Zweifellos einer der einflußreichsten Mathematiker des 20. Jahrhunderts, Michael Atiyah, wird heute 80. (PS: Beinahe-Lösung der Kervaire-Vermutung angekündigt.) Besonders bekannt für den Atiyah-Singer-Indexsatz, für den er 1966 die Fields-Medaille und 2004 noch den Abelpreis erhielt. Der Indexsatz stellt eine Beziehung zwischen der Lösbarkeit von Differentialgleichungen und der Topologie/Geometrie des zugrundeliegenden Raumes her. In seiner einfachsten Form besagt er, daß die Dirac-Gleichung (eine bestimmte Wellengleichung) auf einer geschlossenen Spin-Mannigfaltigkeit (d.i. ein höher-dimensionales Analog von gekrümmten Flächen) nur dann Lösungen haben kann, wenn eine bestimmte topologische Kennzahl der Mannigfaltigkeit, das A-Dach-Geschlecht, nicht 0 ist. (Für die Geometrie folgt daraus zum Beispiel, daß solche Mannigfaltigkeiten nicht positiv gekrümmt sein können: man kann nämlich leicht nachrechnen, daß die Dirac-Gleichung bei positiver Krümmung keine Lösung haben kann.) In der Times ist heute ein Artikel, der auch auf Atiyahs politische Aktivitäten eingeht. PS: Angeblich soll auf der Atiyah80-Konferenz, die zur Zeit in Edinburgh stattfindet, ein Beweis der Kervaire-Vermutung, mit Ausnahme von Dimension 126, durch Hopkins, Hill und Ravenel angekündigt worden sein. Jedenfalls hat J.Morava auf dem ALGTOP-Verteiler mitgeteilt: "from the Atiyah80 conference, where Mike Hopkins announced the proof (joint work with Mike Hill & Doug Ravenel), that there are no Kervaire invariant one elements in above dimension 126. [That case remains open at the moment, but beyond it there are no more.] PPS: Zum Thema Kervaire-Vermutung hat Nick Kuhn ausführlichere Informationen auf ALGTOP mitgeteilt: (1) From a message I sent to my departmental colleagues this morning: Yesterday, at the conference on Geometry and Physics being held in Edinburgh in honor of Sir Michael Atiyah, Harvard Professor Mike Hopkins announced a solution to the 45 year old Kervaire Invariant One problem, one of the major outstanding problems in algebraic and geometric topology. This is joint work with Rochester professor Doug Ravenel and U VA postdoctoral Whyburn Instructor Mike Hill. The solution completes the work on `exotic spheres' begun by John Milnor in the 1950's which led to his Fields Medal. This is a central part of the classification of manifolds (= curves, surfaces, and their higher dimensional analogues). A 1962 Annals of Math paper by Milnor and Michael Kervaire classified exotic differential structures on spheres, subject to one possible ambiguity of order 2 in even dimensions. A 1969 Annals Math paper by Princeton professor William Browder resolved this question, except when the dimension was 2 less than a power of 2. In these dimensions, he translated the problem into one in algebraic topology, specifically one about the existence of certain elements in the stable homotopy groups of spheres. Over the next decade, the elements in dimensions 30, 62, and 126 were shown to exist; equivalently there exist some manifolds in those dimensions with some oddball properties. Significant work on closely related problems was done by Northwestern professor Mark Mahowald. So yesterday's announcement was that in all higher dimensions (254, 510, 1022, etc.), the putative elements do NOT exist. This result is `detected' in a generalized homology theory that is periodic of period 256 built from the complex oriented theory associated to deformations of the universal height 4 formal group law at the prime 2. (By contrast, real K-theory is has period 8 and comes from height 1 deformations, and theories based on elliptic cohomology come from height 2.) The strategy of proof has similarity to work of Ravenel's from the late 1970's, but the success of the strategy now illustrates the power of newly emerging control of subtle number theoretic and group theoretic structure in algebraic topology. (2) Technical stuff, which may or may not be accurate ... Step 1 Using results/methods from [MillerRavWilson], one can show if Theta_j is nonzero, then it is nonzero in pi_*(E_4^{hZ/8}), for some well chosen action of Z/8 on the 4th 2-adic Morava E theory. Step 2 Using a spectral sequence associated to a cleverly chosen filtered equivariant model for E_4 (or similar ??) - and this is the very new bit, I think - one shows that (a) pi_{-2}(E_4^{hZ/8}) = 0 and (b) pi_*(E_4^{hZ/8}) is 256 periodic. Thus thus the Theta_j's cannot exist beginning in dimension 254. Die Folien von Hopkins Vortrag findet man auf https://www.maths.ed.ac.uk/~aar/edinkerv.pdf.