan-Brouwer separation theorem.
      
    • A mathematician of unusual depth and power, Alexander was a principal figure in the American development of algebraic/combinatorial topology.
      
    • Much of his work was of such a basic character that it became common knowledge in topology, with its discoverer being forgotten as a result..
      
    • an imposing figure who possessed great charm and a very "youthful" view of mathematics, being one of the first American mathematicians to fully appreciate the use of modern algebraic methods in topology.
      

36. Hurewicz biography
    
    • Rapidly a strong school of mathematics grew up in the University of Warsaw, with topology being one of the main topics.
      
    • Although Hurewicz knew intimately the topology that was being studied in Poland he chose to go to Vienna to continue his studies.
      
    • Hurewicz died falling off a ziggurat (a Mexican pyramid) on a conference outing at the International Symposium on algebraic topology in Mexico.
      
    • Hurewicz's early work was on set theory and topology and [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
      
    • In the field of general topology his contributions are centred around dimension theory.
      

37. Zarankiewicz biography
    
    • From the time its reopening the university had rapidly become a leading world centre for topology.
      
    • Janiszewski and Mazurkiewicz were conducting a topology seminar there from 1917 onwards, Sierpinski arrived in 1918, and in 1919, the year Zarankiewicz arrived, Kuratowski had just graduated and was beginning his doctoral studies.
      
    • With such a concentration on topology, and the excitement of those studying this new discipline in their newly freed country, it is not surprising that this was the area which attracted Zarankiewicz.
      
    • Zarankiewicz did important work in topology and graph theory.
      
    • His work on triangular numbers inspired Sierpinski to further work on this topic while Zarankiewicz also worked jointly with Kuratowski on topology.
      

38. Dehn biography
    
    • Max Dehn wrote one of the first systematic expositions of topology (1907) and later formulated important problems on group presentations, namely the word problem and the isomorphism problem.
      
    • I recall two cases of failures: Max Dehn, noted for work in topology, got only a weak position.
      
    • In 1907 Dehn wrote one of the first systematic expositions of topology jointly with Heegaard.
      
    • At that time topology was called 'analysis situs'.
      
    • Dehn's work in topology had led him into the study of groups, particularly group presentations which arise naturally from topological considerations.
      

39. Dowker biography
    
    • 16 (5) (1984), 535-541.',4)">4] (or see [Aspects of topology (Cambridge-New York, 1985), xi-xvii.',3)">3]):- .
      
    • Although most of Dowker's work was in topology, his war work set him up well for the applied mathematics post where he continued his research on projectiles.
      
    • James in [I M James (ed.), History of Topology (Amsterdam, 1999), 883-908.',2)">2] summarises his topological work:- .
      
    • While he is best known for his work in point set topology, he also made contributions to category theory, sheaf theory and the theory of knots.
      
    • 16 (5) (1984), 535-541.',4)">4] (or see [Aspects of topology (Cambridge-New York, 1985), xi-xvii.',3)">3]) Strauss describes Dowker's character:- .
      

40. Hausdorff biography
    
    • After 1904 Hausdorff began working in the area for which he is famous, namely topology and set theory.
      
    • Earlier results on topology fitted naturally into the framework set up by Hausdorff as Katetov explains in [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
      
    • He continued to undertake research in topology and set theory but the results could not be published in Germany.
      
    • We also mentioned his work on ordered sets and his masterpiece on set theory and topology Grundzuge der Mengenlehre (1914).
      
    • History Topics: Topology enters mathematics .
      

41. Hirzebruch biography
    
    • He also studied algebraic topology and algebraic geometry with Heinz Hopf at the Eidgenossische Technische Hochschule in Zurich from 1949 to 1950.
      
    • Basically, the role of topology in number theory has progressed beyond the local methods such as p-adic theory to global methods such as intersection numbers of homology classes.
      
    • for outstanding work combining topology, algebraic and differential geometry, and algebraic number theory; and for his stimulation of mathematical cooperation and research.
      
    • For the past three and a half decades, the name of Professor Friedrich Hirzebruch has been connected with famous results in the areas of topology, algebraic geometry, and global differential geometry, results which all mark the beginning of important theories and which have had an enormous influence on the development of modern mathematics.
      
    • the 'topological' proof of the Dedekind reciprocity theorem through 4-manifold theory and other fascinating relations between differential topology and algebraic number theory .
      

42. Jordan biography
    
    • Volumes 1 and 2 contain Jordan's papers on finite groups, Volume 3 contains his papers on linear and multilinear algebra and on the theory of numbers, while Volume 4 contains papers on the topology of polyhedra, differential equations, and mechanics.
      
    • Topology (called analysis situs at that time) played a major role in some of his first publications which were a combinatorial approach to symmetries.
      
    • He introduced important topological concepts in 1866 built on his knowledge of Riemann's work in topology but not the work by Mobius for he was unaware of it.
      
    • However between the editions Jordan had taught more advanced courses on analysis at the College de France and this may have influenced him to put set topology right up front in the second edition.
      
    • History Topics: Topology enters mathematics .
      

43. Mazur Barry biography
    
    • His achievement was already remarkable for by this time he had proved the Schoenflies Conjecture in geometric topology.
      
    • Mazur began his research career in geometric topology but has become one of the world's leading experts in number theory after working in algebraic geometry.
      
    • I came to number theory through the route of algebraic geometry and before that, topology.
      
    • Mazur's work in topology was outstanding and it led to the award of the Veblen Prize in 1966 for his work on the generalized Schoenflies theorem.
      
    • 47 (4) (2000), 477-480.',1)">1] how he moved from topology to algebraic geometry:- .
      

44. Fomin biography
    
    • Kolmogorov suggested problems in the theory of dynamical systems for Fomin to investigate, but Fomin was also advised by Aleksandrov to look at some problems in point-set topology and he also began to work in this area.
      
    • Fomin's topological papers are not numerous, but they are undoubtedly classical pieces of general topology.
      
    • The work for his papers in topology was still going on when World War II broke out and Fomin was conscripted into the Red Army.
      
    • We have already mentioned Fomin's work on topology.
      

45. Hopf Eberhard biography
    
    • Eberhard Hopf, an Austrian mathematician who made significant contributions in topology and ergodic theory, was born in Salzburg.
      
    • While in the Harvard College Observatory he worked on many mathematical and astronomical subjects including topology and ergodic theory.
      
    • In Leipzig Hopf undertook research on quantic mechanics (1937), Geodesics on manifolds of negative curvature (1939), Statistik der geod (1939) and on the influence of curvature of a closed Riemannian manifold on its topology (1941).
      
    • As a result most of his work to ergodic theory and topology was neglected or even attributed to others in the years following the end of World War II.
      

46. Lopatynsky biography
    
    • This book makes the reader familiar with the basic notions and facts of algebra, topology, and functional analysis, and gives a general idea how to apply these notions to the theory of differential equations.
      
    • More detailed presentations can be found in many monographs on topology and functional analysis which will be cited systematically in this text.
      
    • The book contains eight chapters: Sets; Basic algebraic notions; Algebraic equations; Topology; Differentiation and integration; Special linear spaces which are related to Euclidean spaces; Manifolds; and Elements of algebraic topology.
      

47. Singer biography
    
    • Singer is justifiably famous among mathematicians for his deep and spectacular work in geometry, analysis, and topology, culminating in the Atiyah-Singer Index theorem and its many ramifications in modern mathematics and quantum physics.
      
    • They have spawned many developments in differential geometry, differential topology, and analysis ..
      
    • Other significant contributions to geometry were his work with D B Ray on analytic torsion, the precursor of much modern work on "determinant" invariants in geometry, and an influential textbook joint with J A Thorpe, Lecture Notes on Elementary Topology and Geometry ..
      
    • for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics.
      

48. Whitehead Henry biography
    
    • He worked mainly on differential geometry although towards the end of his three years there he became interested in topology.
      
    • As we mentioned Whitehead's interests turned more towards topology near the end of his three years in Princeton when he collaborated with Lefschetz in proving that all analytic manifolds can be triangulated.
      
    • Whitehead also studied Stiefel manifolds and set up a school of topology at Oxford.
      
    • His influence on the development of mathematics during his active lifetime can be partly measured by the innumerable references, implicit and explicit, in current mathematical literature on algebraic and geometric topology; but it could not have been so great without the generosity and enthusiasm which he poured into every mathematical enterprise and which inspired such deep affection in all who knew him well.
      

49. Zeeman biography
    
    • Zeeman's research has been in a variety of areas such as topology, in particular PL topology, dynamical systems and mathematical applications to biology and the social sciences.
      
    • His initial research was in topology and one of his theorems was the unknotting of spheres in five dimensions.
      
    • Certainly his work in topology would make him one of the leading topologists of all time but he may be known principally for other work.
      
    • As early as 1967 he was speaking on (the then) BBC Third Programme on topics such as topology.
      

50. Calugareanu biography
    
    • Calugareanu worked in a number of different mathematical areas such as the theory of functions of one complex variable, geometry, algebra, and topology.
      
    • Guided by this point of view, Calugareanu left us important works on differential geometry and topology.
      
    • In differential topology, starting from an invariant introduced by Gauss, Calugareanu discovered a system of invariants which found applications in knot theory and molecular biology.
      
    • His own work did indeed span several fields, and he recognises that his thread was the idea of invariance which ran through his work in complex variables, differential topology, and modern algebra.
      

51. Wall biography
    
    • Wall's research is mostly in the area of geometric topology and related algebra.
      
    • He has written a number of highly influential books including Surgery on compact manifolds (1970) and A geometric introduction to topology (1972).
      
    • This latter work is an introduction to algebraic topology for a reader without background in general topology.
      
    • From then on, a growing interplay between the fields of topology and algebra contributed to a deeper understanding of singularities.
      

52. Janiszewski biography
    
    • Lebesgue supervised Janiszewski's doctoral studies in topology and in 1911 he submitted his thesis Sur les continus irreductibles entre deux points.
      
    • As early as 1917 [Janiszewski and Mazurkiewicz] were conducting a topology seminar, presumably the first in that new, exuberantly developing field.
      
    • In addition to set theory (which at that time included parts of what we call topology today) Janiszewski produced important results in the foundations of mathematics and other parts of topology.
      

53. Betti biography
    
    • Betti is noted for his contributions to algebra and topology.
      
    • Betti published a memoir on topology in 1871 which contained what we now call the "Betti numbers".
      
    • These were so named by Poincare who was inspired to study topology through Betti's work on the subject.
      
    • History Topics: Topology enters mathematics .
      

54. Griffiths Brian biography
    
    • He published three papers based on the ideas in his thesis Local topological invariants (1953), A mapping theorem in "local" topology (1953), and A contribution to the theory of manifolds (1954).
      
    • As well as illuminating classical mathematics, the book also provides a way forward into more recent topics, clearly demonstrating that subjects like topology and modern algebra have firm classical roots, unlike many expositions which give the impression of self-contained inventions which have superseded older mathematics.
      
    • During these years he published articles on topology and also took a deep interest in mathematical education, publishing important and influential article on that topic.
      
    • For example his work on topology included the text Surfaces published by Cambridge University Press.
      

55. Kolmogorov biography
    
    • There Aleksandrov worked on the topology book which he co-authored with Hopf, while Kolmogorov worked on Markov processes with continuous states and continuous time.
      
    • his ideas in set-theoretic topology, approximation theory, the theory of turbulent flow, functional analysis, the foundations of geometry, and the history and methodology of mathematics.
      
    • In topology Kolmogorov introduced the notion of cohomology groups at much the same time, and independently of, Alexander.
      
    • Another contribution of the highest significance in this area was his definition of the cohomology ring which he announced at the International Topology Conference in Moscow in 1935.
      

56. Thurston biography
    
    • Thurston has fantastic geometric insight and vision: his ideas have completely revolutionised the study of topology in 2 and 3 dimensions, and brought about a new and fruitful interplay between analysis, topology and geometry.
      
    • Thurston's work has had an enormous influence on 3-dimensional topology.
      
    • Direct arguments remain essential, but 3-dimensional topology has now firmly rejoined the main stream of mathematics.
      
    • In 1997 he published Three-dimensional geometry and topology.
      
    • In 1978, W Thurston gave a course at Princeton University, whose subject was the geometry and topology of three-dimensional manifolds.
      
    • It is probably the opinion of all the people working in low-dimensional topology that the ideas contained in these notes have been the most important and influential ideas ever written on the subject.
      
    • The 1978 Princeton lecture notes, although written in an informal style, are self-contained and accessible to graduate students in topology or geometry.
      
    • On 6 January 2005, at the Joint Mathematics Meetings in Atlanta, Georgia, Thurston was awarded the American Mathematical Society Book Prize for Three-dimensional geometry and topology.
      

57. Rudin biography
    
    • from the beginning, it was the set-theoretic aspects of topology which interested me most.
      
    • Her 1952 paper A primitive dispersion set of the plane provided a positive solution to an unsolved problem contained in R L Wilder's book Topology of manifolds (1949).
      
    • In August 1974 Rudin gave a series of lectures on set theoretic topology at the CBMS Regional Conference held at the University of Wyoming, Laramie.
      
    • In the lectures she surveyed what were then the recent results connecting set theory with the problems of general topology.
      

58. Wallace Alexander biography
    
    • He spent the years 1941-47 there building an impressive expertise in algebraic topology [Semigroup Forum 34 (1) (1986), 1-4.',2)">2]:- .
      
    • He began to develop a set of lecture notes on algebraic topology which, although formally unpublished, represents a substantial achievement in research and scholarship.
      
    • These notes, through his students, had a great influence on research in topology and its applications in topological algebra.
      

59. Atiyah biography
    
    • His first major contribution (in collaboration with F Hirzebruch) was the development of a new and powerful technique in topology (K-theory) which led to the solution of many outstanding difficult problems.
      
    • This 'index theorem' had antecedents in algebraic geometry and led to important new links between differential geometry, topology and analysis.
      
    • More recently Atiyah has been influential in stressing the role of topology in quantum field theory and in bringing the work of theoretical physicists, notably E Witten, to the attention of the mathematical community.
      

60. Kurosh biography
    
    • I attended his lectures on the theory of sets, the theory of functions, and topology.
      
    • However, although Kurosh's first results were in topology, solving problems posed by Aleksandrov, he was already interested in the theory of groups.
      
    • As mentioned above, Kurosh's first significant results were in topology, solving a problem set by Aleksandrov.
      

61. Reidemeister biography
    
    • He established a geometry and topology based on group theory without the concept of a limit.
      
    • In particular he wrote an important book Einfuhrung in die kombinatorische Topologie (1932) on combinatorial topology.
      
    • was, above all, a geometer, his book on 'combinatorial topology' contains hardly any drawings.
      

62. Tietze biography
    
    • Tietze contributed to the foundations of general topology and developed important work on subdivisions of cell complexes.
      
    • Among the topics in topology which Tietze worked on were knot theory, Jordan curves and continuous mappings of areas.
      
    • Topics outside topology which Tietze worked on included ruler and compass constructions, continued fractions, partitions, the distribution of prime numbers, and differential geometry.
      

63. Kuperberg biography
    
    • The first course she studied was given by Andrzej Mostowski, then later she attended topology lectures given by Karol Borsuk and found the subject a fascinating one.
      
    • The two published several papers on topology.
      
    • She attended a topology conference at the University of North Carolina, Charlotte in the same year and presented a paper Two Vietoris-type isomorphism theorems in Borsuk's theory of shape, concerning the Vietoris-Cech homology and Borsuk's fundamental groups which was published in the conference proceedings in the following year.
      

64. Behrend biography
    
    • In the same year in Note on the compactification of separated uniform spaces he gave a simple method of obtaining, for any uniform space S, a uniform structure which is totally bounded and compatible with the topology of S.
      
    • His main interest had by now firmly moved from number theory to topology and he is particularly remembered for introducing modern general topology to the University of Melbourne.
      

65. Warner biography
    
    • During this time he had built up an extremely active research group in algebraic topology and Mary made excellent progress in her research in these stimulating conditions.
      
    • The term 'fuzzy' has been used by Poston in his thesis and C T J Dodson [in 1974] to describe a set with a reflexive, symmetric relation, elsewhere [in E C Zeeman's paper" Topology of 3-manifolds and related topics" of 1962] called a tolerance space.
      
    • We conclude with a few general remarks on lattice-valued functions, topology and homology.
      

66. De Groot biography
    
    • In 1964 he became Dean of the Faculty of Science at the University of Amsterdam and, at this time, he gave up his position of Head of Pure Mathematics at the Mathematical Centre but remained associated with the Mathematical Centre as Advisor to Pure Mathematics [General Topology and its Applications 3 (1973), 3-32.',3)">3]:- .
      
    • De Groot worked in topology and group theory.
      
    • Later de Groot worked on set-theoretic topology.
      

67. Pontryagin biography
    
    • In 1934 he became a member of the Steklov Institute and in 1935 he became head of the Department of Topology and Functional Analysis at the Institute.
      
    • Pontryagin worked on problems in topology and algebra.
      
    • lies not merely in its effect on the further development of topology; of equal significance is the fact that his theorem enabled him to construct a general theory of characters for commutative topological groups.
      

68. Chen biography
    
    • The authors of [Contemporary trends in algebraic geometry and algebraic topology, Tianjin (2000, World Sci.
      
    • The goal of Chen's iterated integrals program, which is a de Rham theory for path spaces, was to study the interaction of topology and analysis through path integration.
      
    • Richard Hain, who was Chen's last doctoral student graduating in 1980 with his thesis Iterated Integrals, Minimal Models and Rational Homotopy Theory, is the coauthor of the articles [Contemporary trends in algebraic geometry and algebraic topology, Tianjin (2000, World Sci.
      

69. Heegaard biography
    
    • This counter-example sent Poincare back to the drawing board and thereby contributed to a clarification of some basic notions of algebraic topology.
      
    • Another important Heegaard contribution is his 1907 survey article (with Max Dehn) Analysis Situs where the authors set forth the foundations of combinatorial topology.
      
    • In particular, his interest in topology came from an attempt to study algebraic functions of two complex variables by means of generalized (four dimensional) Riemann surfaces.
      

70. Schmidt biography
    
    • After Schmidt moved to Berlin his interests turned towards topology.
      
    • Schmidt's interest in topology influenced Hopf and, in 1929, he was an examiner of Hopf's doctoral thesis.
      
    • History Topics: Topology enters mathematics .
      

71. Wazewski biography
    
    • Under Zaremba, Wazewski became interested in set theory and topology and decided to study in Paris for his doctorate.
      
    • Wazewski studied in Paris between 1921 and 1923 continuing his interest in topology acquired during his studies at Krakow under Zaremba.
      
    • At about this time his interests shifted away from set theory and topology and he became interested in analysis.
      

72. Nielsen Jakob biography
    
    • Dehn greatly influenced Nielsen and introduced him to the newest ideas in topology and group theory.
      
    • Hansen writes in [I M James (ed.), History of Topology (Amsterdam, 1999), 979-990.',6)">6]:- .
      
    • Jakob Nielsen initiated much of the topology of surfaces and of combinatorial group theory, and for this reason alone he occupies an important place in the history of 20th century mathematics.
      

73. Urbanik biography
    
    • His first work was on topology and he published Sur les espaces complets separables de dimension 0 in 1953 which was a joint paper with B Knaster.
      
    • Urbanik then began to mix an interest in topology with measure theory and probability and his 1954 papers show this mix: Sur un probleme de J F Pal sur les courbes continues; Limit properties of homogeneous Markov processes with a denumerable set of states; Sur la structure non-topologique du corps des operateurs; and Quelques theoremes sur les measures.
      
    • ',3)">3] divides Urbanik's research into five different major areas: topology, measure theory and analysis; probability theory; stochastic processes; information theory and theoretical physics; and general algebras.
      

74. Deligne biography
    
    • He conjectured results about the number of solutions to polynomial equations over the integers using intuition on how algebraic topology should apply in this novel situation.
      
    • A solution to these problems required the development of a new kind of algebraic topology.
      
    • for major contributions to several important domains of mathematics (like algebraic geometry, algebraic and analytic number theory, group theory, topology, Grothendieck theory of motives), enriching them with new and powerful tools and with magnificent results such as his spectacular proof of the "Riemann hypothesis over finite fields" (Weil conjectures).
      

75. Donaldson biography
    
    • This result was published by Donaldson in a paper Self-dual connections and the topology of smooth 4-manifolds which appeared in the Bulletin of the American Mathematical Society in 1983.
      
    • (2) Applications of gauge theory to 4-manifold topology.
      
    • He has created an entirely new and exciting area of research through which much of mathematics passes and which continues to yield mysterious and unexpected phenomena about the topology and geometry of smooth 4-manifolds.
      
    • groundbreaking work in four-dimensional topology, symplectic geometry and gauge theory, and for his remarkable use of ideas from physics to advance pure mathematics.
      

76. Kuiper biography
    
    • Right from the beginning it defines vectors and n-vector space in modern symbolism, gives in concise form the axioms to be utilized throughout, treats the different topics (e.g., affine plane, classification of endomorphisms, quadratic and symmetric bilinear functions, some applications to statistics, motions and affine transformations, and some topology) by up-to-date methods and thus creates a model of a book for the budding research-scientist, ingenious, clear, consistent in structure.
      
    • differential geometry, differential topology, and algebraic topology, and nurturing a number of doctoral students and post-doctoral visitors.
      

77. Sierpinski biography
    
    • From this period Sierpinski worked mostly in the area of set theory but also on point set topology and functions of a real variable.
      
    • It was a great loss for Polish mathematics which was developing favourably in some fields such as set theory and topology ..
      

78. Tait biography
    
    • Topology and Scottish mathematical physics .
      
    • History Topics: Topology and Scottish mathematical physics .
      

79. Tukey biography
    
    • Tukey's research was supervised by Lefschetz and he received his doctorate in 1939 for a dissertation Denumerability in topology which was published in 1940 as Convergence and uniformity in topology .
      

80. Ulam biography
    
    • Then in topology I had a few results.
      
    • Also I had luck with tremendously good collaborators in set theory, in group theory, in topology, in mathematical physics, and in other method, which is not a tremendously intellectual achievement but is very useful, a few things like that.
      

81. Morse biography
    
    • Morse's major works include Calculus of variations in the large (1934), Functional topology and abstract variational theory (1938), Topological methods in the theory of functions of a complex variable (1947) and Lectures on analysis in the large (1947).
      
    • These include papers on minimal surfaces, some on the theory of functions of a complex variable where he was particularly interested in applying topological methods, papers on differential topology and on dynamics.
      

82. Maxwell biography
    
    • Topology and Scottish mathematical physics .
      
    • History Topics: Topology and Scottish mathematical physics .
      

83. Floer biography
    
    • His main interests were in algebraic topology and, in the autumn of 1982, he went to the University of California at Berkeley to continue his research.
      
    • The value of his work was grasped immediately by specialists in differential geometry, topology, and mathematical physics, for whom "Floer homology" has become an essential part of their problem-solving toolkit.
      

84. Freudenthal biography
    
    • He was working on the algebraic characterisation of the topology of the real semisimple Lie groups in 1940 when Germany invaded The Netherlands.
      
    • As we have indicated, Freudenthal's early work was on topology and algebra.
      

85. Williams biography
    
    • Topology and Lie groups (Birkhauser Boston, Inc., Boston, MA, 1994), 3-9.
      
    • Topology and Lie groups (Birkhauser Boston, Inc., Boston, MA, 1994), 3-9.
      

86. Yau biography
    
    • .for his work in nonlinear partial differential equations, his contributions to the topology of differentiable manifolds, and for his work on the complex Monge-Ampere equation on compact complex manifolds.
      
    • His work has had, and will continue to have, a great impact on areas of mathematics and physics as diverse as topology, algebraic geometry, representation theory, and general relativity as well as differential geometry and partial differential equations.
      

87. Euler biography
    
    • Other geometric investigations led him to fundamental ideas in topology such as the Euler characteristic of a polyhedron.
      
    • History Topics: Topology enters mathematics .
      

88. Grothendieck biography
    
    • However it was during this period that his research interests changed and they moved towards topology and geometry.
      
    • During this period Grothendieck's work provided unifying themes in geometry, number theory, topology and complex analysis.
      

89. Ehresmann biography
    
    • Ehresmann followed the moves of the university then, in 1955, a chair of topology was specially created for him in the University of Paris.
      
    • Ehresmann was one of the creators of differential topology.
      

90. Schonflies biography
    
    • In around 1895 Schonflies turned his attention towards set theory and topology.
      
    • Three important papers on plane topology proved the topological invariance of the dimension of the square.
      

91. Young Lai-Sang biography
    
    • The paper Entropy of continuous flows on compact 2-manifolds was published in Topology and shows that a continuous one-dimensional flow on a 2-dimensional manifold has zero topological entropy.
      
    • Today it stands at the crossroads of several areas of mathematics, including analysis, geometry, topology, probability, and mathematical physics.
      

92. Thomson biography
    
    • Topology and Scottish mathematical physics .
      
    • History Topics: Topology and Scottish mathematical physics .
      

93. Lhuilier biography
    
    • His work on Euler's polyhedra formula, and exceptions to that formula, were important in the development of topology.
      
    • History Topics: Topology enters mathematics .
      

94. Begle biography
    
    • His favourite mathematical topic was topology, taught by Raymond Wilder, and after the award of an A.B.
      
    • His early publications were in topology: Homology local connectedness (1941); Locally connected spaces and generalized manifolds (1942); Intersections of contractible polyhedra (1943); Regular convergence (1944); and Duality theorems for generalized manifolds (1945).
      

95. Milnor biography
    
    • This work opened up the new field of differential topology.
      
    • In the 1950s Milnor did a substantial amount of work on algebraic topology which is discussed in [Topological methods in modern mathematics (Houston, TX, 1993), 5-22.',6)">6].
      

96. Freedman biography
    
    • In his paper, The topology of four-dimensional manifolds, published in the Journal of Differential Geometry (1982), Freedman solved this problem, and in particular, the four-dimensional Poincare conjecture.
      
    • My primary interest in geometry is for the light it sheds on the topology of manifolds.
      

97. Helmholtz biography
    
    • For details of the impact of this work, particularly Helmholtz's results on vortices, see the article Topology and Scottish mathematical physics.
      
    • History Topics: Topology and Scottish mathematical physics .
      

98. Stoilow biography
    
    • After this he changed somewhat the direction of his work and began to undertake research on the theory of functions of a real variable and on topology.
      
    • Those who study this deeply original book, epoch-making for topology as well as for function theory, are struck by the exceptional variety and richness of the results and the mastery with which the author passes from the concrete intuition of geometrical facts to the most abstract generalisations.
      

99. Arnold biography
    
    • The areas are Dynamical Systems, Differential Equations, Hydrodynamics, Magnetohydrodynamics, Classical and Celestial Mechanics, Geometry, Topology, Algebraic Geometry, Symplectic Geometry, and Singularity Theory.
      
    • The face of modern mathematics would be unrecognisable without his work in dynamical systems, classical and celestial mechanics, singularity theory, topology, real and complex algebraic geometry, symplectic and contact geometry, hydrodynamics, variation calculus, differential geometry, potential theory, mathematical physics, superposition theory, etc.
      
    • Arnold also produced extremely fruitful ideas, relating classical mechanics to questions of topology.
      

100. Bass biography
     
     • The latter notion was applied by Atiyah and Hirzebruch in order to construct a new cohomology theory which has been enormously fruitful in topology.
       
     • These informal reminiscences, presented at the ICTP 2002 Conference on algebraic K-theory, recount the trajectory in the author's early research, from work on the Serre conjecture (on projective modules over polynomial algebras), via ideas from algebraic geometry and topology, to the ideas and constructions that eventually contributed to the founding of algebraic K-theory.
       

101. Eckmann biography
     
     • It was characteristic of Hopf's views on our science that this meant not only learning algebraic topology - then a very young field - but also getting acquainted with group theory, differential geometry, and algebra in the "abstract" sense of the Emmy Noether school.
       
     • Peter Hilton, who had been a personal friend of Eckmann's for many years spoke in detail of Eckmann's research in topology: continuous solutions of systems of linear equations, a group-theoretical proof of the Hurwitz-Radon theorem, complexes with operators, spaces with means, simple homotopy type.
       

102. Browne biography
     
     • Browne's grant allowed her to study combinational topology at Cambridge University.
       
     • She won a similar fellowship while studying differential topology at Columbia University, New York, in during 1965 - 66.
       

103. Malcev biography
     
     • At the regrettably early age of 57 Malcev died while taking part in the Novosibirsk Topology Conference which he had helped to organise ([Uspekhi Mat.
       
     • Just before his death Malcev had delivered his final lecture at this Novosibirsk Topology Conference.
       

104. Weil biography
     
     • These Weil conjectures, as they came to be called, grew out of his deep insight into the topology of algebraic varieties and provided guiding principles for subsequent developments in the field.
       
     • He contributed substantially to topology, differential geometry and complex analytic geometry.
       

105. Riesz biography
     
     • He also studied orthonormal series and topology.
       
     • History Topics: Topology enters mathematics .
       

106. Krieger biography
     
     • Krieger is best known for her English translation of Sierpinski's Introduction to General Topology (1934) and General Topology (1952).
       

107. Thomason biography
     
     • His supervisor at Princeton was John Moore and he wrote a dissertation on category theory in which he produced results which were to become fundamental tools in topology.
       
     • Few have had the simultaneous grasp of topology, algebraic geometry and K-theory that Thomason did.
       

108. Specker biography
     
     • Geneve, Geneva, 1982), 11-24.',5)">5] where his 32 publications up to 1979 are divided into 10 categories: topology, recursive analysis, combinatorial set theory, type theory, axiomatic set theory, Ramsey's theorem, arithmetic, logic of quantum mechanics, algorithms, and miscellaneous.
       
     • Ernst Specker has made decisive contributions towards shaping directions in topology, algebra, mathematical logic, combinatorics and algorithms over the last 40 years.
       

109. Kurepa biography
     
     • gave talks on the theory of matrices, and held a seminar with topics in set theory, topology and algebra.
       
     • The topics which Kurepa investigated are very varied but lie mostly within topology, set theory and number theory.
       

110. Luzin biography
     
     • He also made significant contributions to descriptive set topology.
       
     • Many of these mathematicians turned to other topics such as topology, differential equations, and functions of a complex variable.
       

111. Nash biography
     
     • In September 1948 Nash entered Princeton where he showed an interest in a broad range of pure mathematics: topology, algebraic geometry, game theory and logic were among his interests but he seems to have avoided attending lectures.
       
     • He was always full of mathematical ideas, not only on game theory, but in geometry and topology as well.
       

112. McDuff biography
     
     • There she attended Frank Adams' topology lectures and around this time her first child was born.
       
     • From the early 1980s McDuff worked on symplectic topology.
       
     • She is the Helen Lyttle Kimmel '42 Professor of Mathematics at Barnard College where she teaches "Introduction to Higher Mathematics" and courses in geometry and topology.
       
     • in recognition of her research in many areas of mathematics and in particular in symplectic topology.
       
     • In the early eighties, shortly before Gromov's work on pseudo-holomorphic curves began to move the subject in new directions, McDuff began her study of symplectic topology and geometry.
       

113. Mazurkiewicz biography
     
     • As early as 1917 [Janiszewski and Mazurkiewicz] were conducting a topology seminar, presumably the first in that new, exuberantly developing field.
       
     • His main work was in topology and the theory of probability.
       

114. Smale biography
     
     • For much of the seventies Steve focused on economics, injecting topology and dynamics into the study of general economic equilibria.
       
     • for his contributions to various aspects of differential topology.
       
     • for his groundbreaking contributions that have played a fundamental role in shaping differential topology, dynamical systems, mathematical economics, and other subjects in mathematics.
       

115. Oleinik biography
     
     • Given Petrovsky's expertise in differential equations, the topology of algebraic curves and surfaces and mathematical physics, it is not difficult to see his influence on the direction that Oleinik's work would take.
       

116. Shnirelman biography
     
     • Shnirelman started research in algebra, geometry and topology as a student but did not consider his results sufficiently important to merit publication.
       

117. Petrovsky biography
     
     • Petrovsky's main mathematical work was on the theory of partial differential equations, the topology of algebraic curves and surfaces, and probability.
       

118. Jones Vaughan biography
     
     • In 1984 Jones discovered an astonishing relationship between von Neumann algebras and geometric topology.
       
     • Jones has made many other contributions to the mathematical community, particularly in his editorial work as editor or associate editor for many journals: the Transactions of the American Mathematical Society, the Pacific Mathematics Journal, the Annals of Mathematics, the New Zealand Journal of Mathematics, Advances in Mathematics, the Journal of Operator Theory, Reviews in Mathematical Physics, the Russian Journal of Mathematical Physics, the Journal of Mathematical Chemistry, Geometry and Topology, and L'Enseignement Mathematique.
       

119. Thom biography
     
     • His work on topology, in particular on characteristic classes, cobordism theory and the Thom transversality theorem led to his being awarded a Fields medal in 1958.
       

120. Zariski biography
     
     • His topological work concentrated mainly on the fundamental group; many of the ideas he pioneered were innovations in topology as well as algebraic geometry and have developed independently in the two fields since then.
       

121. Lyndon biography
     
     • Returning to the United States in 1948 he decided that to make further progress in cohomology theory he needed to learn more about current work in topology and clearly Princeton was the leading institution for research in that area.
       

122. Ostrowski biography
     
     • These are determinants, linear algebra, algebraic equations, multivariate algebra, formal algebra, number theory, geometry, topology, convergence, theory of real functions, differential equations, differential transformations, theory of complex functions, conformal mappings, numerical analysis and miscellany.
       

123. Johnson Barry biography
     
     • Particularly, he reported on applications to noncommutative algebraic topology, noncommutative integration and noncommutative dynamical systems.
       

124. Banach biography
     
     • History Topics: Topology enters mathematics .
       

125. Preston biography
     
     • Topological spaces had not been heard of - though some very popular lectures were given on topology by J H C Whitehead.
       

126. Arf biography
     
     • Much of Arf's most important work was in algebraic number theory and he invented Arf invariants which have many applications in topology.
       

127. Hadamard biography
     
     • History Topics: Topology enters mathematics .
       

128. Lebesgue biography
     
     • He also made major contributions in other areas of mathematics, including topology, potential theory, the Dirichlet problem, the calculus of variations, set theory, the theory of surface area and dimension theory.
       

129. Bortolotti biography
     
     • Bortolotti studied topology at first but later went in the direction of analysis considering the calculus of finite differences, continued fractions, convergence of infinite algorithms, summation of series, the asymptotic behaviour of series and improper integrals.
       

130. Peirce Charles biography
     
     • He then extended his father's work on associative algebras and worked on mathematical logic, topology and set theory.
       

131. Bendixson biography
     
     • He contributed important results in point set topology.
       

132. Serre biography
     
     • Serre's theorem led to rapid progress not only in homotopy theory but in algebraic topology and homological algebra in general.
       

133. Ramanujam biography
     
     • We discussed many topics involving topology and algebraic geometry at that time, and especially Kodaira's Vanishing Theorem.
       

134. Escher biography
     
     • He was fascinated by topology, which only began to be studied during his lifetime, as illustrated by the Mobius strip.
       

135. Iwasawa biography
     
     • A proof of the Riemann-Roch theorem is given, and the theory of Riemann surfaces and their topology is studied.
       

136. Wang biography
     
     • Wang worked on algebraic topology and discovered the 'Wang sequence', an exact sequence involving homology groups associated with fibre bundles over spheres.
       

137. Descartes biography
     
     • History Topics: Topology in mathematics .
       

138. Kodaira biography
     
     • At this time Kodaira was interested in topology, Hilbert spaces, Haar measure, Lie groups and almost periodic functions.
       

139. Speiser biography
     
     • Stiefel's original interests were in topology but his aim was to build an institute where the mathematical implications of computers could be studied [IEEE Annals of the History of Computing 20 (1) (1998), 15-28.',4)">4]:- .
       

140. Walsh Joseph biography
     
     • He studied the relative location of the zeros of pairs of rational functions, zeros and topology of extremal polynomials, the critical points and level lines of Green's functions and other harmonic functions, conformal mappings, Pade approximation, and the interpolation and approximation of continuous, analytic or harmonic functions.
       

141. Artin biography
     
     • He again showed his originality by introducing this new area of research which today is being studied by an increasing number of mathematicians working in group theory, semigroup theory, and topology.
       

142. Cantor biography
     
     • History Topics: Topology enters mathematics .
       

143. Goursat biography
     
     • In [History of topology (North-Holland, Amsterdam, 1999), 111-122.',4)">4] Katz notes that it was Goursat who first noted the generalized Stokes theorem.
       

144. Hodge biography
     
     • During this period he developed the relationship between geometry, analysis and topology and produced some of his best remembered work on the theory of harmonic integrals.
       

145. Weierstrass biography
     
     • History Topics: Topology enters mathematics .
       

146. Blackwell biography
     
     • He had found a game theory proof of the Kuratowski Reduction theorem and connecting the areas of game theory and topology [Mathematical People (Boston, 1985), 18-32.',2)">2]:- .
       

147. Yano biography
     
     • He was in Zurich in June 1960 for the International Symposium on Differential Geometry and Topology, spent April, May and June of 1961 at the University of Washington, then spent a month at the University of Southampton followed by two months at the University of Liverpool in 1962.
       

148. Bolzano biography
     
     • History Topics: Topology enters mathematics .
       

149. Dantzig biography
     
     • One paper classified fields with a locally compact topology.
       

150. Hilbert biography
     
     • History Topics: Topology enters mathematics .
       

151. Hahn biography
     
     • The book is divided into an introduction (containing an exposition of the relevant parts of set theory and point set topology) and five chapters, entitled (I) Additive and totally additive set functions, (II) Measure, (III) Measurable functions, (IV) Integration and (V) Differentiation.
       

152. Dyson biography
     
     • He returned to Trinity College in 1946 as a fellow having written a dissertation from which he published three papers; A theorem on the densities of sets of integers (1945), A theorem in algebraic topology (1948), and On the product of four non-homogeneous linear forms (1948).
       

153. Goldbach biography
     
     • History Topics: Topology enters mathematics .
       

154. Fowler David biography
     
     • A large number of postgraduate students, and a year long topology year which brought many leading mathematicians to spend time there, meant that the department had an outstanding research atmosphere.
       

155. Weyl biography
     
     • It united analysis, geometry and topology, making rigorous the geometric function theory developed by Riemann.
       

156. Stolz biography
     
     • Topics covered are those of interest to Klein and to Stolz and include Klein's Erlangen program, nowhere differentiable continuous functions, geometry, and the topology of the line.
       

157. Saks biography
     
     • Mathematical analysis, and especially those of its branches which used modern methods of set theory and topology, became his main field of interest.
       

158. Cartan Henri biography
     
     • Cartan worked on analytic functions, the theory of sheaves, homological theory, algebraic topology and potential theory, producing significant developments in all these areas.
       

159. Uhlenbeck Karen biography
     
     • I did some very technical work in partial differential equations, made an unsuccessful pass at shock waves, worked in scale invariant variational problems, made a poor stab at three dimensional manifold topology, learned gauge field theory and then some about applications to four dimensional manifolds, and have recently been working n equations with algebraic infinite symmetries.
       

160. Feigl biography
     
     • Feigl worked on geometry, in particular the foundations of geometry and topology.
       

161. Pompeiu biography
     
     • In this work Hausdorff gives a slightly different definition of distance between sets but he also credits Pompeiu's work and shows that the two definitions give the same topology.
       

162. Mahler biography
     
     • At Frankfurt, supported financially by his parents and several members of the Krefeld Jewish community, he attended lectures by Max Dehn on topology, Ernst Hellinger on elliptic functions, Carl Siegel on calculus and Otto Szasz [Biographical Memoirs of Fellows of the Royal Society of London 39 (1994), 265-279.',2)">2]:- .
       

163. Cartan biography
     
     • His work is a striking synthesis of Lie theory, classical geometry, differential geometry and topology which was to be found in all Cartan's work.
       

164. Koszul biography
     
     • The second of the three papers was Sur les operateurs de derivation dans un anneau in which he studied rings having a derivation operator with the formal properties of the coboundary operator of algebraic topology.
       

165. Hsiung biography
     
     • It begins with a review on point-set topology, multi-dimensional calculus and linear algebra.
       

166. Wu Wen-Tsun biography
     
     • Wu extended his mechanization interest, combining it with his original interest in topology to produce computational techniques to handle rational homotopy theory.
       
     • The researches in the first stage, started in 1947, are in pure mathematics, mainly in algebraic topology, occasionally also in algebraic geometry.
       

167. Baer biography
     
     • His mathematical work, some of which has been mentioned above, was wide ranging; topology, abelian groups and geometry.
       

168. Tarski biography
     
     • Tarski made important contributions in many areas of mathematics: set theory, measure theory, topology, geometry, classical and universal algebra, algebraic logic, various branches of formal logic and metamathematics.
       

169. Stone biography
     
     • Then in 1934 he published two papers on Boolean algebras: Boolean algebras and their applications to topology and Subsumption of Boolean algebras under the theory of rings.
       

170. Mobius biography
     
     • History Topics: Topology enters mathematics .
       

171. Lyapin biography
     
     • Analysis, algebra, geometry, and topology being rich in examples of the latter, their abstract theory deserves recognition.
       

172. Von Dyck biography
     
     • Von Dyck made important contributions to function theory, group theory (where a fundamental result on group presentations is named after him), topology (where he was influenced by the work of Riemann), and to potential theory.
       

173. Vranceanu biography
     
     • In 1948 Vranceanu was appointed Head of Geometry and Topology at Bucharest University.
       

174. Kneser Hellmuth biography
     
     • After his doctoral work on quantum theory he turned toward topology and the theory of analytic functions in several indeterminates.
       

175. Segre Beniamino biography
     
     • By 1931 when he was appointed to the chair at Bologna he already had 40 publications in algebraic geometry, differential geometry, topology and differential equations.
       

176. Rado biography
     
     • Their theory was fully explained in the important monograph Continuous transformations in analysis : With an introduction to algebraic topology published in 1955.
       

177. Neumann Bernhard biography
     
     • Hopf had given him a love of topology and this seemed the topic on which he would undertake research.
       

178. Bronowski biography
     
     • Bronowski then returned to Cambridge where he received his doctorate in mathematics for a thesis which looked at problems in geometry and topology.
       

179. Birkhoff Garrett biography
     
     • A course by E C Kemble on quantum mechanics as well as courses on Lebesgue integration and topology gave him a broad education in mathematics.
       

180. Keller biography
     
     • Looking at Keller's work one notices immediately that he made contributions to several, not only all neighbouring, areas of mathematics, the most important being to geometry, to algebraic geometry and to topology; in addition he studied number theoretic and analytic topics as well as those of a more philosophical character.
       

181. Borel Armand biography
     
     • They focus on Lie groups, and their actions, as well as on algebraic and arithmetic groups, and broach core questions regarding many different areas: algebraic topology, differential geometry, analytic geometry, analytic and algebraic geometry, number theory etc.
       

182. Chevalley biography
     
     • In particular algebraic topology has exhibited an insatiable appetite for algebraic gadgets.
       

183. Ribenboim biography
     
     • clearly written introduction to the theory of abelian ordered groups, assuming only an elementary knowledge of abelian group theory and topology.
       

184. Yoccoz biography
     
     • He became a member of the Institut Universitaire de France and a member of the Unite Recherche Associe "Topology and Dynamics" of the Centre National de la Recherche Scientifique at Orsay.
       

185. Von Neumann biography
     
     • Self-adjoint algebras of bounded linear operators on a Hilbert space, closed in the weak operator topology, were introduced in 1929 by von Neumann in a paper in Mathematische Annalen .
       

186. Fourier biography
     
     • History Topics: Topology enters mathematics .
       

187. Marczewski biography
     
     • His main work was in set theory, general topology, and measure theory.
       

188. Iyanaga biography
     
     • He did publish a number of papers, however, which arose through the various courses such as algebraic topology, functional analysis, and geometry, which he taught.
       

189. Van Kampen biography
     
     • Shortly before this Pontryagin, who had been working on problems in topology and algebra, had been studying duality.
       

190. Prufer biography
     
     • The 'Prufer topology' in introduced in the second paper as is a concept which Lefschetz called 'linearly compact groups' in a paper he published in 1942.
       

191. Van der Waerden biography
     
     • He learnt topology from Gerrit Mannoury who was a friend of his father (Mannoury was a Communist and Theo, although not a Communist himself, had many friends in that party).
       
     • At Gottingen, van der Waerden learnt much topology from Hellmuth Kneser.
       
     • from the beginning I was in contact with him, and from him I really learned topology.
       
     • Van der Waerden worked on algebraic geometry, abstract algebra, groups, topology, number theory, geometry, combinatorics, analysis, probability theory, mathematical statistics, quantum mechanics, the history of mathematics, the history of modern physics, the history of astronomy and the history of ancient science.
       

192. Lions Jacques-Louis biography
     
     • By this we mean that approximate solutions are constructed by a reduction of the problem to a finite-dimensional one, and these are then shown to form a relatively compact family in a suitable topology, by means of a priori estimates and other evaluations.
       

193. Gromoll biography
     
     • The first five chapters comprise an introduction to Riemannian geometry, accessible to students with a background in real analysis, linear algebra and first concepts of general topology.
       

194. Jonquieres biography
     
     • History Topics: Topology enters mathematics .
       

195. Eells biography
     
     • In it the whole geometry or topology of the spaces involved play a role, rather than just the equations describing the behaviour or motion in small areas.
       

196. Zorn biography
     
     • In addition to his well known work in infinite set theory, Zorn worked on topology and algebra.
       

197. Gauss biography
     
     • History Topics: Topology enters mathematics .
       

198. Golab biography
     
     • Professor Golab dealt with different fields of mathematics such as geometry, topology, algebra, analysis, logic, functional and differential equations, the theory of numerical methods and various applications of mathematics.
       

199. Almgren biography
     
     • He published a paper of over 40 pages based on the work of his thesis in Topology in the year he received his doctorate.
       

200. Arins biography
     
     • Ludmila Vsevolodovna Keldysh was a professor at Moscow State University and an outstanding mathematician who made important contributions to set theory and geometric topology.
       

201. Schauder biography
     
     • This 1934 paper on topology and partial differential equations is of major importance [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
       

202. Copson biography
     
     • The aim here is to provide a more leisurely approach to the theory of the topology of metric spaces, a subject which is not only the basis of functional analysis but also unifies many branches of classical analysis.
       

History Topics

1. Topology history references
   
   • References for: A history of Topology .
     
   • D E Cameron, The birth of Soviet topology, Proceedings of the 1982 Topology Conference, Topology Proc.
     
   • J W Dauben, Topology : Invariance of dimension, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 939-949.
     
   • J W Dauben, The invariance of dimension : Problems in the early development of set theory and topology, Historia Mathematica 2 (1975), 273-288.
     
   • D C Demaria, Francesco Severi's contribution to topology (Italian), Proceedings of the mathematical congress in celebration of the one hundredth birthday of Guido Fubini and Francesco Severi, Atti Accad.
     
   • J Dieudonne, A History of Algebraic and Differential Topology, 1900-1960 (Basel, 1989).
     
   • J Dieudonne, The beginnings of topology from 1850 to 1914, in Proceedings of the conference on mathematical logic 2 (Siena, 1985), 585-600.
     
   • J J Fingerman, The historical and philosophical significance of the emergence of point set topology (PhD Thesis, University of Chicago, 1981).
     
   • V L Hansen, From geometry to topology (Danish), Normat 36 (2) (1988), 48-60.
     
   • D M Johnson, The problem of the invariance of dimension in the growth of modern topology.
     
   • D M Johnson, The problem of the invariance of dimension in the growth of modern topology.
     
   • S Lefschetz, The early development of algebraic topology, Bol.
     
   • J H Manheim, The genesis of point set topology (New York, 1964).
     
   • E Scholtz, Topology : Geometric, algebraic, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 927-938.
     
   • J C Stillwell, Classical Topology and Combinatorial Group Theory (New York, 1980).
     
   • A Weil, Riemann, Betti and the birth of topology, Archive for History of Exact Sciences 20 (2) (1979), 91-96.
     
   • [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Topology_in_mathematics.html] .
     

2. Topology history references
   
   • References for: A history of Topology .
     
   • D E Cameron, The birth of Soviet topology, Proceedings of the 1982 Topology Conference, Topology Proc.
     
   • J W Dauben, Topology : Invariance of dimension, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 939-949.
     
   • J W Dauben, The invariance of dimension : Problems in the early development of set theory and topology, Historia Mathematica 2 (1975), 273-288.
     
   • D C Demaria, Francesco Severi's contribution to topology (Italian), Proceedings of the mathematical congress in celebration of the one hundredth birthday of Guido Fubini and Francesco Severi, Atti Accad.
     
   • J Dieudonne, A History of Algebraic and Differential Topology, 1900-1960 (Basel, 1989).
     
   • J Dieudonne, The beginnings of topology from 1850 to 1914, in Proceedings of the conference on mathematical logic 2 (Siena, 1985), 585-600.
     
   • J J Fingerman, The historical and philosophical significance of the emergence of point set topology (PhD Thesis, University of Chicago, 1981).
     
   • V L Hansen, From geometry to topology (Danish), Normat 36 (2) (1988), 48-60.
     
   • D M Johnson, The problem of the invariance of dimension in the growth of modern topology.
     
   • D M Johnson, The problem of the invariance of dimension in the growth of modern topology.
     
   • S Lefschetz, The early development of algebraic topology, Bol.
     
   • J H Manheim, The genesis of point set topology (New York, 1964).
     
   • E Scholtz, Topology : Geometric, algebraic, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 927-938.
     
   • J C Stillwell, Classical Topology and Combinatorial Group Theory (New York, 1980).
     
   • A Weil, Riemann, Betti and the birth of topology, Archive for History of Exact Sciences 20 (2) (1979), 91-96.
     
   • http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Topology_in_mathematics.html .
     

3. Topology history
   
   • A history of Topology .
     
   • Geometry and topology index .
     
   • The subject of topology itself consists of several different branches, such as point set topology, algebraic topology and differential topology, which have relatively little in common.
     
   • Perhaps the first work which deserves to be considered as the beginnings of topology is due to Euler.
     Go directly to this paragraph
     
   • Johann Benedict Listing (1802-1882) was the first to use the word topology.
     Go directly to this paragraph
     
   • Listing's topological ideas were due mainly to Gauss, although Gauss himself chose not to publish any work on topology.
     Go directly to this paragraph
     
   • A second way in which topology developed was through the generalisation of the ideas of convergence.
     Go directly to this paragraph
     
   • Cantor also introduced the idea of an open set another fundamental concept in point set topology.
     Go directly to this paragraph
     
   • Riesz, in a paper to the International Congress of Mathematics in Rome (1909), disposed of the metric completely and proposed a new axiomatic approach to topology.
     Go directly to this paragraph
     
   • Geometry and topology index .
     
   • http://www-history.mcs.st-andrews.ac.uk/HistTopics/Topology_in_mathematics.html .
     

4. Knots and physics references
   
   • References for: Topology and Scottish mathematical physics .
     
   • M Epple, Topology, matter, and space.
     
   • M Epple, Geometric aspects in the development of knot theory, in I M James (ed.), History of topology (Amsterdam, 1999), 301-358.
     
   • C Nash, Topology and physics - a historical essay, in I M James (ed.), History of topology (Amsterdam, 1999), 359-416.
     

5. Knots and physics references
   
   • References for: Topology and Scottish mathematical physics .
     
   • M Epple, Topology, matter, and space.
     
   • M Epple, Geometric aspects in the development of knot theory, in I M James (ed.), History of topology (Amsterdam, 1999), 301-358.
     
   • C Nash, Topology and physics - a historical essay, in I M James (ed.), History of topology (Amsterdam, 1999), 359-416.
     

6. Bourbaki 2
   
   • During a Congress, the chapters come up in order of the day, in no particular order, and we never know in advance if we shall be doing only differential topology at this Congress, or if at the next one we shall be doing commutative algebra.
     
   • Topology .
     
   • 1947 Book III Topology: Chapters V, VI, and VII.
     
   • 1948 Book III Topology: Chapter IX.
     
   • This chapter looks at the use of the real numbers in general topology.
     
   • 1949 Book III Topology: Chapter X.
     
   • Book VIII: Elementary Topology .
     
   • That was quite reasonable for general topology and general algebra, which were already solidified around 1950.
     

7. Knots and physics
   
   • Topology and Scottish mathematical physics .
     
   • Again his topology is driven by physical ideas of fluid flow but this notion, similar to the idea of a "cutting surface" which Riemann had introduced and Helmholtz had used, relates to our present day idea of homology, not homotopy.
     

8. Bourbaki 1
   
   • Soon a topology subcommittee was added.
     
   • To many this was a major strength of the highly logical approach but to others it was a major weakness in that real numbers, which seem of fundamental importance, could not be introduced until vast areas of algebra and topology had been set up, of course always in the most general form possible [Math.
     
   • Topology .
     
   • The last section outlines an interesting method of treating structures, such as order, topology, group, ring, etc., on a general basis and having concepts like isomorphism defined quite generally.
     

9. Fractal Geometry
   
   • Geometry and topology index .
     
   • Hausdorff's results from the same paper were important to the field of topology, as well; [Classics on Fractals (Addison-Wesley, 1993).
     
   • He was forced to give up his post as a professor at the University of Bonn in 1935, and even though he continued to work on set theory and topology, his work could only be published outside of Germany.
     
   • Geometry and topology index .
     

10. Non-Euclidean geometry
    
    • Geometry and topology index .
      
    • Geometry and topology index .
      

11. Cubic surfaces
    
    • Geometry and topology index .
      
    • Geometry and topology index .
      

12. Word problems
    
    • In 1910 Max Dehn published Uber die Topologie des dreidimensionalen Raumes (On the topology of 3-dimensional space).
      
    • One is already led to them by necessity with work in topology.
      

13. The four colour theorem
    
    • Geometry and topology index .
      
    • Geometry and topology index .
      

14. History overview
    
    • Lie's work on differential equations led to the study of topological groups and differential topology.
      Go directly to this paragraph
      

15. function concept references
    
    • J H Manheim, The genesis of point set topology (Pergamon Press, Oxford-Paris-Frankfurt; The Macmillan Co., New York, 1964).
      

16. Mathematics and Architecture references
    
    • M Rubin, Architecture and geometry, Structural Topology No.
      

17. Mathematics and Architecture references
    
    • M Rubin, Architecture and geometry, Structural Topology No.
      

18. function concept references
    
    • J H Manheim, The genesis of point set topology (Pergamon Press, Oxford-Paris-Frankfurt; The Macmillan Co., New York, 1964).
      

19. Orbits
    
    • It fact Poincare essentially invented topology in his attempt to answer stability questions in the three body problem.
      Go directly to this paragraph
      

20. Mathematical games
    
    • The Seven Bridges of Konigsberg heralds the beginning of graph theory and topology.
      

Famous Curves

Societies etc

1. European Mathematical Society Prizes
   
   • Using a large deviation principle in the proper topology, Raphael Cerf has established a Wulff construction for the supercritical percolation model in three dimensions.
     
   • The conjecture plays a central role in non-commutative geometry and has far-reaching connections to the Novikov conjecture on higher signatures in topology, to harmonic analysis on discrete groups and the theory of C*-algebras.
     
   • has obtained several strong results on topology and complex analysis.
     
   • became known through his work on symplectic topology.
     
   • This work was basic for later work of Lalonde, McDuff, and Polterovich on the topology of the group of symplectomorphisms.
     
   • has made fundamental and influential contributions to symplectic topology as well as to algebraic geometry and Hamiltonian systems.
     
   • His work is characterised by new depths in the interactions between complex algebraic geometry and symplectic topology.
     
   • A powerful tool in symplectic topology is Biran's decomposition of symplectic manifolds into a disc bundle over a symplectic submanifold and a Lagrangian skeleton.
     
   • Applications of this discovery range from the phenomenon of Lagrangian barriers to surprising novel results on topology of Lagrangian submanifolds.
     
   • The new techniques of working with random partitions invented and successfully developed by Okounkov lead to a striking array of applications in a wide variety of fields: topology of module spaces, ergodic theory, the theory of random surfaces and algebraic geometry.
     

2. International Congress Speakers
   
   • James Waddell Alexander, Some Problems in Topology.
     
   • Samuel Eilenberg, Applications of Homological Algebra in Topology.
     
   • Maxwell Herman Alexander Newman, Geometrical Topology.
     
   • Michael Artin, The Etale Topology of Schemes.
     
   • Charles Terence Clegg Wall, Geometric Topology: Manifolds and Structures.
     
   • Robert Duncan Edwards, The Topology of Manifolds and Cell-Like Maps.
     
   • William Paul Thurston, Geometry and Topology in Dimension Three.
     
   • Robert Duncan MacPherson, Global Questions in the Topology of Singular Spaces.
     
   • Karen Uhlenbeck, Applications of Non-Linear Analysis in Topology.
     
   • Victor Anatolyevich Vassiliev, Topology of Discriminants and Their Complements.
     
   • Helmut Hermann W Hofer, Dynamics, Topology, and Holomorphic Curves.
     
   • Dusa McDuff, Fibrations in Symplectic Topology.
     
   • Michael Jerome Hopkins, Algebraic Topology and Modular Forms.
     

3. AMS Steele Prize
   
   • In 1994 the last of these three categories was put onto a five year cycle of topics: analysis, algebra, applied mathematics, geometry and topology, and discrete mathematics/logic.
     
   • for his many contributions to algebra and algebraic topology, and in particular for his pioneering work in homological and categorical algebra.
     
   • for his fundamental contributions to topology and algebra, in particular for his classic papers on singular homology and his work on axiomatic homology theory which had a profound influence on the development of algebraic topology.
     
   • for having been instrumental in changing the face of geometry and topology, with his incisive contributions to characteristic classes, K-theory, index theory, and many other tools of modern mathematics.
     
   • for his extensive contributions in geometry and topology, the theory of Lie groups, their lattices and representations and the theory of automorphic forms, the theory of algebraic groups and their representations and extensive organizational and educational efforts to develop and disseminate modern mathematics.
     
   • for his paper "Pseudo-holomorphic curves in symplectic manifolds", which revolutionized the subject of symplectic geometry and topology and is central to much current research activity, including quantum cohomology and mirror symmetry.
     
   • in recognition of a lifetime of expository contributions ranging across a wide spectrum of disciplines including topology, symmetric bilinear forms, characteristic classes, Morse theory, game theory, algebraic K-theory, iterated rational mapsÉand the list goes on.
     

4. AMS Veblen Prize
   
   • It is given in recognition for work in geometry or topology by a member of the American Mathematical Society which is published in a North American journal.
     
   • for his contributions to various aspects of differential topology.
     
   • for his work in nonlinear partial differential equations, his contributions to the topology of differentiable manifolds, and for his work on the complex Monge-Ampere equation on compact complex manifolds.
     
   • for his work on the topology of low-dimensional manifolds, and to Clifford H Taubes for his foundational work in Yang-Mills theory.
     
   • for his work in differential geometry, to Yakov Eliashberg for his work in symplectic and contact topology, and to Michael J Hopkins for his work in homotopy theory.
     
   • in recognition of his work in geometric topology, in particular, the topology of 3-dimensional manifolds.
     
   • for their joint contributions to both three- and four-dimensional topology through the development of deep analytical techniques and applications.
     
   • for their contributions to 3- and 4-dimensional topology through their Heegaard Floer homology theory.
     

5. BMC 1997
   
   • Clarke, F W Formal group laws on algebraic topology .
     
   • Fenn, R A Advances in algebraic topology over the last 30 years .
     
   • Fenn, RAdvances in algebraic topology over the last thirty years .
     

6. BMC 2002
   
   • Tillmann, U The topology of surface bundles .
     
   • Special session: Geometry topology and mechanics Organiser: M Robertsn .
     

7. BMC 2007
   
   • Stacey, AAlgebraic objects inalgebraic topology .
     

8. Minutes for 1989
   
   • A R Pears LMS Topology .
     
   • R J Steiner EMS Topology .
     

9. Minutes for 1990
   
   • A R Pears LMS Topology .
     
   • R J Steiner EMS Topology .
     

10. Abel Prize
    
    • for playing a key role in shaping the modern form of many parts of mathematics, including topology, algebraic geometry and number theory.
      
    • for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics.
      

11. Minutes for 1964
    
    • (c) The Topology speakers had obtained expenses from other sources; they were guests of the Colloquium in the Residence at Leicester.
      
    • It was agreed to revert to separate mornings for Analysis and Functional Analysis; Algebra and Theory of Numbers; Geometry and Topology.
      

12. BMC 1960
    
    • Eckmann, BGroup concepts in topology and algebra .
      
    • Dowker, C HUniform topology .
      

13. SCminutes2002.html
    
    • Analytic Topology, organiser Chris Good, speakers: Prof Steve Watson, York University, North York, Toronto, Canada, Prof Jan van Mill, Vrije Universiteit, Amsterdam, The Netherlands.
      
    • This special session will be part of the satellite meeting ÒGalway Topology", which will run in Birmingham from the Monday morning to the Thursday afternoon.
      

14. BMC 1984
    
    • Rees, E GThe topology of algebraic varieties .
      
    • Steiner, REquivalences between topology and algebra .
      

15. Scientific Committee 2002
    
    • Analytic Topology, organiser Chris Good, speakers: Prof Steve Watson, York University, North York, Toronto, Canada, Prof Jan van Mill, Vrije Universiteit, Amsterdam, The Netherlands.
      
    • This special session will be part of the satellite meeting "Galway Topology", which will run in Birmingham from the Monday morning to the Thursday afternoon.
      

16. Minutes for 2000
    
    • There were satellite conferences on harmonic maps; model theory; regular dynamics; rings and quantum groups; there was also the British Topology Meeting.
      
    • Geometry and Topology: .
      

17. BMC 1967
    
    • Zeeman, E CPiecewise-linear topology .
      
    • Brown, RGroupoids in topology and algebra .
      

18. BMC 1986
    
    • Thurston, W P Three-dimensional geometry and topology .
      

19. BMC 1987
    
    • Johnstone, P T Another view of fibrewise topology .
      

20. BMC 1962
    
    • Wall, C T CRecent developments in differential topology .
      

21. BMC 1991
    
    • Ray, N Sequences of polynomials in topology, combinatorics and formal calculus .
      

22. BMC 2003
    

23. Minutes for 1956
    
    • The programme for the 1957 Colloquium was discussed and it was agreed that one day should be devoted to each of (i) Analysis (ii) Algebra and Number Theory (iii) Geometry and Topology.
      

24. BMC 1949
    
    • Discussion on topology .
      

25. Mathematics 2005
    
    • Geometry and Topology .
      

26. BMC 1971
    
    • Sanderson, B JA geometric view of algebraic topology .
      

27. BMC 1979
    
    • Craw, I GCommutative Banach algebras and the topology of the maximal ideal space .
      

28. Minutes for 1954
    
    • It was agreed that one day should be devoted to each of Probability with Logic and Foundations, Analysis with Number Theory, and Topology.
      

29. Minutes for 1957
    
    • (b) that one day should be devoted to each of (i) Analysis and Statistics (ii) Algebra and Logic (iii) Geometry and Topology.
      

30. Minutes for 1955
    
    • (c) Geometry and Topology .
      

31. Minutes for 1998
    
    • These include specialist meetings such as the British Topology Meeting and LMS Invited lectures, as well as the BMC and the BAMC (British Applied Mathematics Colloquium).
      

32. BMC 1980
    
    • Fuglede, B The fine topology in potential theory .
      

33. Minutes for 1958
    
    • nnn(iii) Geometry and Topology; .
      

34. Report 2007
    
    • In addition we had 12 morning speakers, 2 special sessions (in Fourier Analysis and in Topology), and 12 splinter groups.
      

35. BMC 1964
    
    • Hirzebruch, FGroup structures in topology .
      

36. Minutes for 1997
    
    • Geometry/Topology *Dr B H Bowditch (Southampton), *Dr A D King (Lancaster), Dr V V Goryunov (Liverpool), Dr W J Harvey (King's), Dr J C Wood (Leeds), Dr D Singerman (Southampton), Dr A J Baker (Glasgow), *Dr D Joyce (Oxford).
      

37. Ukrainian Academy of Sciences
    
    • Research priorities in the institute include: theory of differential equations, mathematics of physics, statistical theory, theory of functions, topology, algebra, the dynamics of special mechanical systems, computer programming and the institute develops various fields of mathematics for the natural sciences and technology.
      

38. Ukrainian Academy of Sciences
    
    • Research priorities in the institute include: theory of differential equations, mathematics of physics, statistical theory, theory of functions, topology, algebra, the dynamics of special mechanical systems, computer programming and the institute develops various fields of mathematics for the natural sciences and technology.
      

39. Young Mathematician prize
    
    • for works on topology in codimension two.
      

40. Shaw Prize
    
    • for his initiation of the field of global differential geometry and his continued leadership of the field, resulting in beautiful developments that are at the centre of contemporary mathematics, with deep connections to topology, algebra and analysis, in short, to all major branches of mathematics of the last sixty years.
      

41. Serbian Academy of Sciences
    
    • In the 1980s geometry and topology moved into leading roles, while in the 1990s the original topics from the 1950s of analysis and mechanics again became among the most widely studied.
      

42. New York Academy of Sciences
    
    • Similarly summer conferences were held on general topology and applications with the proceedings published in the Annals.
      

43. NAS Award in Mathematics
    
    • for his role in the introduction and application of radically new approaches to the topology of singular spaces, including characteristics classes, intersection homology, perverse sheaves, and stratified Morse theory.
      

44. LMS Presidential Addresses
    
    • Duality in topology.
      

45. Fields Medal
    

46. Czech Academy of Sciences
    
    • The Institute is concerned mainly with mathematical analysis (differential equations, numerical analysis, functional analysis, theory of functions, mathematical physics), probability theory and mathematical statistics, mathematical logic, theoretical computer science and graph theory, numerical algebra, topology (general and algebraic) and theory of teaching mathematics.
      

47. AMS Satter Prize
    
    • for her outstanding work in 3-dimensional topology.
      

48. Scientific Committee 2005
    
    • Special sessions will be on Topology and Classical Harmonic Analysis, as already decided in Liverpool.
      

49. BMC 2006
    
    • Farber, M Topology and robot motion planning .
      

50. Minutes for 1977
    
    • 5 Analysis, 3 Topology, 4 Algebra, 2 Number Theory, 1 Geometry, 1 Combinatorics, 1 Logic, 1 Probability.
      

51. BMC 1996
    
    • Casson, A The geometric topology of 3-manifolds .
      

52. BMC-BAMC meeting 2005
    
    • Special sessions will be on Topology and Classical Harmonic Analysis, as already decided in Liverpool.
      

53. BMC 1974
    
    • Brieskorn, E Topology of singularities and related topics .
      

54. Minutes for 1998
    
    • Geometry/Topology .
      

55. Minutes for 1997
    
    • Geometry/Topology *Dr B H Bowditch (Southampton), *Dr A D King (Lancaster), Dr V V Goryunov (Liverpool), Dr W J Harvey (King's), Dr J C Wood (Leeds), Dr D Singerman (Southampton), Dr A J Baker (Glasgow), *Dr D Joyce (Oxford).
      

56. BMC 2005
    
    • Lackenby, M The geometry anf topology of finite Cayley grphs .
      

57. Minutes for 1997
    
    • Geometry/Topology .
      

58. Minutes for 1963
    
    • Dr Adams undertook to arrange a programme of three related morning lectures on Topology.
      

59. Minutes for 2002
    
    • Suggested themes: number theory, analysis, groups, algebra / groups, topology, logic/etc, history/etc.
      

60. Minutes for 1952
    
    • The programme for the 1953 Colloquium was discussed and it was agreed that if possible one day should be devoted to each of Analysis, Topology and Differential Geometry, Algebra and Number Theory.
      

61. Minutes for 2008
    
    • In addition we had 12 morning speakers, 2 special sessions (in Fourier Analysis and in Topology), and 12 splinter groups.
      

62. BMC 1968
    
    • Johnson, B ERelations between algebras and topology in Banach algebras .
      

63. Minutes for 2005
    
    • There will be Special Sessions on Topology and Classical Harmonic Analysis.
      

64. Scientific Committee 2006
    
    • Dr Catharina Stroppel (Glasgow) (Algebraic Lie Theory and Topology) .
      

References

1. References for Bing
   
   • M Brown, The mathematical work of R H Bing, in Proceedings of the 1987 Topology Conference, Birmingham, AL, 1987, Topology Proc.
     
   • F Burton Jones, R H Bing, in Proceedings of the 1987 Topology Conference, Birmingham, AL, 1987, Topology Proc.
     
   • S Singh, Publications of R H Bing classified by the year, in Proceedings of the 1987 Topology Conference, Birmingham, AL, 1987, Topology Proc.
     
   • S Singh, R H Bing (1914-1986) : a tribute, Special volume in honor of R H Bing (1914-1986), Topology Appl.
     
   • M Starbird, R H Bing's human and mathematical vitality, in Handbook of the history of general topology, Vol.
     
   • L Whyburn, R H Bing 1949-50, in Proceedings of the 1987 Topology Conference, Birmingham, AL, 1987, Topology Proc.
     

2. References for Eilenberg
   
   • I M James, Some topologists, in I M James (ed.), History of Topology (Amsterdam, 1999), 883-908.
     
   • S Mac Lane, Samuel Eilenberg and topology (Polish), Wiadom.
     
   • S Mac Lane, The work of Samuel Eilenberg in topology, in Algebra, topology, and category theory : a collection of papers in honor of Samuel Eilenberg (New York, 1976), 133-144.
     

3. References for Brouwer
   
   • Geometry, Analysis, Topology and Mechanics (Amsterdam, 1976).
     
   • T Crilly and D Johnson, The emergence of topological dimension theory, in I M James (ed.), History of topology (Amsterdam, 1999), 1-24.
     
   • H Freudenthal, The first beginnings of modern topology, according to unpublished works from the estate of L E J Brouwer (Dutch), Nederl.
     
   • W P van Stigt, L E J Brouwer : intuitionism and topology, Proceedings, Bicentennial Congress Wiskundig Genootschap (Amsterdam, 1979), 359-374.
     

4. References for Urysohn
   
   • L K Arboleda, Origins of the Soviet school of topology : Remarks about the letters of P S Aleksandrov and P S Uryson to Maurice Frechet (Russian), Istor.-Mat.
     
   • T Crilly and D Johnson, The emergence of topological dimension theory, in I M James (ed.), History of topology (Amsterdam, 1999), 1-24.
     
   • M Katetov, P S Uryson and the beginnings of general topology (Czech), Pokroky Mat.
     
   • D van Dalen, Luitzen Egbertus Jan Brouwer, in I M James (ed.), History of topology (Amsterdam, 1999), 947-964.
     

5. References for Bott
   
   • Topology and Lie groups (Birkhauser Boston, Inc., Boston, MA, 1994).
     
   • Topology and Lie groups (Birkhauser Boston, Inc., Boston, MA, 1994), 3-9.
     
   • Topology and Lie groups (Birkhauser Boston, Inc., Boston, MA, 1994), 11-26.
     
   • Topology and Lie groups (Birkhauser Boston, Inc., Boston, MA, 1994), i-xii.
     

6. References for Jones Burton
   
   • P Nyikos, F Burton Jones's contributions to the normal Moore space problem, in Topology Conference,1979, Greensboro, N.C., 1979 (Greensboro, N.C., 1980), 27-38.
     
   • Publications of F Burton Jones, in Topology Conference, 1979, Greensboro, N.C., 1979 (Greensboro, N.C., 1980), ii-v.
     
   • J T Rogers Jr, F Burton Jones (1910-1999) - an appreciation, in Proceedings of the 1999 Topology and Dynamics Conference, Salt Lake City, UT, Topology Proc.
     

7. References for Smale
   
   • J Palis, On the contribution of Smale to dynamical systems, in From Topology to Computation : Proceedings of the Smalefest (New York, 1993), 165-178.
     
   • M M Peixoto, Some recollections of the early work of Steve Smale, in From Topology to Computation : Proceedings of the Smalefest (New York, 1993, 73-75.
     
   • S Smale, A survey of some recent developments in differential topology, Bull.
     
   • A J Tromba, Smale and nonlinear analysis : a personal perspective, in From Topology to Computation : Proceedings of the Smalefest (New York, 1993), 481-492.
     

8. References for Whyburn
   
   • G T Whyburn, Analytic Topology (New York, 1942).
     
   • L Whyburn, Letters from the R L Moore papers, Topology Proc.
     
   • L Whyburn, A visit with E H Moore, Topology Proc.
     
   • G T Whyburn, Dynamic topology, Amer.
     

9. References for Moore Robert
   
   • L Whyburn, R L Moore's first doctoral student at Texas, Algebraic and geometric topology (Berlin, 1978), 33-37.
     
   • L Whyburn, Letters from the R L Moore papers, Proceedings of the 1977 Topology Conference I, Topology Proc.
     

10. References for Stoilow
    
    • C Andreian Cazacu and T M Rassias, On Stoilow's work and its influence, in Analysis and topology (World Sci.
      
    • M Jurchescu, Simion Stoilow and the Romanian mathematical school, in Analysis and topology (World Sci.
      
    • S Marcus, Stoilow's work in real analysis : its significance and its impact, in Analysis and topology (World Sci.
      

11. References for Lefschetz
    
    • R F Brown, Fixed Point Theorems, in History of Topology (Oxford, 1999), 271-300.
      
    • I M James, Some topologists, in History of Topology (Oxford, 1999), 883-908.
      

12. References for De Rham
    
    • H Cartan, Les travaux de Georges de Rham sur les varietes differentiables, in A Haefliger and R Narasimhan (eds.), Essays on Topology and Related Topics : Memoires dedies a Georges de Rham (Berlin - Heidelberg - New York, 1970).
      
    • A Haefliger and R Narasimhan (eds.), Essays on Topology and Related Topics : Memoires dedies a Georges de Rham (Berlin - Heidelberg - New York, 1970).
      

13. References for Hausdorff
    
    • E Eichhorn, Felix Hausdorff/Paul Mongre: some aspects of his life and the meaning of his death, Recent developments of general topology and its applications (Berlin, 1992), 85-117.
      
    • G Preuss, Felix Hausdorff (1868-1942), Handbook of the history of general topology 1 (Dordrecht, 1997), 1-19.
      

14. References for Dowker
    
    • I M James, Some topologists, in I M James (ed.), History of Topology (Amsterdam, 1999), 883-908.
      
    • D Strauss, Obituary : Clifford Hugh Dowker, in Aspects of topology (Cambridge-New York, 1985), xi-xvii.
      

15. References for Gelfand
    
    • S G Gindikin, A A Kirillov and D B Fuks, Work of I M Gelfand in functional analysis, algebra and topology, Russian Math.
      
    • S G Gindikin, A A Kirillov and D B Fuks, Work of I M Gelfand in functional analysis, algebra and topology (Russian), Uspekhi Mat.
      

16. References for Descartes
    
    • H Samelson, Descartes and differential geometry, in Geometry, topology, and physics (Cambridge, MA, 1995), 323-328.
      
    • M Tarina, La geometrie de Descartes en perspective historique, in The XVIIIth National Conference on Geometry and Topology (Cluj-Napoca, 1988), 207-208.
      

17. References for Milnor
    
    • J Sondow, An aroma of paradox and audacity : Milnor's work in differential topology, in Topological methods in modern mathematics (Houston, TX, 1993), 23-30.
      
    • J Stasheff, Milnor's work from the perspective of algebraic topology, in Topological methods in modern mathematics (Houston, TX, 1993), 5-22.
      

18. References for Hurewicz
    
    • K Borsuk, Witold Hurewicz - life and work, in Handbook of the history of general topology 1 (Dordrecht, 1997), 79-84.
      
    • E Fadell, The contributions of Witold Hurewicz to algebraic topology, Collected works of Witold Hurewicz (Providence, R.I., 1995), xxxiii-xl.
      

19. References for James
    
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20. References for Nielsen Jakob
    
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21. References for Borsuk
    
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22. References for Steenrod
    
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23. References for Vranceanu
    
    • I D Teodorescu, Gheorghe Vranceanu-le fondateur de l'ecole de geometrie differentielle a Bucarest, in Differential geometry and topology applications in physics and technics, Bucharest, 1991, Polytech.
      
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24. References for Poincare
    
    • P S Aleksandrov, Poincare and topology (Russian), Uspekhi Mat.
      
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25. References for Riemann
    
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26. References for Chow
    
    • S S Chern, Wei-Liang Chow, 1911-1995, in Contemporary trends in algebraic geometry and algebraic topology, Tianjin, 2000 (World Sci.
      
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27. References for Moore Eliakim
    
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28. References for Tietze
    
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29. References for Wilder
    
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30. References for Heegaard
    
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31. References for Aristotle
    
    • M J White, On continuity : Aristotle versus topology?, Hist.
      

32. References for Rudin
    
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33. References for Eckmann
    
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34. References for Tait
    
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35. References for Weil
    
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36. References for Patodi
    
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37. References for Rey Pastor
    
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38. References for Listing
    
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39. References for Noether Emmy
    
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40. References for Thomae
    
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41. References for Hopf
    
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42. References for Escher
    
    • D Schattschneider, Escher: A mathematician in spite of himself, Structural Topology 15 (1988), 9-22.
      

43. References for Kuratowski
    
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44. References for Goursat
    
    • V Katz, Differential forms, in History of topology (North-Holland, Amsterdam, 1999), 111-122.
      

45. References for Hedrick
    
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46. References for Novikov Sergi
    
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47. References for Adams Frank
    
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48. References for De Groot
    
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49. References for Donaldson
    
    • S Donaldson, Remarks on gauge theory, complex geometry and 4-manifold topology, in M Atiyah and D Iagolnitzer (eds.), Fields Medallists Lectures (Singapore, 1997), 384-403.
      

50. References for Uhlenbeck Karen
    
    • S Donaldson, Remarks on gauge theory, complex geometry and 4-manifold topology, in M Atiyah and D Iagolnitzer (eds.), Fields Medallists Lectures (Singapore, 1997), 384-403.
      

51. References for Sierpinski
    
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52. References for Betti
    
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53. References for Severi
    
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54. References for Delone
    
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55. References for Netto
    
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56. References for Seifert
    
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57. References for Freudenthal
    
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58. References for Bolyai
    
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59. References for Williams
    
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60. References for Chen
    
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61. References for Frechet
    
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62. References for Tikhonov
    
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Additional material

1. Kuratowski: 'Introduction to Topology
   
   • Kuratowski: Introduction to Topology .
     
   • Sets and Topology .
     
   • Introduction to part II : Topology .
     
   • Topology is the study of those properties of geometric configurations which remain invariant when these configurations are subjected to one-to-one bicontinuous transformations, or homeomorphisms (see Chapter XII, § 3).
     
   • As can already be seen from the above example topology operates with more general concepts than analysis; differential properties of a given transformation are nonessential for topology, but bicontinuity is essential.
     
   • As a consequence, topology is often suitable for the solution of problems to which analysis cannot give the answer.
     
   • a topology, and hence - to a geometrisation or rather to a topologisation - of the investigation.
     
   • theorems on the existence of a solution of certain types of differential equations can be expressed as theorems on the existence of invariant points of a function space (the space of continuous functions) under continuous transformations; these theorems can be proved by topological methods in a more general form and in a simpler way than was formerly done without the aid of topology.
     
   • How much more general ought the spaces considered in topology be in order that they suffice for applications and yet, because of undue generality, they do not become too artificial? The answer to this question depends on the aims which a given topological work is to serve.
     
   • In Chapters IX-XII we give the fundamental concepts with which we must deal in all parts of topology.
     
   • Chapter XXI contains, in a very general outline, an introduction to homology theory which forms a fundamental part of algebraic topology.
     
   • This is the origin of the name algebraic topology in contrast to set-theoretic topology, in which we make use of the concepts and theorems of set theory.
     
   • Worthy of remark is the relation of the individual branches of mathematics which we observe here: topology, being a powerful tool for functional analysis and for various branches of classical analysis, which in its turn is connected, because of its applications, with technology and the natural sciences, itself makes use of the methods of algebra and the theory of sets.
     
   • In its initial stages, set-theoretic and algebraic topology developed entirely independently and possessed completely different thematic.
     
   • Set-theoretical topology, formerly called the theory of point sets, and concerning arbitrary subsets of Euclidean space, was begun by G Cantor, the creator of the theory of sets (circa 1880).
     
   • Algebraic topology was created by H Poincare in the last years of the past century; its objects were n-dimensional polygons and polyhedra.
     
   • This period was preceded by the transition from the investigation of subsets of Euclidean space in set-theoretic topology to the investigation of arbitrary topological spaces.
     
   • This extension of the thematics of topology appeared to a significant degree in connection with the new mathematical investigations concerning the concept of function space and infinite-dimensional spaces introduced by Hilbert.
     
   • In the last thirty years or so there has appeared an unusually rich flourishing of topology; many fundamental problems of topology have been solved and new methods developed.
     
   • Topology, which until recently was a conglomeration of loosely related theorems, became a systematic science, and topological methods penetrated into many other domains of mathematics.
     
   • http://www-history.mcs.st-andrews.ac.uk/Extras/Kuratowski_Topology.html .
     

2. Kuratowski: 'Set Theory and Topology' Foreword
   
   • Kuratowski: Set Theory and Topology Foreword .
     
   • Kazimierz Kuratowski's main work was in the area of topology and set theory.
     
   • He wrote an important textbook Set Theory and Topology for beginners.
     
   • The ideas and methods of set theory and topology permeate modern mathematics.
     
   • The purpose of the present volume is to give an accessible presentation of the fundamental concepts of set theory and topology; special emphasis being placed on presenting the material from the viewpoint of its applicability to analysis, geometry, and other branches of mathematics such as probability theory and algebra.
     
   • Consequently, results important for set theory and topology but not having close connection with other branches of mathematics, are given a minor role or are omitted entirely.
     
   • Here is Kuratowski's Introduction to the Topology part of the text.
     

3. Kuratowski: 'Introduction to Set Theory
   
   • Sets and Topology .
     
   • Although these subjects form at the present time an important part of mathematics and are being actively developed, the discussion of them in this book lies outside the principal goal of the book which is: the presentation of the most important branches of set theory and topology from the point of view of their applications to other branches of mathematics.
     
   • In the course of years, however, when set theory showed its usefulness in many branches of mathematics such as the theory of analytic functions or theory of measure, and when it became an indispensable basis for new mathematical disciplines (such as topology, the theory of functions of a real variable, the foundations of mathematics), it became an especially important branch and tool of modern mathematics.
     

4. Gordon Preston on semigroups
   
   • Topological spaces had not been heard of - though some very popular lectures were given on topology by J H C Whitehead (Henry Whitehead).
     
   • For the first year I worked with him on algebraic topology.
     
   • Until then I had come from a background in algebraic topology and algebraic geometry.
     

5. André Weil: 'Algebraic Geometry
   
   • A history of enumerative geometry could be a fascinating chapter in the general history of mathematics during the previous and present centuries, provided it brought to light the connections with related subjects, not merely with projective geometry, but with group-theory, the theory of Abelian functions, topology, etc.; this would require another book and a more competent writer.
     
   • In the hope of doing away once for all with the resulting confusion (for those who will adopt our language, or at any rate an equivalent one which may be translated into ours term for term), two separate terms will be used here for these two kinds of entities, instead of the one term "reducible variety" which has previously been applied to both: "bunches of varieties" for the former, and (as in modem topology) "cycles" for the latter, while the word "variety" will be reserved for "absolutely irreducible algebraic varieties", i.e.
     

6. Donald C Spencer's publications
   
   • K Kodaira and D C Spencer, On the variation of almost-complex structure, Algebraic Geometry and Topology (Princeton University Press, 1957), 139-150.
     
   • on Manifolds and Related Topics in Topology (University of Tokyo Press, 1975), 303-311.
     

7. Kurosh: 'Lectures on general algebra' Introduction
   
   • How great, and sometimes decisive, the impact of this modern algebra was on the development of many domains of mathematics, among which we mention, in the first instance, topology and functional analysis, is common knowledge.
     
   • Within the framework of the classical parts of general algebra, independent trends arose: homological algebra, which has already led to numerous results in topology and algebraic geometry; projective algebra, including the elements of projective geometry; and differential algebra, where general algebra yielded direct results in the theory of differential equations.
     

8. NAS Award in Mathematics
   
   • for his role in the introduction and application of radically new approaches to the topology of singular spaces, including characteristics classes, intersection homology, perverse sheaves, and stratified Morse theory.
     

9. F A Behrend's LMS Obituary by B H Neumann
   
   • Felix Behrend's sympathies within pure mathematics were wide, and his creativeness ranged over theory of numbers, algebraic equations, topology, and foundations of analysis.
     

10. Jacobson: 'Structure of Rings
    
    • In Chapter IX we define a topology of the set of primitive ideals of a ring and we use this to obtain representations of rings as rings of continuous functions on topological spaces.
      

11. Einar Hille: 'Analytic Function Theory
    
    • On the other hand, this is not a textbook in topology: if intuition helps, an appeal is made to intuition.
      

12. Von Neumann: 'The Mathematician' Part 2
    
    • The same may be said about topology.
      

13. Kurosh's book The theory of groups 1st edition
    
    • It was of particular importance that this restriction very soon led to conflicts with the needs of neighbouring branches of mathematics: in several parts of geometry, the theory of automorphic functions, topology, in all of these one again and again came across algebraic structures similar to groups, but infinite, and so demands were made upon the theory of groups that the theory of finite groups was not in a position to satisfy.
      

14. George Temple's Inaugural Lecture II
    
    • I see a whole topology, .
      

15. The Shaw Prize in Mathematical Sciences
    
    • for his initiation of the field of global differential geometry and his continued leadership of the field, resulting in beautiful developments that are at the centre of contemporary mathematics, with deep connections to topology, algebra and analysis, in short, to all major branches of mathematics of the last sixty years.
      

16. Halmos: creative art
    
    • No one can call himself a mathematician nowadays who doesn't have at least a vague idea of homological algebra, differential topology, and functional analysis, and every mathematician is probably somewhat of an expert on at least one of these subjects and yet when I studied mathematics in the 1930's none of those phrases had been invented, and the subjects they describe existed in seminal forms only.
      

17. Bronowski and retrodigitisation
    
    • Although Jacob Bronowski's name is most remembered in association with the BBC television documentary series The Ascent of Man he made at the end of his life - it inspired discussion [17, 18] in the Gazette of a tessellation found at the Alhambra - he read mathematics at the University of Cambridge, where he went on to take a doctorate with a thesis in geometry and topology.
      

18. Heinz Hopf Collected papers' Preface
    
    • It is indeed astonishing to realise that this oeuvre of a whole scientific life consists of only about 70 writings, Comptes Rendus Notes and survey articles included, and of course the book Topology I written jointly with Paul Aleksandrov.
      

19. P G Tait's obituary of Listing
    
    • The term Topology was introduced by Listing to distinguish what may be called qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated.
      

20. Selected papers of Edward Marczewski' Preface
    
    • Until the late fifties his main fields of interest were measure theory, descriptive set theory, general topology and probability theory.
      

21. Kurosh: 'The theory of groups' 1st edition
    
    • It was of particular importance that this restriction very soon led to conflicts with the needs of neighbouring branches of mathematics: in several parts of geometry, the theory of automorphic functions, topology, in all of these one again and again came across algebraic structures similar to groups, but infinite, and so demands were made upon the theory of groups that the theory of finite groups was not in a position to satisfy.
      

22. A D Aleksandrov's view of Mathematics
    
    • Algebra, geometry, analysis and topology are well represented and there are chapters on number theory, numerical analysis and computing machines.
      

Quotations

1. Quotations by Weyl
   
   • In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics.
     

2. Quotations by Poincare
   
   • Point set topology is a disease from which the human race will soon recover.
     

3. A quotation by Lefschetz
   
   • If it's just turning the crank it's algebra, but if it's got an idea in it, it's topology.
     

Chronology

1. Mathematical Chronology
   
   • Betti publishes a memoir on topology which contains the "Betti numbers".
     
   • Poincare begins work on algebraic topology.
     
   • Poincare publishes Analysis situs his first work on topology which gives an early systematic treatment of the topic.
     
   • He is the originator of algebraic topology publishing six papers on the topic.
     
   • Heegaard and Dehn publish Analysis Situs which marks the beginnings of combinatorial topology.
     
   • Weyl publishes Die Idee der Riemannschen Flache which brings together analysis, geometry and topology.
     
   • Milnor publishes On manifolds homeomorphic to the 7-sphere which opens up the new field of differential topology.
     
   • Thom is awarded a Fields Medal for his work on topology, in particular on characteristic classes, cobordism theory and the "Thom transversality theorem".
     
   • Sergi Novikov's work on differential topology, in particular in calculating stable homotopy groups and classifying smooth simply-connected manifolds, leads him to make the "Novikov Conjecture".
     
   • Grothendieck receives a Fields Medal for his work on geometry, number theory, topology and complex analysis.
     
   • Donaldson publishes Self-dual connections and the topology of smooth 4-manifolds which leads to totally new ideas concerning the geometry of 4-manifolds.
     
   • Krystyna Kuperberg solves the "Seifert Conjecture" about the topology of dynamical systems.
     
   • Borcherds is awarded a Fields Medal for his work in automorphic forms and mathematical physics; Gowers receives one for his work in functional analysis and combinatorics; Kontsevich receives one for his work in algebraic geometry, algebraic topology, and mathematical physics; and McMullen receives one for his work on holomorphic dynamics and geometry of 3-dimensional manifolds.
     

2. Chronology for 1890 to 1900
   
   • Poincare begins work on algebraic topology.
     
   • Poincare publishes Analysis situs his first work on topology which gives an early systematic treatment of the topic.
     
   • He is the originator of algebraic topology publishing six papers on the topic.
     

3. Chronology for 1950 to 1960
   
   • Milnor publishes On manifolds homeomorphic to the 7-sphere which opens up the new field of differential topology.
     
   • Thom is awarded a Fields Medal for his work on topology, in particular on characteristic classes, cobordism theory and the "Thom transversality theorem".
     

4. Chronology for 1960 to 1970
   
   • Sergi Novikov's work on differential topology, in particular in calculating stable homotopy groups and classifying smooth simply-connected manifolds, leads him to make the "Novikov Conjecture".
     
   • Grothendieck receives a Fields Medal for his work on geometry, number theory, topology and complex analysis.
     

5. Chronology for 1990 to 2000
   
   • Krystyna Kuperberg solves the "Seifert Conjecture" about the topology of dynamical systems.
     
   • Borcherds is awarded a Fields Medal for his work in automorphic forms and mathematical physics; Gowers receives one for his work in functional analysis and combinatorics; Kontsevich receives one for his work in algebraic geometry, algebraic topology, and mathematical physics; and McMullen receives one for his work on holomorphic dynamics and geometry of 3-dimensional manifolds.
     

6. Chronology for 1870 to 1880
   
   • Betti publishes a memoir on topology which contains the "Betti numbers".
     

7. Chronology for 1910 to 1920
   
   • Weyl publishes Die Idee der Riemannschen Flache which brings together analysis, geometry and topology.
     

8. Chronology for 1900 to 1910
   
   • Heegaard and Dehn publish Analysis Situs which marks the beginnings of combinatorial topology.
     

9. Chronology for 1980 to 1990
   
   • Donaldson publishes Self-dual connections and the topology of smooth 4-manifolds which leads to totally new ideas concerning the geometry of 4-manifolds.