2003 Jean-Pierre Serre: an Overview of His Work
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A Tribute to Henri Cartan
A Tribute to Henri Cartan This collection of articles paying tribute to the mathematician Henri Cartan was assembled and edited by Pierre Cartier, IHÉS, and Luc Illusie, Université Paris-Sud 11, in consultation with Jean-Pierre Serre, Collège de France. The collection begins with the present introductory article, which provides an overview of Cartan's work and a short contribution by Michael Atiyah. This overview is followed by three additional articles, each of which focuses on a particular aspect of Cartan's rich life. —Steven G. Krantz This happy marriage, which lasted until his death Jean-Pierre Serre (followed, a few months later, by that of his wife), Henri Cartan produced five children: Jean, Françoise, Étienne, 8 July 1904–13 August 2008 Mireille, and Suzanne. In September 1939, at the beginning of the Henri Cartan was, for many of the younger gen- war, he moved to Clermont-Ferrand, where the eration, the symbol of the resurgence of French University of Strasbourg had been evacuated. A mathematics after World War II. He died in 2008 year later he got a chair at the Sorbonne, where he at the age of 104 years. was given the task of teaching the students of the Personal Life ENS. This was a providential choice that allowed the “normaliens” (and many others) to benefit for Henri was the eldest son of the mathematician more than twenty-five years (1940–1965) from Élie Cartan (1869–1951), born in Dolomieu (Isère), his courses and seminars. In fact there was a two- and of his wife Marie-Louise Bianconi, of Corsican year interruption when he returned to Strasbourg origin. -
Arxiv:1006.1489V2 [Math.GT] 8 Aug 2010 Ril.Ias Rfie Rmraigtesre Rils[14 Articles Survey the Reading from Profited Also I Article
Pure and Applied Mathematics Quarterly Volume 8, Number 1 (Special Issue: In honor of F. Thomas Farrell and Lowell E. Jones, Part 1 of 2 ) 1—14, 2012 The Work of Tom Farrell and Lowell Jones in Topology and Geometry James F. Davis∗ Tom Farrell and Lowell Jones caused a paradigm shift in high-dimensional topology, away from the view that high-dimensional topology was, at its core, an algebraic subject, to the current view that geometry, dynamics, and analysis, as well as algebra, are key for classifying manifolds whose fundamental group is infinite. Their collaboration produced about fifty papers over a twenty-five year period. In this tribute for the special issue of Pure and Applied Mathematics Quarterly in their honor, I will survey some of the impact of their joint work and mention briefly their individual contributions – they have written about one hundred non-joint papers. 1 Setting the stage arXiv:1006.1489v2 [math.GT] 8 Aug 2010 In order to indicate the Farrell–Jones shift, it is necessary to describe the situation before the onset of their collaboration. This is intimidating – during the period of twenty-five years starting in the early fifties, manifold theory was perhaps the most active and dynamic area of mathematics. Any narrative will have omissions and be non-linear. Manifold theory deals with the classification of ∗I thank Shmuel Weinberger and Tom Farrell for their helpful comments on a draft of this article. I also profited from reading the survey articles [14] and [4]. 2 James F. Davis manifolds. There is an existence question – when is there a closed manifold within a particular homotopy type, and a uniqueness question, what is the classification of manifolds within a homotopy type? The fifties were the foundational decade of manifold theory. -
Gromov Receives 2009 Abel Prize
Gromov Receives 2009 Abel Prize . The Norwegian Academy of Science Medal (1997), and the Wolf Prize (1993). He is a and Letters has decided to award the foreign member of the U.S. National Academy of Abel Prize for 2009 to the Russian- Sciences and of the American Academy of Arts French mathematician Mikhail L. and Sciences, and a member of the Académie des Gromov for “his revolutionary con- Sciences of France. tributions to geometry”. The Abel Prize recognizes contributions of Citation http://www.abelprisen.no/en/ extraordinary depth and influence Geometry is one of the oldest fields of mathemat- to the mathematical sciences and ics; it has engaged the attention of great mathema- has been awarded annually since ticians through the centuries but has undergone Photo from from Photo 2003. It carries a cash award of revolutionary change during the last fifty years. Mikhail L. Gromov 6,000,000 Norwegian kroner (ap- Mikhail Gromov has led some of the most impor- proximately US$950,000). Gromov tant developments, producing profoundly original will receive the Abel Prize from His Majesty King general ideas, which have resulted in new perspec- Harald at an award ceremony in Oslo, Norway, on tives on geometry and other areas of mathematics. May 19, 2009. Riemannian geometry developed from the study Biographical Sketch of curved surfaces and their higher-dimensional analogues and has found applications, for in- Mikhail Leonidovich Gromov was born on Decem- stance, in the theory of general relativity. Gromov ber 23, 1943, in Boksitogorsk, USSR. He obtained played a decisive role in the creation of modern his master’s degree (1965) and his doctorate (1969) global Riemannian geometry. -
Part III Essay on Serre's Conjecture
Serre’s conjecture Alex J. Best June 2015 Contents 1 Introduction 2 2 Background 2 2.1 Modular forms . 2 2.2 Galois representations . 6 3 Obtaining Galois representations from modular forms 13 3.1 Congruences for Ramanujan’s t function . 13 3.2 Attaching Galois representations to general eigenforms . 15 4 Serre’s conjecture 17 4.1 The qualitative form . 17 4.2 The refined form . 18 4.3 Results on Galois representations associated to modular forms 19 4.4 The level . 21 4.5 The character and the weight mod p − 1 . 22 4.6 The weight . 24 4.6.1 The level 2 case . 25 4.6.2 The level 1 tame case . 27 4.6.3 The level 1 non-tame case . 28 4.7 A counterexample . 30 4.8 The proof . 31 5 Examples 32 5.1 A Galois representation arising from D . 32 5.2 A Galois representation arising from a D4 extension . 33 6 Consequences 35 6.1 Finiteness of classes of Galois representations . 35 6.2 Unramified mod p Galois representations for small p . 35 6.3 Modularity of abelian varieties . 36 7 References 37 1 1 Introduction In 1987 Jean-Pierre Serre published a paper [Ser87], “Sur les representations´ modulaires de degre´ 2 de Gal(Q/Q)”, in the Duke Mathematical Journal. In this paper Serre outlined a conjecture detailing a precise relationship between certain mod p Galois representations and specific mod p modular forms. This conjecture and its variants have become known as Serre’s conjecture, or sometimes Serre’s modularity conjecture in order to distinguish it from the many other conjectures Serre has made. -
Roots of L-Functions of Characters Over Function Fields, Generic
Roots of L-functions of characters over function fields, generic linear independence and biases Corentin Perret-Gentil Abstract. We first show joint uniform distribution of values of Kloost- erman sums or Birch sums among all extensions of a finite field Fq, for ˆ almost all couples of arguments in Fq , as well as lower bounds on dif- ferences. Using similar ideas, we then study the biases in the distribu- tion of generalized angles of Gaussian primes over function fields and primes in short intervals over function fields, following recent works of Rudnick–Waxman and Keating–Rudnick, building on cohomological in- terpretations and determinations of monodromy groups by Katz. Our results are based on generic linear independence of Frobenius eigenval- ues of ℓ-adic representations, that we obtain from integral monodromy information via the strategy of Kowalski, which combines his large sieve for Frobenius with a method of Girstmair. An extension of the large sieve is given to handle wild ramification of sheaves on varieties. Contents 1. Introduction and statement of the results 1 2. Kloosterman sums and Birch sums 12 3. Angles of Gaussian primes 14 4. Prime polynomials in short intervals 20 5. An extension of the large sieve for Frobenius 22 6. Generic maximality of splitting fields and linear independence 33 7. Proof of the generic linear independence theorems 36 References 37 1. Introduction and statement of the results arXiv:1903.05491v2 [math.NT] 17 Dec 2019 Throughout, p will denote a prime larger than 5 and q a power of p. ˆ 1.1. Kloosterman and Birch sums. -
Abragam a & Bleaney B. Electron Paramagnetic Resonance of Transition Ions. Oxford, England: Oxford University Press, 1970. 7
® CC/NUMBER 28 This Week's Citation Classic JULY 13, 1992 Abragam A & Bleaney B. Electron paramagnetic resonance of transitionions. Oxford, England: Oxford University Press, 1970. 700 p. [Ecole Normale. Paris, France, and Clarendon Laboratory, University of Oxford. England] This book covers both experimental and theoretical His proposal was for a gigantic vol- aspects of electron paramagnetic resonance, for ume, including spin resonance stud- the 3d, 4d, 5d, 4f, and 5f transition groups. It has ies of ions in semiconductors and at remained the standard work on the subject for 20 years. [The SCI® indicates that this book has been defect sites. This would have been far cited in more than 3,160 publications.] larger than the 700 pages that were published by the Oxford University Press in 1970. Electron Paramagnetic Resonance As professor at the Ecole Normale in Paris, he gave a different set of lec- Brebis Bleaney tures every year. (He complained that if Clarendon Laboratory he lectured on his own subject of mag- University of Oxford netic resonance, the students were not Oxford OX1 3PU interested, while if he lectured on one England of their subjects, they knew much more The first experiments on electron about it than he did.) Nevertheless, he paramagnetic resonance in Oxford had ready sets of lecture notes cover- 1 were made by Desmond M.S. Bagguley ing the theory of electron paramag- in 1945 and extended to low tempera- netic resonance of transition ions. 2 tures by R.P. Penrose and myself in These formed the second part of our 1947. Anatole Abragam came to Ox- book, apart from a chapter on the Jahn- ford in 1948 from Paris, and his doc- Teller effect that he added later, before toral thesis under Maurice Pryce de- I had finished writing the first part of veloped the theory of magnetic experimental aspects. -
A Short Survey of Cyclic Cohomology
Clay Mathematics Proceedings Volume 10, 2008 A Short Survey of Cyclic Cohomology Masoud Khalkhali Dedicated with admiration and affection to Alain Connes Abstract. This is a short survey of some aspects of Alain Connes’ contribu- tions to cyclic cohomology theory in the course of his work on noncommutative geometry over the past 30 years. Contents 1. Introduction 1 2. Cyclic cohomology 3 3. From K-homology to cyclic cohomology 10 4. Cyclic modules 13 5. Thelocalindexformulaandbeyond 16 6. Hopf cyclic cohomology 21 References 28 1. Introduction Cyclic cohomology was discovered by Alain Connes no later than 1981 and in fact it was announced in that year in a conference in Oberwolfach [5]. I have reproduced the text of his abstract below. As it appears in his report, one of arXiv:1008.1212v1 [math.OA] 6 Aug 2010 Connes’ main motivations to introduce cyclic cohomology theory came from index theory on foliated spaces. Let (V, ) be a compact foliated manifold and let V/ denote the space of leaves of (V, ).F This space, with its natural quotient topology,F is, in general, a highly singular spaceF and in noncommutative geometry one usually replaces the quotient space V/ with a noncommutative algebra A = C∗(V, ) called the foliation algebra of (V,F ). It is the convolution algebra of the holonomyF groupoid of the foliation and isF a C∗-algebra. It has a dense subalgebra = C∞(V, ) which plays the role of the algebra of smooth functions on V/ . LetA D be a transversallyF elliptic operator on (V, ). The analytic index of D, index(F D) F ∈ K0(A), is an element of the K-theory of A. -
Jacques Tits
Jacques Tits Jacques Tits was born in Uccle, in the southern outskirts of Brussels, Belgium, on August 12, 1930. He retired from his professorship at the Collège de France in Paris in 2000 and has since then been Professor Emeritus. His father a mathematician, Jacques’s mathematical talent showed early. At the age of three he was able to do all the operations of arithmetic. He skipped several years at school. His father died when Jacques was only 13 years old. Since the family had very little to live on, Jacques started tutoring students four years older to contribute to the household expenses. He passed the entrance exam at the Free University of Brussels at the age of 14, and received his doctorate in 1950 at 20 years of age. Tits was promoted to professor at the Free University of Brussels in 1962 and remained in this position for two years before accepting a professorship at the University of Bonn in 1964. In 1973 he moved to Paris, taking up a position as Chair of Group Theory in the Collège de France. Shortly after, in 1974, he became a naturalised French subject. Tits held this chair until he retired in 2000. Jacques Tits has been a member of the French Académie des Sciences since 1974. In 1992 he was elected a Foreign Member of the US National Academy of Sciences and the American Academy of Arts and Sciences. In addition he holds memberships of science academies in Holland and Belgium. He has been awarded honorary doctorates from the Universities of Utrecht, Ghent, Bonn and Leuven. -
Arxiv:1003.6025V1 [Math.HO] 31 Mar 2010 Karl Stein (1913-2000)
Karl Stein (1913-2000) Alan Huckleberry Karl Stein was born on the first of January 1913 in Hamm in Westfalen, grew up there, received his Abitur in 1932 and immediately thereafter be- gan his studies in M¨unster. Just four years later, under the guidance of Heinrich Behnke, he passed his Staatsexam, received his promotion and became Behnke’s assistant. Throughout his life, complex analysis, primarily in higher dimensions (“mehrere Ver¨anderliche”), was the leitmotif of Stein’s mathematics. As a fresh Ph.D. in M¨unster in 1936, under the leadership of the master Behnke, he had already been exposed to the fascinating developments in this area. The brilliant young Peter Thullen was proving fundamental theorems, Henri Cartan had visited M¨unster, and Behnke and Thullen had just writ- ten the book on the subject. It must have been clear to Stein that this was the way to go. Indeed it was! The amazing phenomenon of analytic continuation in higher dimensions had already been exemplified more than 20 years be- fore in the works of Hartogs and E. E. Levi. Thullen’s recent work had gone much further. In the opposite direction, Cartan and Thullen had arXiv:1003.6025v1 [math.HO] 31 Mar 2010 proved their characterization of domains in Cn which admit a holomor- phic function which can not be continued any further. Behnke himself was also an active participant in mathematics research, always bringing new ideas to M¨unster. This was indeed an exciting time for the young researcher, Karl Stein. Even though the pest of the Third Reich was already invading academia, Behnke kept things going for as long as possible. -
Pierre Deligne
www.abelprize.no Pierre Deligne Pierre Deligne was born on 3 October 1944 as a hobby for his own personal enjoyment. in Etterbeek, Brussels, Belgium. He is Profes- There, as a student of Jacques Tits, Deligne sor Emeritus in the School of Mathematics at was pleased to discover that, as he says, the Institute for Advanced Study in Princeton, “one could earn one’s living by playing, i.e. by New Jersey, USA. Deligne came to Prince- doing research in mathematics.” ton in 1984 from Institut des Hautes Études After a year at École Normal Supériure in Scientifiques (IHÉS) at Bures-sur-Yvette near Paris as auditeur libre, Deligne was concur- Paris, France, where he was appointed its rently a junior scientist at the Belgian National youngest ever permanent member in 1970. Fund for Scientific Research and a guest at When Deligne was around 12 years of the Institut des Hautes Études Scientifiques age, he started to read his brother’s university (IHÉS). Deligne was a visiting member at math books and to demand explanations. IHÉS from 1968-70, at which time he was His interest prompted a high-school math appointed a permanent member. teacher, J. Nijs, to lend him several volumes Concurrently, he was a Member (1972– of “Elements of Mathematics” by Nicolas 73, 1977) and Visitor (1981) in the School of Bourbaki, the pseudonymous grey eminence Mathematics at the Institute for Advanced that called for a renovation of French mathe- Study. He was appointed to a faculty position matics. This was not the kind of reading mat- there in 1984. -
Homogenization 2001, Proceedings of the First HMS2000 International School and Conference on Ho- Mogenization
in \Homogenization 2001, Proceedings of the First HMS2000 International School and Conference on Ho- mogenization. Naples, Complesso Monte S. Angelo, June 18-22 and 23-27, 2001, Ed. L. Carbone and R. De Arcangelis, 191{211, Gakkotosho, Tokyo, 2003". On Homogenization and Γ-convergence Luc TARTAR1 In memory of Ennio DE GIORGI When in the Fall of 1976 I had chosen \Homog´en´eisationdans les ´equationsaux d´eriv´eespartielles" (Homogenization in partial differential equations) for the title of my Peccot lectures, which I gave in the beginning of 1977 at Coll`egede France in Paris, I did not know of the term Γ-convergence, which I first heard almost a year after, in a talk that Ennio DE GIORGI gave in the seminar that Jacques-Louis LIONS was organizing at Coll`egede France on Friday afternoons. I had not found the definition of Γ-convergence really new, as it was quite similar to questions that were already discussed in control theory under the name of relaxation (which was a more general question than what most people mean by that term now), and it was the convergence in the sense of Umberto MOSCO [Mos] but without the restriction to convex functionals, and it was the natural nonlinear analog of a result concerning G-convergence that Ennio DE GIORGI had obtained with Sergio SPAGNOLO [DG&Spa]; however, Ennio DE GIORGI's talk contained a quite interesting example, for which he referred to Luciano MODICA (and Stefano MORTOLA) [Mod&Mor], where functionals involving surface integrals appeared as Γ-limits of functionals involving volume integrals, and I thought that it was the interesting part of the concept, so I had found it similar to previous questions but I had felt that the point of view was slightly different. -
Interview with Henri Cartan, Volume 46, Number 7
fea-cartan.qxp 6/8/99 4:50 PM Page 782 Interview with Henri Cartan The following interview was conducted on March 19–20, 1999, in Paris by Notices senior writer and deputy editor Allyn Jackson. Biographical Sketch in 1975. Cartan is a member of the Académie des Henri Cartan is one of Sciences of Paris and of twelve other academies in the first-rank mathe- Europe, the United States, and Japan. He has re- maticians of the twen- ceived honorary doc- tieth century. He has torates from several had a lasting impact universities, and also through his research, the Wolf Prize in which spans a wide va- Mathematics in 1980. riety of areas, as well Early Years as through his teach- ing, his students, and Notices: Let’s start at the famed Séminaire the beginning of your Cartan. He is one of the life. What are your founding members of earliest memories of Bourbaki. His book Ho- mathematical inter- mological Algebra, writ- est? ten with Samuel Eilen- Cartan: I have al- berg and first published ways been interested Henri Cartan’s father, Élie Photograph by Sophie Caretta. in 1956, is still in print in mathematics. But Cartan. Henri Cartan, 1996. and remains a standard I don’t think it was reference. because my father The son of Élie Cartan, who is considered to be was a mathematician. I had no doubt that I could the founder of modern differential geometry, Henri become a mathematician. I had many teachers, Cartan was born on July 8, 1904, in Nancy, France. good or not so good.