Activities This Week

We consider the family of periodic measures for the full diagonal action on the space of unimodular lattices. This family is important and natural due to its tight relation to class groups in number fields. We show that many natural families of measures on the space of lattices can be approximated using this family (in the weak sense). E.g., we show that for any 0<c\leq 1, the measure cm_{X_n} can be approximated this way, where m_{X_n} denotes the Haar probability measure on X_n. Moreover, we show that non ergodic measures can be approximated. Our proof is based on the equidistribution of Hecke neighbors and on constructions of special number fields. We will discuss the results, alternative ways to attack the problem, and our method of proof. This talk is based on a joint work with Omri Solan.

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