Measure equivalence was introduced by Gromov as a measure-theoretic analogue to quasi-isometry between finitely generated groups. In this talk I will present measure equivalence classification results for right-angled Artin groups, and more generally graph products. This is based on joint works with Jingyin Huang and with Amandine Escalier. 

Magic angles refer to a remarkable theoretical (Bistritzer--MacDonald, 2011) and experimental (Jarillo-Herrero et al 2018) discovery, that two sheets of graphene twisted by a certain (magic) angle display unusual electronic properties such as superconductivity.

Mathematically, this is related to having flat bands of nontrivial topology for the corresponding periodic Hamiltonian and their existence be shown for the chiral model of twisted bilayer graphene (Tarnopolsky-Kruchkov-Vishwanath, 2019). A spectral characterization of magic angles (Becker--Embree--Wittsten--Z, 2021, Galkowski--Z, 2023) also produces complex values and the distribution of their reciprocals looks remarkably like a distribution of scattering resonances for a two dimensional problem, with the real magic angles corresponding to anti-bound states. I will review various results on that distribution as well as on the properties of the associated eigenstates.

The talk is based on joint works with S Becker, M Embree, J Galkowski, M Hitrik, T Humbert and J Wittsten

The handlebody group is defined to be the mapping class group of a handelbody (rel. boundary). It is a subgroup of the mapping class group of the surface of the handlebody, and maps onto the outer automorphism group of its fundamental group (the free group of rank equal to its genus). 

Recently Hainaut and Petersen described a subspace of moduli space forming an orbifold classifying space for the handlebody group, and combined this with work of Chan-Galatius-Payne to construct cohomology classes in the group. I will talk about how one can build on their ideas to define a cocompact EG for the handlebody group inside Teichmüller space. This is a manifold with boundary and comes with a filtration by labelled disk systems which we call the `RGB (red-green-blue) disk complex.' I will describe this filtration, use it to describe the boundary of the manifold, and speculate about potential applications to duality results. Based on work-in-progress with Dan Petersen.

A structure is omega-categorical if its theory has a unique countable model (up to isomorphism). We will survey some old results concerning the Apps-Wilson structure theory for omega-categorical groups and state a conjecture of Wilson from the 80s on omega-categorical characteristically simple groups. We will also discuss the analogous of Wilson’s conjecture for Lie algebras and present some connections with the restricted Burnside problem.

I am going to discuss main results of my paper "Physics over a finite field and Wick rotation", arxiv 2306.15698. It introduces a structure over a pseudo-finite field which might be of interest in Foundations of Physics. The main theorem establishes an analogue of the polar co-ordinate system in the pseudo-finite field. A stability classification status of the structure is an open question.

Let $S$ be a smooth projective surface with $p_g>0$ and $H^1(S,{\mathbb Z})=0$. 
We consider the moduli spaces $M=M_S^H(r,c_1,c_2)$ of $H$-semistable sheaves on $S$ of rank $r$ and 
with Chern classes $c_1,c_2$. Associated a suitable class $v$ the Grothendieck group of vector bundles
on $S$ there is a deteminant line bundle $\lambda(v)\in Pic(M)$, and also a tautological sheaf $\tau(v)$ on $M$.

In this talk we derive a conjectural generating function for the virtual Verlinde numbers, i.e. the virtual holomorphic 
Euler characteristics of all determinant bundles $\lambda(v)$ on M, and for Segre invariants associated to $\tau(v)$ . 
The argument is based on conjectural blowup formulas and a virtual version of Le Potier's strange duality. 
Time permitting we also sketch a common refinement of these two conjectures, and their proof for Hilbert schemes of points.
 

The notion of biexactness for groups was introduced by Ozawa in 2004 and has since become a major tool used for studying solidity of von Neumann algebras. We introduce the notion of biexactness for von Neumann algebras, which allows us to place many previous solidity results in a more systematic context, and naturally leads to extensions of these results. We will also discuss examples of solid factors that are not biexact. This is a joint work with Jesse Peterson.