Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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Geometric Spectral Theory
No full textSpectral geometry is a rapidly developing field with new classes of operators, boundary value problems and geometric objects arising in different applications. At the same time, classical problems continue gaining novel flavors. The main focus of the workshop was on some of the most significant recent developments in geometric spectral theory including geometry of eigenvalues and eigenfunctions, singular spectral problems, and spectral optimization. The talks were complemented by three thematic open problem sessions on the main topics of the meeting
Optimization Problems for PDEs in Weak Space-Time Form
No full textOptimization problems constrained by time-dependent Partial
Differential Equations (PDEs) are challenging from a computational
point of view: even in the simplest case, one needs to solve a system
of PDEs coupled globally in time and space for the unknown solutions
(the state, the costate and the control of the system). Typical and
practically relevant examples are the control of nonlinear heat
equations as they appear in laser hardening or the thermic control of
flow problems (Boussinesq equations). Specifically for PDEs with a
long time horizon, conventional time-stepping methods require an
enormous amount of computer memory allocations for the respective
other variables. In contrast, adaptive-in-time-and-space methods aim
at distributing the available degrees of freedom in an a-posteriori
fashion to capture singularities and are, therefore, most
promising. Recently, well-posed weak variational formulations have
been introduced for time-dependent PDEs such as the heat equation,
linear transport and the wave equation. Those formulations also allow
for a sharp relation between the approximation error and the residual,
which is particularly relevant for model reduction. Moreover, for
those tensor-basis formulations, advanced algebraic solvers
designed to take into account these multiarray
(tensorial) formulations appear to be particularly competitive with
respect to time-marching schemes, especially in higher dimensions. We
plan to discuss whether these techniques can be extended to nonlinear
PDEs like Hamilton-Jacobi-Bellman equations, or stochastic PDEs and
variational inequalities. Another topic will be adaptive schemes
which, when properly designed, inherit the stability of the continuous
formulation.
The central goals of the workshop are the analysis, fast solvers and model
reduction for PDE-constrained control and optimization problems based on
weak formulations of the underlying PDE(s)
Free Boundary Problems Arising in Fluid Mechanics
No full textFluid mechanics is one of the classical areas in the study of partial differential equations and has been a vast subject of research in the last centuries. A relevant class of problems are those in which the evolution of fluids of different nature and their interaction is described through the dynamics of their common boundary. Such problems are called free-boundary problems. The key topic of this workshop deals with recent advances on the analysis of free-boundary problems which open up a whole new area of research activity. More precisely, we will address problems as the vortex-patch, the study of water waves, interface flows in porous media and Hele-Shaw cells as well as atmospheric front dynamics where the formation of finite time singularities is a fundamental open question
Arbeitsgemeinschaft: Twistor D-Modules and the Decomposition Theorem
No full textThe purpose of this Arbeitsgemeinschaft is to introduce the notion of twistor -modules and their main properties. The guiding principle leading this discussion is Simpson's "meta-theorem", which gives a heuristic for generalizing (mixed) Hodge-theoretic results into (mixed) twistor-theoretic results. The strength of the twistor approach is that it enables to enlarge the scope of Hodge theory not only to arbitrary semi-simple perverse sheaves, equivalently semi-simple regular holonomic -modules via the Riemann-Hilbert correspondence, but also to possibly semi-simple irregular holonomic -modules. An overarching goal for this session is Mochizuki's proof of the decomposition theorem for semi-simple holonomic -modules on a smooth complex projective variety, first conjectured by Kashiwara in 1996
Tensor-Triangular Geometry and Interactions
No full textThe workshop brought together experts in a rapidly growing field of tensor triangular geometry highlighting applications to and techniques coming from homotopy theory, algebraic geometry, modular representation theory, motivic homotopy theory and noncommutative algebra
Patterns and Waves in Theory, Experiment, and Application
No full textIn this snapshot of modern mathematics we describe some of the most prevalent waves and patterns that can arise in mathematical models and which are used to describe a number of biological, chemical, physical, and social processes. We begin by focussing on two types of patterns that do not change in time: space-filling patterns and localized patterns. We then discuss two types of waves that evolve predictably as time goes on: spreading waves and rotating waves. All our examples are motivated with real-world applications and we highlight some of the main lines of research that mathematicians pursue to better understand them
Mini-Workshop: Nonlinear Approximation of High-dimensional Functions in Scientific Computing
No full textApproximation techniques for high dimensional PDEs are crucial for contemporary scientific computing tasks and gained momentum in recent years due to the renewed interest in neural networks.
It seems that especially nonlinear parametrizations will play an essential role in efficient and tractable approximations of high dimensional problems.
We held a mini-workshop on the relation and possible synergy of neural networks and tensor product approximation.
To reliably evaluate the prospect of different numerical experiments, the traditional talks were accompanied by live coding sessions
Hypoelliptic Operators in Geometry
No full textThe workshop titled "Hypoelliptic Operators in Geometry" was coorganized by Davide Barilari (Padova), Xiaonan Ma (Paris),
Nikhil Savale (Köln) and Yi Wang (Baltimore). It was well attended by 55 participants, with 45 of them being present in person and 10 being online. The participants
came from several continents, age groups and included male as well as female researchers. Several interesting themes were discussed including:
analysis around Kohn's Laplacian in CR geometry, analogous covariant operators arising in conformal geometry, the spectral theory of the sub-Riemannian Laplacian, pseudodifferential calculi in non-commutative geometry and the geometric applications of Bismut's hypoelliptic Laplacians
Mathematical Foundations of Biological Organisation
No full textThe workshop aimed to explore the use of new mathematical and computational approaches to investigate the fundamental principles governing the organization and dynamics of biological systems. This necessitated conversations among mathematical biologists working at different scales, from molecular to organismal levels. The meeting aimed to encourage interdisciplinary collaborations and showcase recent advances in diverse areas
Arithmetic of Shimura Varieties
No full textThe aim of this workshop was to discuss recent developments
on the arithmetic of Shimura varieties and on related topics within the Langlands program, and to initiate and support further research in this direction. The talks presented new methods and results covering topics ranging from geometric questions on the reduction of Shimura varieties to representations in their cohomology, automorphic forms, and questions on the geometry and arithmetic of moduli spaces of bundles on the Fargues-Fontaine curve