Termine

In the talk we will outline the recently obtained proof of GLC for D-modules. This is a collaborative project with D. Arinkin, D. Beraldo, L. Chen, J. Faergeman, K. Lin, S. Raskin and N. Rozenblyum.

TBA

In the complex plane, there is a correspondence between solutions of the Monge?Ampère equation and solutions of a certain differential inclusion associated to SO(2). Under this correspondence, the W^{2,1+epsilon} regularity of solutions to Monge?Ampère is rephrased as a quantitative unique continuation principle for solutions of the differential inclusion. We will sketch a proof of the latter, which relies on quasiconformal maps and the rigidity estimate for SO(2). Based on joint work with G. De Philippis and R. Tione

One-dimensional Calogero-Moser-Sutherland particle models with N particles can be regarded as diffusions on suitable subsets of $\mathbb R^N$ like Weyl chambers and alcoves with second order differential operators as generators which are singular on the boundaries of the state spaces. The most relevant examples are multivariate Bessel processes and Heckman-Opdam processes which are related to special functions associated with root systems. These models include Dyson's Brownian motions and multivariate Jacobi processes and, for fixed times, $\beta$-Hermite, Laguerre, and Jacobi ensembles. The processes depend on parameters which have the interpretation of an inverse temperature. We review several freezing limits for fixed N when one or several parameters tend to $\infty$. Usually, the limits are normal distributions and, in the process case, Gaussian processes where the parameters of the limit distributions are described in terms of solutions of ordinary differential equations which appear as frozen versions of the particle diffusions. We also discuss connections of these ODEs with the zeros of the classical orthogonal polynomials and polynomial solutions of some associated one-dimensional inverse heat equations.

While attention for and usage of Deep Neural Network (DNN) based applications skyrocket, the mathematical understanding of their behavior and capabilities is still in its infancy. Contrary to traditional approaches, that depend on training by loss minimization algorithms, a method will be presented to explicitly construct DNNs that emulate multivariate Chebyshev polynomials and can be used to approximate a large class of functions. The theory of this method, it's accuracy and it's bounds on depth and size will be introduced as well as an implementation and comparison to training-based DNNs.

Infinitesimal Einstein deformations are solutions of the linearised Einstein equation. They can be considered as potential tangent vectors to curves of Einstein metrics. An important question is to decide for a given infinitesimal Einstein deformations whether it is integrable, i.e. indeed tangent to such a curve. In 1981 Koiso introduced an obstruction against integrability of infinitesimal Einstein deformations. However, so far the obstruction was computed only in very few cases. In my talk I will present a new formulation of Koiso's obstruction which makes it more accessible to calculations, in particular on Kähler manifolds. I will demonstrate this for the symmetric metric on the complex 2-plane Grassmannians. Here it turns out that in half of the cases all infinitesimal Einstein deformations are obstructed, i.e. the metric is isolated in the space of Einstein metrics. My talk is based on joint work with Paul-Andi Nagy