AGNT Ical Atom

This talk is based on my ongoing work with Steffen Muller and Padma Srinivasan. I will explain the idea of the Quadratic Chabauty method for finding rational points on curves and how one reinterprets previous work by Balakrishnan and Dogra, on their own and with collaborators, using p-adic adelic metrics on line bundles. Time permitting I will explain how one can use this to compute the “local contributions away from p” for Quadratic Chabauty, which are crucial for computations.

Continuing with the topics of last week’s talk, I will explain how one can use this to compute the “local contributions away from p” for Quadratic Chabauty, which are crucial for computations.

An effective approach to the Diophantine problem of enumerating all points on curves with non-abelian fundamental groups, such as those of genus greater than 1, is provided (conjecturally always) by the Chabauty-Kim method. The central object in this method is a Selmer scheme associated to the initial curve of interest and generalizing the association of Selmer groups to elliptic curves. In this talk, we’ll show that arithmetic dualities produce (derived) symplectic and Lagrangian structures on associated spaces which reflect certain expectations coming from “arithmetic topology”. In addition to some Diophantine utility, this should be viewed as foundational towards a “TQFT” approach to L-functions and related invariants analogous to a parallel story producing knot invariants from structures on character varieties which will be elaborated upon.

Topological recursion is a remarkable universal recursive procedure that has been found in many enumerative geometry problems, from combinatorics of maps, to random matrices, Gromov-Witten invariants, Hurwitz numbers, Mirzakhani’s hyperbolic volumes of moduli spaces, knot polynomials. A recursion needs an initial data: a spectral curve, and the recursion defines the sequence of invariants of that spectral curve. There is a duality in topological recursion which allows one to obtain closed formulas for the invariants of the recursion and which has implications in free probability theory and integrable hierarchies. In the talk I will survey recent progress in the topic with the examples from Hurwitz numbers theory, Hodge integrals and combinatorics of maps.

The talk is based on the joint works with A. Alexandrov, P. Dunin-Barkowski, M. Kazarian and S. Shadrin.

The Gelfand-Kazhdan formal geometry is a way of describing geometric structures on a smooth manifold M in terms of the jet bundle. The works of Fedosov, Nest-Tsygan and Bezrukavnikov-Kaledin put the problem of classifying deformation quantizations of, respectively, smooth, holomorphic and algebraic symplectic manifolds into the context of formal geometry. They showed that, if the Hodge filtration on the cohomology of the symplectic manifold splits, the set of deformation quantizations of M could be identified with a certain subset of $H^2(M)[[h]]$ via the so-called period map. In the talk I want to describe an upgrade of the period map from a map between sets to a morphism between suitably defined deformation functors. This upgrade could be used to reprove the Fedosov-Nest-Tsygan-Bezrukavnikov-Kaledin theorems, to help classify quantizations without the Hodge filtration splitting condition, and to connect the period map with the so-called Rozansky-Witten invariants.