Various notions of metric approximation for countable groups have been introduced and studied in the last decade, with sofic and hyperlinear approximations being two notable examples among them. The class of linear sofic groups was introduced by Glebsky and Rivera and was subsequently studied by Arzhantseva and Paunescu. This mode of approximation uses the general linear group over, say, the field of complex numbers as model groups, equipped with the distance defined using the normalized rank. Among their other interesting results, Arzhantseva and Paunescu prove that every linear sofic group is 1/4-linear sofic, where the constant 1/4 quantifies how well non-identity elements can be separated from the identity matrix. In this talk, which is based on joint work with Maryam Mohammadi Yekta, we will address the question of optimality of the constant 1/4 and report on some progress in this direction.

For a function $m$ on the real line, its Fourier multiplier $T_m$ is the operator which acts on a function $f$ by first multiplying the Fourier transform of $f$ by $m$, and then taking the inverse Fourier transform of the product. These are well-studied objects in classical harmonic analysis. Of particular interest is when the Fourier multiplier defines a bounded operator on $L_p$. Fourier multipliers can be generalized to arbitrary locally compact groups. If the group is non-abelian, the $L_p$ spaces involved are now the non-commutative $L_p$ spaces associated with the group von Neumann algebra. Fourier multipliers also have a natural extension to the multilinear setting. However, their behaviour can differ markedly from the linear case, and there is much that is unknown even about multilinear Fourier multipliers on the reals.

One question of interest is this: If $m$ is a function on a group $G$ which defines a bounded $L_p$ multiplier, is the restriction of m to a subgroup $H$ also the symbol of a bounded $L_p$ multiplier on $H$? De Leeuw proved that the answer is yes, when $G$ is $\mathbb{R}^n$. This was extended to the commutative case by Saeki and to the non-commutative case (provided the group $G$ is sufficiently nice) by Caspers, Parcet, Perrin and Ricard. In this talk, I will show how to extend these De Leeuw type theorems to multilinear Fourier multipliers on non-commutative groups. This is part of joint work with Martijn Caspers, Bas Janssens and Lukas Miaskiwskyi.

In this talk, I will discuss radial and one-dimensional symmetry properties for the stationary incompressible Euler equations in dimension 2 and some related semilinear elliptic equations. I will show that a steady flow of an ideal incompressible fluid with no stagnation point and tangential boundary conditions in an annulus is a circular flow. The same conclusion holds in complements of disks as well as in punctured disks and in the punctured plane, with some suitable conditions at infinity or at the origin. If possible, I will also discuss the case of parallel flows in two-dimensional strips, in the half-plane and in the whole plane. The proofs are based on the study of the geometric properties of the streamlines of the flow and on radial and one-dimensional symmetry results for the solutions of some elliptic equations satisfied by the stream function. The talk is based on joint works with N. Nadirashvili.

Motivated by the study of greedy algorithms for graph coloring, we introduce a new graph parameter, which we call weak degeneracy. This notion formalizes a particularly simple way of ``saving" colors while coloring a graph greedily. It turns out that many upper bounds on chromatic numbers follow from corresponding bounds on weak degeneracy. In this talk I will survey some of these bounds as well as state a number of open problems. This is joint work with Eugene Lee.