Veranstaltungen

Chern-Simons theories with gauge supergroups appear naturally in string theory and they possess interesting applications in mathematics, e.g. for the construction of knot and link invariants. In this talk we explain the combinatorial quantisation of Chern-Simons theories and also the GL(1|1) generalisation of it, for punctured Riemann surfaces of arbitrary genus. We construct the algebra of observables, and study their representations and applications to the construction of 3-manifold invariants. This is based on the joint work https://arxiv.org/abs/1811.09123 and part 2 in progress with A. Gainutdinov, M. Pawelkiewicz, V Schomerus.

It is more difficult to find non-tautological algebraic cycles on moduli spaces of smooth curves than on moduli spaces of stable curves. In fact, there were only 11 known pairs (g,n) for which M_{g,n} was known to have a non-tautological algebraic cycle, due to Graber--Pandharipande and van Zelm. I will explain how to produce non-tautological algebraic cycles in infinitely many more cases. In particular, M_g has non-tautological algebraic cycles whenever g is at least 16. This is joint work with V. Arena, E. Clader, R. Haburcak, A. Li, S.C. Mok, and C. Tamborini.

Optimization problems with random objective functions are central in computer science, probability, and modern data science. Despite their ubiquity, finding efficient algorithms for solving these problems remains a major challenge. Interestingly, many random optimization problems share a common feature, dubbed as statistical-computational gap: while the optimal value can be pinpointed non-constructively, all known polynomial-time algorithms find strictly sub-optimal solutions. That is, an optimal solution can only be found through brute force search which is computationally expensive. In this talk, I will discuss an emerging theoretical framework for understanding the computational limits of random optimization problems, based on the Overlap Gap Property (OGP). This is an intricate geometrical property that achieves sharp algorithmic lower bounds against the best known polynomial-time algorithms for a wide range of random optimization problems. I will focus on two models to demonstrate the power of the OGP framework: (a) the symmetric binary perceptron, a simple neural network classifying/storing random patterns and a random constraint satisfaction problem, widely studied in probability, statistics, and computer science, and (b) the random number partitioning problem as well as its planted counterpart, which is closely related to the design of randomized controlled trials. In addition to yielding sharp algorithmic lower bounds, our techniques also give rise to new toolkits for the study of statistical-computational gaps in other models, including the online setting.

We consider a random bistochastic matrix of size N of the form (1-r)M + rQ where 0<r<1, where M is a uniformly distributed permutation and Q is a given bistochastic matrix. We take interest in the spectral behavior for large dimension N. Under sparsity and regularity assumptions on the *-distribution of Q (that is the normalized trace of polynomials in Q and Q*), we can prove that the second largest eigenvalue of (1-r)M + rQ is essentially bounded by an approximation of the spectral radius of a deterministic asymptotic equivalent given by free probability theory. This upper-bound has application in graph theory, for instance for the construction of graph expander.

Solvency II requires that to be solvent, insurance and reinsurance undertakings that adopt the internal model should hold their own funds able to cover losses in excess of expected ones at a given confidence level over a one-year period. This Solvency Capital Requirement (SCR) is defined as the Value-at-Risk of the Net Asset Value probability distribution at a 99.5% confidence level over a one-year period. Estimating the SCR involves nested simulations, incurring prohibitive computational costs. While machine and deep learning methods exhibit accuracy, their lack of explainability impedes adoption in the highly regulated insurance sector. This paper introduces an extension of the Least Square Monte Carlo method based on recent advances in explainable deep learning known as ‘localGLMnet’. The proposed approach allows for an accurate estimation of the SCR without compromising model explainability. It allows for deriving some interesting insights into the impact of risk factors on the value of the insurance liabilities. Numerical experiments performed on two realistic insurance portfolios validate our proposal. Additionally, we illustrate that the ElasticNet regularisation can be applied to enhance the model’s performance further.

In a series of papers, Loeffler-Zerbes and their collaborators constructed a non-zero Euler system for Siegel modular forms, providing the first evidence for the BSD conjecture for abelian surfaces of rank 0. I will present a proof of the non-triviality of the Euler system for the minimal cohomological weight by the generalized explicit reciprocity law for the critical twist, following Kato. This is a joint work with Tadashi Ochiai (Tokyo Institute of Technology).

In this talk, I extend well-known results on the generating function of Euler characteristics of Hilbert schemes of points from complex manifolds to complex supermanifolds. These are two-variable generating functions in general: since functions on a zero dimensional subscheme of a supermanifold are Z_2-graded, there is an even degree and an odd degree. For supermanifolds of a given superdimension (n|m), the generating function can be expressed in terms of the corresponding generating function for C^{n|m}, which can in turn be computed by an analogue of box-counting. These basic generating functions are worked in low superdimension and can be expressed as rational functions of two variables.