Veranstaltungen

Abstract: Motivated by heat kernel renormalization of perturbative quantum field theory, we study the cutting and gluing behaviour of the heat kernel on a Riemannian manifold $M$ which is cut along a compact hypersurface $\gamma$ into two Riemannian manifolds $M_1$, $M_2$. Under a certain assumption on $(M,\gamma)$ (which we conjecture to be true for all Riemannian manifolds), we prove a gluing formula for the heat kernel which involves only the heat kernels on $M_1, M_2$ and $\gamma$. This is a joint work with Pavel Mnev, arxiv:2404.00156.

Floer homology in its Monopole, Heegard and ECH version’s has proved a powerful tool for the study of the geometry and topology of three manifolds. Their older cousin instanton Floer homology also has its share of trophies but remains more mysterious and harder to compute. This talk will survey a few highlights of applications of the theory and problems that remain to be solved and prospects for the future.

Moduli spaces of stable maps and admissible covers of a curve give rise to interesting cycles on moduli spaces of stable curves. The talk will be about a wall-crossing formula relating these two types of cycles. Based on a joint work with Max Schimpf. 

manifolds, both from a complex analytic and algebraic point of view. This kind of manifolds plays a central role in classification problems in complex algebraic geometry, being fundamental building blocks of Ricci-flat manifolds, and their theory has become a very popular research topic in the last years. Together with their definition(s) and their main properties, we will see the four known families of deformation classes of this kind of manifolds and we will approach the problem of their bimeromorphic classification.

I will show how the study of non-tautological classes on the moduli space of abelian varieties helps explain the structure of the tautological ring of the moduli space of curves of compact type. On the curves side, this is joint work with Hannah Larson and Johannes Schmitt, and on the abelian varieties side, it is joint with Dragos Oprea and Rahul Pandharipande.

The Square Peg Problem (SPP), formulated by Toeplitz in 1911 and still unsolved, asks whether every Jordan curve contains four points at the vertices of a square. We shall discuss how, when the Jordan curve is smooth, SPP and related peg problems can be interpreted as questions in symplectic geometry, and deduce some consequences both for smooth curves and for other classes. Joint work with Josh Greene.

We study a new notion of trace and extension operators for abstract Hilbert complexes.

In this talk we will look at some recent developments about integer partitions, including coloured partitions, semi-prime partitions and r-prime partitions. We will give asymptotic formulae for the number of these partitions. We will also look at some interesting elements of the proof of these formulae, without going into too much technical detail. In particular, we have a look at strange and pseudo-differentiable functions.

In this talk, we study dependence uncertainty and the resulting effects on tail risk measures, which play a fundamental role in modern risk management. We introduce the notion of a regular dependence measure, defined on multi-marginal couplings, as a generalization of well-known correlation statistics such as the Pearson correlation. The first main result states that even an arbitrarily small positive dependence between losses can result in perfectly correlated tails beyond a certain threshold and seemingly complete independence before this threshold. In a second step, we focus on the aggregation of individual risks with known marginal distributions by means of arbitrary nondecreasing left-continuous aggregation functions. In this context, we show that under an arbitrarily small positive dependence, the tail risk of the aggregate loss might coincide with the one of perfectly correlated losses. A similar result is derived for expectiles under mild conditions. In a last step, we discuss our results in the context of credit risk, analyzing the potential effects on the value at risk for weighted sums of Bernoulli distributed losses. The talk is based on joint work with Corrado De Vecchi and Jan Streicher.