COLUMBIA UNIVERSITY Department of Mathematics   ELLIS R. KOLCHIN MEMORIAL LECTURES   Robert MacPherson IAS Topology and the Langlands Program   Abstract: 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.                                                                                                 Friday, March 24 4:30 p.m., 312 Mathematics Building  Tea will be served at 3:45 p.m., 508 Mathematics Building   Directions to Columbia University