Weekly Bulletin

The Toda lattice prescribes the evolution of many particles interacting under certain Hamiltonian dynamics; it is an archetypal example of a completely integrable system. In this talk we describe several results providing the long-time asymptotics of this system, under random, dense initial data. The proofs proceed by finding a way to interpret the Toda lattice as a collection of "quasi-particles" that behave similarly to solitons, and providing a framework to study how these quasi-particles asymptotically evolve in time. The arguments use ideas from random matrix theory, particularly the analysis of Lyapunov exponents governing the decay rates of eigenvectors of random tridiagonal matrices.

Moduli spaces often admit several natural compactifications. The comparison of different compactifications is referred to as wall-crossing. Some compactifications are simpler than others, making wall-crossing one of the most powerful established techniques, for example, in the study of moduli spaces of sheaves. I will discuss wall-crossing phenomena for moduli spaces of maps (Gromov–Witten theory) and other related spaces. The first glimpses of wall-crossing in Gromov–Witten theory go back to its beginnings, namely to Givental’s proof of mirror symmetry for the quintic in 1990s. Later, independently of Givental’s work, Manin and Bayer were the first to explore the idea of wall-crossing in the setting of maps, extending Hassett’s ideas on weighted marked curves. Only with the work of Ciocan-Fontanine, Kim, and Maulik from 2010s, which in turn builds on the work of many others, did it become possible to exploit different compactifications of moduli spaces of maps with considerable flexibility for GIT quotients, thereby revealing Givental’s proof as a wall-crossing calculation. Zhou then made the next major step by proving the Ciocan-Fontanine–Kim wall-crossing formulas for GIT quotients in full generality. The goal is to explain how the ideas of Ciocan-Fontanine, Kim, and Zhou can be applied in three different settings: configuration spaces of points, maps to GIT quotients, and Hurwitz theory. Based on these examples, I will try to formulate some general principles underlying such wall-crossing phenomena. The final goal is to combine all three settings to provide a wall-crossing between spaces of maps and Hilbert schemes associated to threefolds of the form Surface × Curve.

TBA

<p>Abstract: The aim of the talk is to present a sub-Riemannian analogue of the celebrated Zador theorem, a cornerstone of the theory of quantization of measures. After a brief introduction to sub-Riemannian geometry and Carnot groups, I will outline the key ideas behind the proof of the Euclidean Zador theorem and its sub-Riemannian version. Talk based on a joint work with Mikaela Iacobelli.</p>