Vorträge in der Woche 04.07.2022 bis 10.07.2022

Montag, 04.07.2022: Structure-preserving Reduced Order Modelling and Scientific Machine Learning

Dr. Michael Kraus (MPI für Plasmaphysik, Garching)

In many applications, such as optimisation, uncertainty quantification and inverse problems, it is required to perform repeated simulations of high-dimensional physical systems for different choices of parameters. In order to save computational cost, surrogate models can be constructed by expressing the solution in a low-dimensional basis, obtained from training data. This is referred to as model reduction. Past investigations have shown that, when performing model reduction of Lagrangian or Hamiltonian systems, it is crucial to preserve the symplectic structure associated with the system in order to ensure long-term numerical stability and restrict error growth. In this talk, we will review structure-preserving reduced basis methods and discuss novel approaches to structure-preserving hyper-reduction, i.e. the treatment of nonlinear operators, and apply these techniques to construct reduced order models of the Vlasov-Poisson system. Further, we propose a new neural network architecture in the spirit of autoencoders that provide more general reduced solution spaces. While traditional reduced basis methods such as proper orthogonal decomposition provide linear solution spaces, autoencoders provide solution manifolds, requiring a lower number of degrees of freedom to yield the same expressivity.

Montag, 04.07.2022: High-dimensional stability of groups

Prof. Dr. Roman Sauer (KIT)

ABSTRACT: In 1967 in one of the most impactful 3-page mathematical papers, David Kazhdan defined a property of groups, nowadays called (Kazhdan) property T. Property T for a group G is defined in terms of the unitary representations of G. It can be alternatively characterized by the first cohomology of G. We study a higher-dimensional variant of property T and prove it for a large class of arithmetic groups. We discuss consequences for the cohomology of arithmetic groups and the approximation and stability properties. The methods we use are not so much of arithmetic origin but come from functional analysis and geometric group theory. This talk is based on joint work with Uri Bader and with Uri Bader and Alex Lubotzky. Ein Vorkolloquium unter dem Titel Introduction to Kazhdan T ist geplant. Moderator: Claudius Kamp

Mittwoch, 06.07.2022: tba

Johannes Rau ( Universidad de los Andes, Bogotá, Colombia )

tba

Mittwoch, 06.07.2022: Real algebraic varieties close to smooth tropical limits

Johannes Rau, Universidad de los Andes, Bogota, Columbia

This talk is based on joint work with Kris Shaw and Arthur Renaudineau. I will present a combinatorial setup, based on smooth tropical varieties and real phase structures, which after "unfolding" produces a certain class of PL-manifolds. We have two motivations in mind: Firstly, in generalisation of Viro's combinatorial patchwoking to arbitrary codimension, the arising PL-geometries can be used to describe the topology of real algebraic varieties close to the tropical limit. Secondly, even if not "realisable" by real algebraic varieties, real phase structures provide a geometric framework for combinatorial structures such as oriented matroids. Hybrid participation is possible, please send an email to Hannah Markwig to request the zoom link.

Donnerstag, 07.07.2022: TBA

Dr. Thomas Körber (Universität Wien)

Donnerstag, 07.07.2022: Effective Dynamics of Interacting Fermions: Hartree-Fock Theory and Beyond

Niels Benedikter (Milan)

Almost 100 years after the publication of the Schrödinger equation, the quantum many-body problem still remains a mathematical challenge. The derivation of effective evolution equations provides a tool for an approximate solution, deriving physical predictions in certain scaling limits. I will discuss the coupled mean-field and semiclassical scaling limit for high-density fermionic systems, and sketch the derivation of the time-dependent Hartree-Fock equation in this limit. I will also present more recent results that can be seen as a next-order correction to Hartree-Fock theory, based on bosonization.