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.
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