Weekly Bulletin

Using Clausen-Scholze's theory of condensed mathematics, we construct a full 6-functor formalism for p-adic sheaves on rigid-analytic varieties. As a special case of this formalism we obtain Poincaré duality for the étale F_p-cohomology of smooth proper rigid-analytic varieties. By applying the formalism to classifying stacks of p-adic groups, we obtain new insights into the p-adic Langlands program.

In the unit tangent bundle of a finite volume Riemannian manifold with negative curvature, a closed strong unstable leaf pushed by the geodesic flow equidistributes towards the maximal entropy measure. Fixing a family of discrete points with geometric origin (intersection with divergent orbits of the geodesic flow) on these unstable leaves, and having care of taking neither too many nor too few points (using a prescribed density), we prove that the family of points equidistributes towards a measure supported on a truncated weak stable leaf. We give arithmetic applications by varying arithmetic hyperbolic manifolds. This is a joint work with Jouni Parkkonen.

By Delzant's Theorem, symplectic toric manifolds are classified by their unimodular moment polytopes. In this talk, we will discuss the bijection between the faces of the moment polytope and the symplectic toric submanifolds of the corresponding symplectic toric manifold. To study recursive aspects like this, it is useful to generalise the notion of a symplectic toric manifold. First we present how this generalisation works in the local picture, given by symplectic toric representations and unimodular polyhedral cones. Then, we will show how this can be used to establish the global correspondence between submanifolds and faces.

In 2008, Fusco, Maggi, and Pratelli proved the sharp stability of the isoperimetric inequality, an open problem originally raised by Hall in 1992. This inequality states that for any set of finite perimeter $E\subset \mathbb R^n$ with $|E|=|B|$ and a barycenter at the origin, one has $ P(E)-P(B)\ge c(n) |E\Delta B|^2.$ Here, the power $2$ on the right-hand side is optimal. During my talk, I will review some earlier results on improving their proof and share some of my recent joint work with A. Figalli in this direction.

In this talk we will introduce quantum topology, which means using representations of quantum groups to construct topological invariants. The starting point of quantum topology is the Jones polynomial for knots and the Reshetikhin Turaev invariant for 3-manifolds. By studying these examples we can see how the concept of quantum groups naturally arises.

Cohomological information about arithmetic groups is of great interest to algebraists, geometers, and number theorists alike. In this talk we will review some important cohomological properties of (S-)arithmetic groups, looking at concrete examples and some famous results. Among these is a thorem of Lee--Szczarba, stating that the cohomology of SL(n,Z) with rational coefficients is zero in dimension n(n-1)/2. Shifting focus to arithmetic subgroups of classical semisimple groups, we shall see how to obtain --- with the aid of Coxeter groups and buildings --- a generalization of the theorem of Lee--Szczarba: the rational cohomology of such arithmetic groups vanishes in their virtual cohomological dimension. Based on joint work with B. Brück and R. Sroka.

It is well known that BG carries a 2-shifted symplectic structure. In this talk, I will study the shifted lagrangian groupoids of BG. I will show how many constructions on Poisson geometry unify using the language of shifted symplectic groupoids. This is work in progress with Daniel Alvarez and Henrique Bursztyn.

I will report on a research program to use ideas from stochastic analysis in the context of constructive quantum field theories. Stochastic analysis can be summarized as the study of measures on path space via push-forward from Gaussian measures. The basic example is the Ito map which sends Brownian motion to a Markov diffusion process solution to a stochastic differential equation. Parisi-Wu stochastic quantisation can be understood as a stochastic analysis of an Euclidean quantum field, in the above sense. In this talk I will focus on another way to introduce such an "Ito map" which has connection to the continuous renormalization group a la Polchinski and which uses a forward-backwards stochastic differential equation. In order to be able to give a full non-perturbative construction I will focus on the case of Grassmann measures seen as instances of non-commutative random fields.

The planning problem in Mean Field Games (MFG) was introduced by P.-L. Lions in his lessons, to describe models in which a central planner would like to steer a population to a predetermined final configuration while still allowing individuals to choose their own strategies. In a recent variational approach, see Graber, Mészáros, Silva and Tonon (2019) and Orrieri, Porretta and Savaré (2019) the authors studied the well-posedness of this problem in case of merely summable initial and final measures, using techniques, coming from optimal transport, introduced by Benamou and Brenier in 2000, extended to the congestion case in Carlier, Cardaliaguet and Nazaret (2013), and already used to show the existence and uniqueness of weak solutions for classical MFGs by Cardaliaguet and collaborators. The case of less regular initial and final measures is now studied via techniques introduced by Jimenez in 2008, for the analogous problem in optimal transport.

The topic of potential discrimination in statistical and machine learning (ML) is being increasingly discussed, both by the scientific community and the wider public. In the insurance industry specifically, customers and regulators demand that individuals are treated fairly. Regulations of the European Union, for example, mandate that gender is not be used as a factor in determining the prices of policies. <br> This talk is based on a method introduced in a series of papers by Lindholm et al. on "discrimination-free insurance pricing" that address specifically the issue of potential indirect discrimination [1, 2, 3]. <br> In the first part of the talk, we outline the solution by Lindholm et al., and we discuss that indirect discrimination is a topic that is subtle and can be difficult to understand for decision takers. In the second part of our talk, we present a Swiss Re internally built toolbox implementing the methodology. In the third part of our talk, we apply the methodology to model human mortality. We use public data of the German association of actuaries, which we interpret in terms of a Bayesian network describing the relation between age, gender, the smoker status and the mortality of individuals. <br> [1] M. Lindholm, R. Richman, A. Tsanakas and M. V. Wuthrich, "Discrimination-free insurance pricing," ASTIN Bulletin: The Journal of the IAA, vol. 52, pp. 55 - 89, 2022. <br> [2] M. Lindholm, R. Richman, A. Tsanakas and M. V. Wuthrich, "A Multi-Task Network Approach for Calculating Discrimination-Free Insurance Prices," 2 11 2022. [Online]. Available: https://ssrn.com/abstract=4155585. <br> [3] M. Lindholm, R. Richman, A. Tsanakas and M. V. Wuthrich, "A Discussion of Discrimination and Fairness in Insurance Pricing," 2 09 2022. [Online]. Available: https://ssrn.com/abstract=4207310.