Activities This Week

In 1981, Drinfeld enumerated the number of irreducible l-adic local systems of rank two on a projective smooth curve in positive characteristic fixed by the Frobenius endomorphism. Interestingly, this number bears resemblance to the number of points on a variety over a finite field. Deligne proposed conjectures to extend and comprehend Drinfeld’s result. In this talk, I will present Deligne’s conjectures and discuss some mysterious phenomena that have emerged in various cases where this number is related to the number of stable Higgs bundles.

The Teichmuller space of geometric structures of a given type is a quotient of the (generally, infinite-dimensional) space of geometric structures by the group of isotopies, that is, by the connected component of the diffeomorphism group. In several important qand smooth.uestions, such as for symplectic, hyperkahler, Calabi-Yau, G2 structures, this quotient is finite-dimenisional and even smooth. The mapping class group acts on the Teichmuller space by natural diffeomorphisms, and this action is in many important situations ergodic (in particular, it has dense orbits), bringing strong consequences for the geometry. I would describe the Teichmuller space for the best understood cases, such as symplectic and hyperkahler manifolds, and give a few geometric applications.

בהרצאה נדון בהעתקות מעניינות מחבורה כלשהיא לתוך הממשיים שנקראות קואזי-מורפיזמים. נראה שימושים שלהן בהרבה ענפים של מתמטיקה כגון: תורת הקשרים, תורת החבורות, גאומטריה וטופולוגיה

To any group, and more generally a C*-algebra, is associated its simplex of traces. The extreme points are called characters, which are a central notion in harmonic analysis. For a Kazhdan group, the simplex of traces is Bauer - the extreme points are closed. For the free group Fn the simplex of traces is Poulsen - the extreme points are dense. What about the simplex of Out(Fn)-invariant traces on Fn (n>3)? Is it Bauer, Poulsen or something in between? What about free products of finite groups, and free product of matrix algebras? Some answers and proofs will be provided, after an introduction on traces and characters. The talk is based on works with Levit, Orovitz and Slutsky.