Asymptotic spectra and Applications I

Jeroen Zuiddam
Member, School of Mathematics
October 8, 2019

The first lecture in this series is an introduction to the theory of asymptotic spectra. This theory describes asymptotic behavior of basic objects in mathematics like graphs and tensors. Example applications that we will see are the matrix multiplication problem, the cap set problem, the sunflower problem, the quantum entanglement problem, and the problem of efficient communication over a noisy channel. We will start from scratch.

Logarithmic concavity of Schur polynomials

June Huh
Visiting Professor, School of Mathematics
October 7, 2019

Schur polynomials are the characters of finite-dimensional irreducible representations of the general linear group. We will discuss both continuous and discrete concavity property of Schur polynomials. There will be one theorem and eight conjectures. No background beyond basic representation theory will be necessary to enjoy the talk. Based on joint work with Jacob Matherne, Karola Mészáros, and Avery St. Dizier.

Bourgeois contact structures: tightness, fillability and applications.

Agustin Moreno
University of Augsburg
October 7, 2019
Starting from a contact manifold and a supporting open book decomposition, an explicit construction by Bourgeois provides a contact structure in the product of the original manifold with the two-torus. In this talk, we will discuss recent results concerning rigidity and fillability properties of these contact manifolds. For instance, it turns out that Bourgeois contact structures are, in dimension 5, always tight, independent on the rigid/flexible classification of the original contact manifold.

Weak solutions of the Navier-Stokes equations may be smooth for a.e. time

Maria Colombo
École Polytechnique Fédérale de Lausanne; von Neumann Fellow, School of Mathematics
October 7, 2019

In a recent result, Buckmaster and Vicol proved non-uniqueness of weak solutions to the Navier-Stokes equations which have bounded kinetic energy and integrable vorticity. 


We discuss the existence of such solutions, which in addition are regular outside a set of times of dimension less than 1.