25th Anniversary - Abstracts

R. MacPherson - Topology and the Langlands program

An analogy dating to the nineteenth century goes like this:  Number ring <--> Ring of functions on an affine curve over a finite field <--> Ring of functions on a complex curve.  So problems in number theory have analogues in complex geometry.

A lot of recent activity recently uses this analogy to go from ideas in the Langlands program to objects in complex geometry, where topological methods apply. This talk will look at two examples. The first is the interpretation of Hecke operators in terms of Schubert varieties. The Langlands dual group emerges naturally from topological considerations by the Drinfeld-Ginsburg-Lusztig-Mirkovic-Vilonen theorem. This is an ingredient of Geometric Langlands. The second is the geometric interpretation of transfer factors, in terms of Lefschetz numbers on affine Springer fibers. This leads to cases of the Fundamental Lemma, proved jointly with Goresky and Kottwitz, and then much more generally by Laumon and Ngo.

This talk will emphasize the beautiful topological objects that arise from the Langlands program, rather than the technicalities of Geometric Langlands or the Fundamental Lemma.


W. Nahm - Quantum field theory and Mathematics

Anders als bei Gravitationstheorie und Quantenmechanik hatte die Entwicklung der Quantenfeldtheorie um 1930 keinen Widerhall in der Mathematik gefunden, trotz der Anstrengung von Hermann Weyl. Die durch das Thirring-Modell gebotene Chance wurde lange nicht genutzt, erst ueber den Zusammenhang von affinen Kac-Moody-Algebren und konformer Feldtheorie kam es wieder zu einer engeren Zusammenarbeit, mit wichtigen Konsequenzen wie der Erforschung der Mirror-Symmetrie. Die konform invarianten Theorien versprechen weiterhin viele neue Einsichten, aber die Quantenfeldtheorie sollte auch insgesamt zu einem Kerngebiet der Mathematik werden, gerade weil sie zeigt, wie wenig natuerlich die traditionellen Grenzen zwischen Algebra, Analysis und Geometrie heute geworden sind.


M. Rapoport - Non-archimedean period domains

Period domains over the complex numbers are open subsets of partial flag varieties, which parametrize Hodge structures. Non-archimedean period domains are their analogues over p-adic fields and parametrize p-adic Hodge structures. The most famous non-archimedean period domain is the Drinfeld half space (the complement of all Q_p-rational hyperplanes in projective space). This talk will give more examples and report on recent results on the cohomology of these spaces.