Chiral fermion anomalies in any spacetime dimension are computed by evaluating an eta-invariant on a closed manifold in one higher dimension. The APS index theorem then implies that both local and global gauge anomalies are detected by bordism invariants, each being classified by certain abelian groups that I will identify. Mathematically, these groups are connected via a short exact sequence that splits non-canonically. This enables one to relate global anomalies in one gauge theory to local anomalies in another, by which we revive (from the bordism perspective) an old idea of Elitzur and Nair for deriving global anomalies. As an example, I will show how the SU(2) anomaly in 4d can be derived from a local anomaly by embedding SU(2) in U(2).

We explain how group analogues of Slodowy slices arise by interpreting certain Weyl group elements as braids. Such slices originate from classical work by Steinberg on regular conjugacy classes, and different generalisations recently appeared in work by Sevostyanov on quantum group analogues of W-algebras and in work by He-Lusztig on Deligne-Lusztig varieties.

Our perspective furnishes a common generalisation, essentially solving the problem. We also give a geometric criterion for Weyl group elements to yield strictly transverse slices.

We will discuss several versions of Ito’s formula in the case where the function is path dependent and only concave or C1 in the sense of Dupire. In particular, we will show that it can be used to solve (super) hedging problems, in the context of market impact or under volatility uncertainty. 

I will discuss regularity properties for solutions of linear second order non-uniformly elliptic equations in divergence form. Assuming certain integrability conditions on the coefficient field, we obtain local boundedness and validity of Harnack inequality. The assumed integrability assumptions are sharp and improve upon classical results due to Trudinger from the 1970s.

As an application of the local boundedness result, we deduce a quenched invariance principle for random walks among random degenerate conductances. If time permits I will discuss further regularity results for nonlinear non-uniformly elliptic variational problems.

We consider the homogenization of a Stokes system in a domain having many small random holes. This model mainly arises from problems of solid-fluid interaction (e.g. the flow of a viscous and incompressible fluid through a porous medium). We aim at the rigorous derivation of the homogenization limit both in the Brinkmann regime and in the one of Darcy’s law. In particular, we focus on holes that are distributed according to probability measures that allow for overlapping and clustering phenomena.

We explore one facet of an old problem: the approximation of hyperbolic conservation laws by viscous counterparts. While qualitative convergence results are well-known, quantitative rates for the inviscid limit are less common. In this talk, we consider the simplest case: a one-dimensional scalar strictly-convex conservation law started from "generic" smooth initial data. Using a matched asymptotic expansion, we quantitatively control the inviscid limit up to the time of first shock. We conclude that the inviscid limit has a universal character near the first shock. This is joint work with Sanchit Chaturvedi.

Every finite group $G$ has a normal series each of whose factors is either a solvable group or a direct product of non-abelian simple groups. The minimum number of nonsolvable factors, attained on all possible such series in G, is called the nonsolvable length $\lambda(G)$ of $G$. In recent years several authors have investigated this invariant and its relation to other relevant parameters. E.g. it has been conjectured by Khukhro and Shumyatsky (as a particular case of a more general conjecture about non-$p$-solvable length) and Larsen that, if $\nu(G)$ is the length of the shortest law holding in the finite group G, the nonsolvable length of G can be bounded above by some function of $\nu(G)$. In a joint work with Francesco Fumagalli and Felix Leinen we have confirmed this conjecture proving that the inequality $\lambda(G) < \nu(G)$ holds in every finite group $G$. This result is obtained as a consequence of a result about permutation representations of finite groups of fixed nonsolvable length. In this talk I will outline the main ideas behind the proof of our result.

Recent developments on the black hole information paradox have shown that Euclidean wormholes — so called “replica wormholes’’  — can dominate the von Neumann entropy as computed by a gravitational path integral, and that inclusion of these wormholes results in a unitary Page curve. This development raises some puzzles from the perspective of factorization, and has raised questions regarding what the gravitational path integral is computing. In this talk, I will focus on understanding the relationship between the gravitational path integral and the partition function via the gravitational free energy (more generally the generating functional). A proper computation of the free energy requires a replica trick distinct from the usual one used to compute the entropy. I will show that in JT gravity there is a regime where the free energy computed without replica wormholes is pathological. Interestingly, the inclusion of replica wormholes is not quite sufficient to resolve the pathology: an alternative analytic continuation is required. I will discuss the implications of this for various interpretations of the gravitational path integral (e.g. as computing an ensemble average) and also mention some parallels with spin glasses.