## März 2018

Recently several conjectures were resolved by using a classical method for constructing vectors in irreducible representations of coordinate rings of group varieties. We explain the method and several applications, for example a proof of Weintraub’s 1990 conjecture on plethysm coefficients (with Bürgisser and Christandl) and a disproof of Mulmuley and Sohoni’s 2008 occurrence obstruction conjecture in geometric complexity theory (with Bürgisser and Panova). Moreover, we explain how we were able to shed more light on several other conjectures by running the method on cluster computers.

Generalized exponents are important graded multiplicities in representation theory of simple Lie algebras. Notably, they are particular Kazhdan-Lusztig polynomials. In type A, they admit a nice combinatorial description in terms of Lascoux-Schützenberger’s charge statistics on semistandard tableaux. In this talk I will recall their definition and explain how to get similar statistics beyond type A. This will give a combinatorial proof of the positivity of their coefficients but also some other interesting properties. This is a work in collaboration with Cristian Lenart.

Vortrag in der Reihe: Oberseminar ANALYSIS – PROBABILITY

I will present some recent results obtained in collaboration with V. Banica and F. de la Hoz on the evolution of vortex filaments according to the so called Localized Induction Approximation (LIA). This approximation is given by a non-linear geometric partial differential equation, that is known under the name of the Vortex Filament Equation (VFE). The aim of the talk is threefold. First, I will recall the Talbot effect of linear optics. Secondly, I will give some explicit solutions of VFE where this Talbot effect is also present. Finally, I will consider some questions concerning the transfer of energy and momentum for these explicit solutions.

Vortrag in der Reihe: Oberseminar ANALYSIS – PROBABILITY

Vortrag in der Reihe: Arbeitsgemeinschaft ANGEWANDTE ANALYSIS

In this talk, I plan to study convergence rates in $L^2$ norm for elliptic homogenization problems in Lipschitz domains. It involves some new weighted-type inequalities for the smoothing operator at scale $varepsilon$, as well as, layer and co-layer type estimates, and the related details will be touched. In order to obtain a sharp result, a duality argument will be imposed. Here we do not require any smoothness assumption on the coefficients, and the main ideas may be extended to other models, such as Stokes systems and parabolic systems, arising in the periodic homogenization theory.

Vortrag in der Reihe: Arbeitsgemeinschaft ANGEWANDTE ANALYSIS

The Stochastic Reconfiguration algorithm has been recently proposed to efficiently train Neural-Network Quantum States, i.e., Restricted Boltzmann Machines (RBMs) with complex parameters built to simulate the ground state of a quantum many-body problem. The SR algorithm is not only a convenient algorithm for the training these RBMs, but it is also theoretically justified once a non-Euclidean manifold structure, based on Quantum Information Geometry, is defined over the search space associated to a RBM. I will show that the gradient descent optimization algorithm used for the many-body problem is the quantum generalization of the Riemannian natural gradient introduced by Amari. Moreover, I will compare the geometry of the Neural-Network Quantum States with the one of the Quantum Boltzmann Machine.

Vortrag in der Reihe: Arbeitsgemeinschaft ANGEWANDTE ANALYSIS