Activities This Week

TBA

Surface holonomy and the Wess–Zumino phase are fundamental in string theory and Chern–Simons theory, but giving a fully analytic description of their nonabelian versions has been a longstanding challenge. In this talk, I will explain how Yekutieli’s theory of nonabelian multiplicative integration on surfaces provides such a framework. The starting point is a smooth 2-connection (α,β) on a Lie crossed module. I will describe how one constructs multiplicative integrals associated to this data, and then show that these integrals satisfy the axioms of a transport 2-functor in the sense of Schreiber and Waldorf, providing an explicit model of nonabelian surface holonomy. I will conclude by discussing the resulting three-dimensional Stokes theorem and its relation to the Wess–Zumino phase, including the abelian case as a special instance.

We count the number of minimally ramified quartic dihedral exten- sions of the rational numbers by discriminant unconditionally, and by the product of ramified primes assuming the Generalized Riemann Hypothe- sis.

In 1981, Diaconis and Shahshahani showed that roughly n*log(n) random transpositions are required to mix a deck of n cards (namely, to produce an approximately random permutation in S_n). In their work, they translated this problem into a question in the representation theory of the symmetric group S_n, about bounding the values of irreducible characters at a transposition t = (i j). In this talk, I will explain how this perspective extends far beyond card shuffling; For a finite or compact group G, one can ask how quickly repeated multiplication by a fixed conjugacy class becomes uniformly distributed, and how this problem is controlled by general character estimates. I will describe this general framework, survey some known results, and discuss recent progress on character bounds in unitary groups. Based on a joint work (in progress) with Nir Avni, Peter Keevash and Noam Lifshitz and on a work with Nir Avni and Michael Larsen (arXiv:2402.11108).