AGNT Ical Atom

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 examine the arithmetic properties of eigenvalues of random matrices with integer entries, focusing on the irreducibility of their characteristic polynomials and their Galois groups. Rivin, Jouve-Kowalski-Zywina, and Lubotzky-Rosenzweig previously studied characteristic polynomials arising from random walks on the Cayley graphs of Zariski-dense finitely generated subgroups of linear groups, such as SL_n(Z) . Eberhard, resolving conjectures of Babai and Vu-Wood assuming ERH, analyzed discrete random matrices and established that the characteristic polynomial of a matrix with independent entries (say taking the values 0,1 with equal probabilities) is irreducible and has a large Galois group with high probability as the matrix dimension grows. Ferber, Jain, Sah, and Sawhney proved a counterpart of these results to symmetric matrices.

In this talk, I will present recent results joint with Daniele Garzoni and Sasha Sodin on random tridiagonal matrices where the main diagonal consists of independent Bernoulli entries, the superdiagonal and subdiagonal entries are identically one, and all other entries are zero. We show that the characteristic polynomial of such matrices is irreducible and analyze the structure of its Galois group. If time permits, we will discuss applications to the localization of the eigenstates in Anderson’s 1-dim localization model.

A key feature of our approach lies in combining techniques from both the above random walk framework and the above discrete matrix setting. The latter leverages the Extended Riemann Hypothesis (ERH) to reduce the problem to analyzing the distribution of eigenvalues modulo primes p. To achieve strong error bounds in these computations, we exploit the powerful mixing properties of simple groups such as PSL_2(p) , a central tool in the above-mentioned random walk results.

We will describe a general strategy for proving the algebraicity of the Hodge Weil classes on abelian varieties of Weil type. The latter are even dimensional abelian varieties admitting a suitable embedding of a CM number field in their rational endomorphism ring. We will describe the implementation of the strategy for abelian varieties of dimension 4 and 6, and why it implies the Hodge conjecture for abelian varieties of dimension at most 5.

Harish Chandra developed representation theory on real and p-adic groups using analytic objects such as matrix coefficients and his distributional characters. He found that density properties of these distributions in the class of all invariant distributions plays an important role in Establishing basic results of Harmonic analysis on the group. Moving to G-Homogeneous spaces, spherical characters are distributions that play an important role in the relative trace formula. These objects were studied extensively in special cases and important results were obtained by Rallis and Rader who formulated natural density problems regarding these distributions. In a joint work with A. Aizenbud (Weizmann) and J. Bernstein (Tel-Aviv), we introduce some algebraic methods based on the concept of Cohen-Macaulay and Bernstein’s theory of representations of p-adic groups to tackle some of these density problems in the -adic case. In my presentation, I will not assume knowledge of Bernsein’s theory or knowledge of Harmonic analysis on p-adic groups.

The Kazhdan–Lusztig isomorphism, relating the affine Hecke algebra of a p-adic group to the equivariant K-theory of the Steinberg variety of its Langlands dual, played a key role in the proof of the Deligne–Langlands conjectures concerning the classification of tamely ramified irreducible representations. In this talk, I will recall the statement of the Kazhdan–Lusztig isomorphism. I will also introduce the relative Langlands duality and propose a conjectural relative version of the Kazhdan–Lusztig isomorphism. I will focus on specific examples which we can prove.

Chromatic homotopy theory aims to study cohomology theories through a hierarchy of simpler layers, organized by a notion called height. In this talk I will introduce the basic ideas behind this viewpoint and explain two approaches to analyzing these monochromatic layers: the classical K(n)-local category, which is closely related to one-dimensional formal group laws, and the T(n)-local or telescopic category, which is more directly tied to periodic phenomena in the stable homotopy groups of spheres. I will then describe a framework for understanding periodicity inside the chromatic layers, and explain how this allows one to lift Picard elements from the K(n)-local setting to the telescopic setting. Finally, I will present an application to chromatic Galois theory, leading to the construction of a first example of a non-abelian Galois extension in the T(n)-local world.