Plenary Speakers

Caucher Birkar

Tsinghua University & University of Cambridge

Geometry and Numbers

In this talk we will discuss some recent interesting connections between properties of sets of non-negative integers with different kinds of geometries. Starting with a primitive integer vector, we examine certain associated functions and relate the setting to statements in convex, toric, and birational geometries.

Hugo Duminil-Copin

IHES & University of Geneva

Critical phenomena through the lens of the Ising model

The Ising model is one of the most classical lattice models of statistical physics undergoing a phase transition. Initially imagined as a model for ferromagnetism, it revealed itself as a very rich mathematical object and a powerful theoretical tool to understand cooperative phenomena. Over one hundred years of its history, a profound understanding of its critical phase has been obtained. While integrability and mean-field behavior led to extraordinary breakthroughs in the two-dimensional and high-dimensional cases respectively, the model in three and four dimensions remained mysterious for years. In this talk, we will present recent progress in these dimensions based on a probabilistic interpretation of the Ising model relating it to percolation models.

Alan Edelman

MIT

Symmetric Spaces, Random Matrix Theory, and Numerical Linear Algebra

We really enjoy when different areas of pure and applied mathematics interact. We take a close look at the connection of symmetric spaces, random matrix theory, the Cartan decomposition, and numerical linear algebra. Among our results are 53 Matrix Factorizations most of which appear new,  a completed symmetric space based theory of classical random matrix ensembles, visualizations of generalizations of classical Lie Groups with Julia, and which Lie Groups provide structured singular value decompositions.

Joint work with Sungwoo Jeong (Cornell University)

Tom Scanlon

UC Berkeley

Differential algebra for Diophantine geometry and functional transcendence

Differential algebra has long appeared in works on Diophantine geometry over function fields and functional transcendence for special functions. Notable instances include Manin's 1963 proof of the Mordell conjecture over function fields and Ax's 1971 theorem on a functional version of Schanuel's conjecture about algebraic relations involving the exponential function. Recent generalizations of Ax's theorem by several teams of mathematicians vastly expand the reach of differential algebra. The aim of this lecture will be to trace some of this history illustrating the applications through my own work in separate projects with Pila, Eterovič, and Dupuy and Freitag.

Nina Snaith

University of Bristol

Every moment brings a treasure: random matrix theory and the Riemann zeta function

It has been over 50 years since random matrix theory and number theory met over a cup of tea at the Institute for Advanced Study at Princeton and 25 years since we learned how to model values of the Riemann zeta function using characteristic polynomials of random unitary matrices, and yet new bricks are still being added to this construction. I will give an introduction to this connection and discuss why moments of characteristic polynomials are important in number theory.

Corinna Ulcigrai

University of Zurich

Flows on surfaces: dynamics and rigidity

Flows on surfaces are one of the most studied examples of dynamical systems, starting from the work of Poincaré at the end of the 19th century. Many models of systems of physical origin are described by flows on surfaces, e.g. in celestial mechanics, polygonal billiard dynamics, or solid-state physics.

While the topological structure of trajectories has been well understood already in the last century, the ergodic theory and the fine chaotic properties of flows which preserve area have been an active area of research in the last decades. In this talk, we will survey some of the results, as well as recent breakthroughs on linearisation and rigidity questions in higher genus.