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.
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.
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.