Weekly Bulletin

I will talk about the proof of the Hecke orbit conjecture for Shimura varieties of Hodge type. This is a conjecture proposed by Chai and Oort on the geometry of the reduction modulo p of Shimura varieties. After recalling the statement, I will explain how to linearise the problem using some "generalised Serre-Tate coordinates" on central leaves. Subsequently, I will explain how the monodromy groups of F-isocrystals enter into the picture. I will end the talk with some ideas on how to prove a certain local monodromy theorem, which is a crucial step in the proof of the conjecture. This is a joint work with Pol van Hoften.

Origamis are special translation surfaces which are obtained as patchwork from finitely many Euclidean squares. An important algebraic invariant is their Veech group which is a finite index subgroup of \(SL(2,\mathbb{Z})\). It is unknown which subgroups of \(SL(2,\mathbb{Z})\) occur as Veech groups. However for congruence groups of prime level there is a constructive approach which shows that all but five special cases occur. We find four of the missing five cases and study the question whether the result can be generalized to general congruence groups. The question is intimately related to the study of orbit spaces of \(SL(2,\mathbb{Z}/n\mathbb{Z})\).

Recent years have seen the appearance of a plethora of symplectically-meaningful metrics on collections of Lagrangian submanifolds. Indeed, on top of the better-known Lagrangian Hofer metric and the spectral metric, Biran, Cornea, and Shelukhin have for example introduced the so-called shadow metrics on these spaces. These new metrics are particularly interesting, as they allow to compare Lagrangian submanifolds which may not even have the same homotopy type. It is however unclear what it means for two Lagrangian submanifolds to be nearby in these metrics. I will explain how—under the presence of bounds coming from Riemannian geometry—being close can be understood in terms of the classical Hausdorff metric, giving thus a much better understanding of the situation.

We will introduce the space of Vector Fields with Integer Valued Fluxes (VFIVF) and discuss some of its analytical properties. In particular we will address the weak and strong closure of this space in L^p and the relationship of VFIVFs with integer rectifiable 1-currents. We will also see how VFIVFs arise naturally in some problems in the Calculus of Variations. This is based on a joint work with Riccardo Caniato.

The main example of a polynomial structure on the unit sphere S^n is a non-constant polynomial map to another sphere S^r. We will discuss the possible values of (n, r) and the related constructions of polynomial structures. As an application, we will exhibit a link with two other apriori unrelated problems: 1) in spectral theory: ``Can one hear the shape of a drum?" for the Bochner (connection Laplacian) and 2) in dynamical systems: ergodicity of certain (partially hyperbolic) systems arising as flows of frames. This is joint work with Thibault Lefeuvre.

We will discuss a complete computation of the geometric Tevelev degrees of projective spaces, in terms of Schubert calculus. That is, we enumerate maps from a general pointed curve to a projective space of any dimension passing through the appropriate number of general points on the target. Previously, the answers were known for covers of the projective line, or for maps of sufficiently large degree (in which case the answers agree with virtual counts in Gromov-Witten theory). The proof employs the moduli space of complete collineations in an essential way, and also goes through a result of Klyachko on torus orbit closures in the Grassmannian.

I will talk about a result with Tom Hutchcroft, stating that the number of ends of the uniform spanning tree (UST) is almost surely equal to the number of ends of the underlying graph in the context of recurrent stationary random rooted graphs. Together with previous results in the transient case, this completely resolves the problem of the number of ends of wired uniform spanning forest components in stationary random rooted graphs and confirms a conjecture of Aldous and Lyons (2006). Our work elaborates on work with Nathanael Berestycki, in which we relate the end-structure of the UST on recurrent graphs to potential theoretic properties of the underlying graphs.

In joint work with Christophe Reutenauer (UQAM) we explicitly computed the zeta function of the Hilbert scheme of <I>n</I> points on the two-dimensional torus, which amounts to the same as computing the number of ideals of codimension <I>n</I> of the algebra of two-variable Laurent polynomials over a finite field. On the way we found a family <I>P_n(q)</I> of polynomials with nice properties: they are palindromic, their coefficients are non-negative integers and their values at 1 and at roots of unity of order 2, 3, 4 and 6 can be expressed in terms of modular forms related to Dedekind's eta function.

Causal models can provide good predictions even under distributional shifts. This observation has led to the development of various methods that use causal learning to improve the generalization performance of predictive models. In this talk, we consider this type of approach for instrumental variable (IV) models. IV allows us to identify a causal function between covariates X and a response Y, even in the presence of unobserved confounding. In many practical prediction settings the causal function is however not fully identifiable. We consider two approaches for dealing with this under-identified setting: (1) By adding a sparsity constraint and (2) by introducing the invariant most predictive (IMP) model, which deals with the under-identifiability by selecting the most predictive model among all feasible IV solutions. Furthermore, we analyze to which types of distributional shifts these models generalize.

A salient feature of topological insulators and symmetry protected topological phases of quantum lattice systems is the spectral gap above the ground state. I will review recent results of two types. The first are proofs of lower bounds for the spectral gap for a variety of specific models. The second type of results is concerned with the stability of the spectral gap and other fundamental characteristics of the system under uniformly small perturbations of the interactions in the model Hamiltonian. I will discuss the basic strategies used to approach both types of problems.

In this talk we investigate the use of Restricted Boltzmann Machines (RBMs) in credit risk management. RBMs are stochastic neural networks capable of universal approximation of loss distributions. We test this model on an empirical dataset of default probabilities of 30 investment-grade US companies and show that it outperforms commonly used parametric factor copula models across several credit risk management tasks. In particular, the model leads to better out-of-sample fits for the empirical loss distribution and more accurate risk measure estimations. We introduce an importance sampling procedure which allows risk measures to be estimated at high confidence levels in a computationally efficient way. Furthermore, we show that the statistical factors extracted by the model admit an interpretation in terms of the underlying portfolio sector structure and provide practitioners with quantitative tools for the management of concentration risk. Finally, we showcase the usefulness of the model for stress testing by estimating stressed risk measures (e.g. stressed VaR) for our empirical portfolio under various macroeconomic stress test scenarios, such as those specified by the FRB's Dodd-Frank Act stress test.

Let A_g denote the moduli stack of principally polarized abelian varieties of dimension g. The arithmetic volume of the compactification of A_g is defined to be the arithmetic degree of the metrized Hodge bundle. In 1999, Kühn proved a beautiful formula for the arithmetic volume of the compactification of A_1 in terms of special values of the Riemann zeta function. In this talk, we report on joint work with Barbara Jung generalizing Kühn's result to the case g=2.

In this talk, Gaudenz Koeppel, Chief Analytics Officer at Axpo Trading & Sales, will talk us through their journey of building machine learning models for power trading applications and taking them into 24/7 operation. Gaudenz will expand on some of the learnings, the importance of explainers as well as how and what aspects of such models must be monitored and how this monitoring information creates new insights. This will be a very practical, hands-on talk.

Hilbert schemes of points on a smooth projective curve are simply symmetric powers of the curve itself; they are smooth and we know essentially everything about them. We propose a variation by studying double nested Hilbert schemes of points, which parametrize flags of 0-dimensional subschemes satisfying certain nesting conditions dictated by Young diagrams. These moduli spaces are almost never smooth but admit a virtual structure à la Behrend-Fantechi. We explain how this virtual structure plays a key role in (re)proving the correspondence between Gromov-Witten invariants and stable pair invariants for local curves, and say something on their K-theoretic refinement.

Ramsey theory asks questions of the form 'can we find some order in enough disorder'. In the geometric setting, a very attractive theory emerges, called 'Euclidean' Ramsey theory. We will survey this area, giving classical background as well as some recent developments.