Free convolution is a fundamental operation in free probability. It expresses the distribution of the sum of two freely independent random variables in terms of the distributions of the summands. Compared to classical convolution of probability measures, free convolution is considerably more difficult to analyze and calculate. To untangle this complicated operation, we introduce a precise functional comparison between free and classical convolutions. This comparison states that the expectation of f w.r.t. classical convolution is larger than the expectation w.r.t. free convolution as long as f has non-negative fourth derivative. The comparison is based on the existence of convolution comparison measures, novel measures on the plane whose positivity depends on a peculiar identity involving Hermitian matrices.

Classical Ehrhart theory studies lattice point enumeration in integer dilates of lattice polytopes. We discuss a new and conjecture-laden q-analog of Ehrhart theory involving the orbit harmonics deformation of algebraic combinatorics. A new and somewhat subtle `harmonic algebra' attached to a lattice polytope plays a key role. Joint with Vic Reiner.

We discuss recent developments around Hilbert's sixth problem about the passage from kinetic models to macroscopic fluid equations. We employ the technique of slow spectral closure to rigorously establish the existence of hydrodynamic manifolds and derive new non-local fluid equations for rarefied flows independent of Knudsen number. We show the singularity of certain scaled solutions, including the divergence of the Chapman--Enskog series for an explicit example, and apply neural nets to learn the optimal hydrodynamic closure from data. The new dynamically optimal constitutive laws are applied to a rarefied flow problem and we discuss the classical problem of the number of macroscopic rarefied fluid fields from a data-driven point of view.

Bio: Florian Kogelbauer is a Senior Research Fellow at ETH Zürich’s Department of Mathematics, affiliated with RiskLab and the Finsure Tech Hub. His research centres on nonlinear dynamical systems, kinetic theory, and fluid dynamics, with recent work on hydrodynamic closures and spectral theory for kinetic equations. He previously held academic and research roles at the University of Vienna and AIST-Tohoku University in Japan, alongside consulting positions at KPMG Austria.

In 1966, Mark Kac asked the famous question “Can one hear the shape of a drum?”
In his article with this question as the title, he translated it into eigenvalue problems for planar domains.
This question highlighted the relationship between eigenvalues and geometry.
One can then ask how eigenvalues are related to the geometry of the boundary.
In this talk, we consider a special type of eigenvalues, called Steklov eigenvalues, that are closely tied to boundary geometry.
We will introduce Steklov eigenvalues and explain their basic background and applications.
Then we will discuss our recent results on inequalities relating Steklov eigenvalues to the boundary area of compact manifolds.

This dissertation consists of four papers that develop computational and structural results in equivariant stable homotopy theory. The results include the computation of the reduced ring of the $RO(C_2)$-graded $C_2$-equivariant stable stems, the construction of the first family of $C_{p^n}$-equivariant ``$v_1$''-self maps, the computation of the $C_{p^n}$-equivariant Mahowald invariants of all elements in the Burnside ring, extending the classical computations of Bredon--Landweber and Iriye, and the computation of the spoke-graded $C_3$-equivariant stable stems.