Plenary Lectures

Monday, 09:45

Euler systems and the Birch—Swinnerton-Dyer conjecture

Sarah Zerbes, ETH Zürich

L-functions are one of the central objects of study in number theory. There are many beautiful theorems and many more open conjectures linking their values to arithmetic problems. The most famous example is the conjecture of Birch and Swinnerton-Dyer, which is one of the Clay Millenium Prize Problems. I will discuss this conjecture and some related open problems, and I will describe some recent progress on these conjectures, using tools called „Euler systems“.

Monday, 11:15

Challenges for non-selfadjoint spectral problems in analysis and computation

Christiane Tretter, Universität Bern

Non-selfadjoint spectral problems appear frequently in a wide range of applications. Reliable information about their spectra is therefore crucial, yet extremely difficult to obtain. This talk focuses on tools to master these challenges such as spectral pollution or spectral invisibility. In particular, the concept of essential numerical range for unbounded linear operators is introduced and studied, including possible equivalent characterizations and perturbation results. Compared to the bounded case, new interesting phenomena arise which are illustrated by some striking examples. A key feature of the essential numerical range is that it captures, in a unified and minimal way, spectral pollution which may affect e.g. spectral approximations of PDEs by projection methods or domain truncation methods. As an application, Maxwell's equations with conductivity will be considered.

Tuesday, 11:30

Matrix Concentration and Free Probability

Afonso S. Bandeira, ETH Zürich

Matrix Concentration inequalities such as Matrix Bernstein inequality have played an important role in many areas of pure and applied mathematics. These inequalities are intimately related to the celebrated noncommutative Khintchine inequality of Lust-Piquard and Pisier. In the middle of the 2010's, Tropp improved the dimensional dependence of this inequality in certain settings by leveraging cancellations due to non-commutativity of the underlying random matrices, giving rise to the question of whether such dependency could be removed. In this talk we leverage ideas from Free Probability to fully remove the dimensional dependence in a range of instances, yielding optimal bounds in many settings of interest. As a byproduct we develop matrix concentration inequalities that capture non-commutativity (or, to be more precise, „freeness“), improving over Matrix Bernstein in a range of instances. No background knowledge of Free Probability will be assumed in the talk. Joint work with March Boedihardjo and Ramon van Handel.

Wednesday, 11:30

K3 surfaces and their avatars in conformal field theory

Katrin Wendland, Trinity College Dublin

K3 surfaces are compact, complex surfaces, which have fascinated algebraic geometers since the mid nineteenth century. Over the past few decades, their appearance in conformal field theory has triggered new interest in these surfaces. The talk showcases this perspective on K3 surfaces and provides an overview of recent developments in the field.

Wednesday, 14:00

On the Hilbert Transform

Stefanie Petermichl, Julius-Maximilians-Universität Würzburg

The Hilbert transform is a central, classical singular operator in harmonic analysis. It is a frequency filter and gives access to the boundary values of the conjugate function, thus producing orthogonal level sets. It is best understood as an integral operator via a singular kernel or as a multiplier operator on the Fourier transform side. Over twenty years ago my dyadic Hilbert transform model appeared, had many applications, but seemed to come out of nowhere. In this lecture, we consider a slightly modified dyadic model and show that it is an extremely natural extension of the concept of analyticity. As a consequence, we get linear two sided norm bounds if these operators act on functions with values where orthogonality is lost - UMD Banach spaces.

Wednesday, 15:30

Equations over groups

Andreas Thom, TU Dresden

The study of equations in the language of groups has a long history and many applications. I will explain how topological methods can be applied to solve equations over groups and also mention some recent advances in the study of identities for finite and infinite groups.

Thursday, 09:00

Packing graphs, large and small

Julia Böttcher, The London School of Economics and Political Science

Graph packing is an area in discrete mathematics that has seen swift and exciting progress in the last decade, with a number of breakthrough results and the development of an array of important new techniques. In this talk I will introduce this area and its motivation, survey some of the classic and some of the new results, and point out a number of ideas that were crucial for recent advances.

Thursday, 10:30

Model Order Reduction and Learning for PDE Constrained Optimization and Inverse Problems

Mario Ohlberger, University of Münster

Model order reduction for parameterized partial differential equations is a very active research area that has seen tremendous development in recent years from both theoretical and application perspectives. A particular promising approach is the reduced basis method that relies on the approximation of the solution manifold of a parameterized system by tailored low dimensional approximation spaces that are spanned from suitably selected particular solutions, called snapshots. With speedups that can reach several orders of magnitude, reduced basis methods enable high fidelity real-time simulations for certain problem classes and dramatically reduce the computational costs in many-query applications. While the „online efficiency“ of these model reduction methods is very convincing for problems with a rapid decay of the Kolmogorov n-width, there are still major drawbacks and limitations. Most importantly, the construction of the reduced system in a so called „offline phase“ is extremely CPU-time and memory consuming for large scale systems. For practical applications, it is thus necessary to derive model reduction techniques that do not rely on a classical offline/online splitting but allow for more flexibility in the usage of computational resources. In this talk we focus on learning based reduction methods in the context of PDE constrained optimization and inverse problems and evaluate their overall efficiency. We discuss learning strategies, such as adaptive enrichment as well as a combination of reduced order models with machine learning approaches in the contest of time dependent problems. Concepts of rigorous certification and convergence will be presented, as well as numerical experiments that demonstrate the efficiency of the proposed approaches.