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- ItemLow-dimensional Topology(Zürich : EMS Publ. House, 2020) Moriah, Yoav; Purcell, Jessica; Schleimer, SaulThe workshop brought together experts from across all areas of low-dimensional topology, including knot theory, mapping class groups, three-manifolds and four-manifolds. In addition to the standard research talks we had five survey talks by Burton, Minsky, Powell, Reid, and Roberts leading to discussions of open problems. Furthermore we had three sessions of five-minute talks by a total of thirty-five participants.
- ItemChallenges in Optimization with Complex PDE-Systems (hybrid meeting)(Zürich : EMS Publ. House, 2021) Kunisch, Karl; Leugering, Günter; Rocca, ElisabettaThe workshop concentrated on various aspects of optimization problems with systems of nonlinear partial differential equations (PDEs) or variational inequalities (VIs) as constraints. In particular, discussions around several keynote presentations in the areas of optimal control of nonlinear or non-smooth systems, optimization problems with functional and discrete or switching variables leading to mixed integer nonlinear PDE constrained optimization, shape and topology optimization, feedback control and stabilization, multi-criteria problems and multiple optimization problems with equilibrium constraints as well as versions of these problems under uncertainty or stochastic influences, and the respectively associated numerical analysis as well as design and analysis of solution algorithms were promoted. Moreover, aspects of optimal control of data-driven PDE constraints (e.g. related to machine learning) were addressed.
- ItemMini-Workshop: Variable Curvature Bounds, Analysis and Topology on Dirichlet Spaces (hybrid meeting)(Zürich : EMS Publ. House, 2021) Güneysu, Batu; Keller, Matthias; Kuwae, KazuhiroA Dirichlet form $\mathcal{E}$ is a densely defined bilinear form on a Hilbert space of the form $L^2(X,\mu)$, subject to some additional properties, which make sure that $\mathcal{E}$ can be considered as a natural abstraction of the usual Dirichlet energy $\mathcal{E}(f_1,f_2)=\int_D (\nabla f_1,\nabla f_2) $ on a domain $D$ in $\mathbb{R}^m$. The main strength of this theory, however, is that it allows also to treat nonlocal situations such as energy forms on graphs simultaneously. In typical applications, $X$ is a metrizable space, and the theory of Dirichlet forms makes it possible to define notions such as curvature bounds on $X$ (although $X$ need not be a Riemannian manifold), and also to obtain topological information on $X$ in terms of such geometric information.
- ItemNew Perspectives and Computational Challenges in High Dimensions(Zürich : EMS Publ. House, 2020) Prochno, Joscha; Thäle, Christoph; Werner, ElisabethHigh-dimensional systems are frequent in mathematics and applied sciences, and the understanding of high-dimensional phenomena has become increasingly important. The mathematical subdisciplines most strongly related to such phenomena are functional analysis, convex geometry, and probability theory. In fact, a new area emerged, called asymptotic geometric analysis, which is at the very core of these disciplines and bears a number of deep connections to mathematical physics, numerical analysis, and theoretical computer science. The last two decades have seen a tremendous growth in this area. Far reaching results were obtained and various powerful techniques have been developed, which rather often have a probabilistic flavor. The purpose of this workshop was to explored these new perspectives, to reach out to other areas concerned with high-dimensional problems, and to bring together researchers having different angles on high-dimensional phenomena.
- ItemMini-Workshop: Scattering Amplitudes, Cluster Algebras, and Positive Geometries (hybrid meeting)(Zürich : EMS Publ. House, 2021) Thomas, Hugh; Williams, LaurenCluster algebras were developed by Fomin and Zelevinsky about twenty years ago. While the initial motivation came from within algebra (total positivity, canonical bases), it quickly became clear that cluster algebras possess deep links to a host of other subjects in mathematics and physics. In a separate vein, starting about ten years ago, Arkani-Hamed and his collaborators began a program of reformulating the bases of quantum field theory, motivated by a desire to discover the basic rules of quantum mechanics and spacetime as arising from deeper mathematical principles. Their approach to the fundamental problem of particle scattering amplitudes entails encoding the solution in geometrical objects, "positive geometries'' and "amplituhedra''. Surprisingly, cluster algebras have been found to be tightly woven into the mathematics needed to describe these geometries. The purpose of this workshop is to explore the various connections between cluster algebras, scattering amplitudes, and positive geometries.
- ItemModuli spaces and Modular forms (hybrid meeting)(Zürich : EMS Publ. House, 2021) van der Geer, Gerard; Gritsenko, ValeryThe relation between moduli spaces and modular forms goes back to the theory of elliptic curves. On the one hand both topics experience their own growth and development, but from time to time new unexpected links show up and usually these lead to progress on both sides. One subject where there has been a lot of progress concerns the moduli of abelian varieties and K3 surfaces and especially on the Kodaira dimension of these spaces. The idea of the workshop was to bring together the experts of the two areas in the hope that discussion, interaction and lectures would spur the development of new ideas. The lectures of the workshop gave ample evidence of the interaction and provided opportunities for further interaction. Besides the lectures participants interacted via zoom in smaller groups.
- ItemMini-Workshop: Non-semisimple Tensor Categories and Their Semisimplification (online meeting)(Zürich : EMS Publ. House, 2021) Etingof, Pavel; Schweigert, ChristophFinite tensor categories are, despite their many applications and great interest, notoriously hard to classify. Among them, the semisimple ones (called fusion categories) have been intensively studied. Those with non-integral dimensions form a remarkable class. Already more than 20 years ago, tilting modules have been proposed as a source of such fusion categories. In this way, the Verlinde categories associated to the pair of a simple complex Lie algebra $\mathfrak g$ and an integer level $k$ have been recovered in a purely algebraic framework - called semisimplification of tensor categories. Recently efforts to understand how to go beyond these examples emerged. This mini-workshop aims at bringing together experts from various branches of representation theory and topological field theory to deepen our understanding of finite tensor categories and to compare new ways to understand semisimplification.
- ItemCombinatorial Optimization (hybrid meeting)(Zürich : EMS Publ. House, 2021) Iwata, Satoru; Kaibel, Volker; Svensson, OlaCombinatorial Optimization deals with optimization problems defined on combinatorial structures such as graphs and networks. Motivated by diverse practical problem setups, the topic has developed into a rich mathematical discipline with many connections to other fields of Mathematics (such as, e.g., Combinatorics, Convex Optimization and Geometry, and Real Algebraic Geometry). It also has strong ties to Theoretical Computer Science and Operations Research. A series of Oberwolfach Workshops have been crucial for establishing and developing the field. The workshop we report about was a particularly exciting event - due to the depth of results that were presented, the spectrum of developments that became apparent from the talks, the breadth of the connections to other mathematical fields that were explored, and last but not least because for many of the particiants it was the first opportunity to exchange ideas and to collaborate during an on-site workshop since almost two years.
- ItemBoundary Element Methods(Zürich : EMS Publ. House, 2020) Hiptmair, Ralf; Sayas, Francisco-Javier; Steinbach, OlafThe field of boundary element methods (BEM) relies on recasting boundary value problems for (mostly linear) partial differential equations as (usually singular) integral equations on boundaries of domains or interfaces. Its main goal is the design and analysis of methods and algorithms for the stable and accurate discretization of these integral equations, the data-sparse representation of the resulting systems of equations, and their efficient direct or iterative solution. Boundary element methods play a key role in important areas of computational engineering and physics addressing simulations in acoustics, electromagnetics, and elasticity. Thus progress in boundary element method, both theoretical and algorithmic, is definitely relevant beyond mathematics. Boundary element methods had been developed for many decades, but during the past two decades the field has seen a surge in research activity, spurred by algorithmic and theoretical breakthroughs concerning BEM for electromagnetics, time-domain methods, new approaches to eigenvalue problems, adaptivity, local low-rank matrix compression, and frequency-explicit analysis, to name only a few. The contributions in this report give an impressive panorama of the many and diverse current research activities in BEM. They range profound mathematical analyses with striking results to new algorithmic developments. On the one hand, the results are based on a large variety of tools from many areas of mathematics. On the other hand, research in BEM blazes the trail for progress in the numerical treatment of non-local operators, a field that is rapidly gaining importance.
- ItemStatistics of Stochastic Differential Equations on Manifolds and Stratified Spaces (hybrid meeting)(Zürich : EMS Publ. House, 2021) Li, Xue-Mei; Pokern, Yvo; Sturm, AnjaStatistics for stochastic differential equations (SDEs) attempts to use SDEs as statistical models for real-world phenomena. This involves an understanding of qualitative properties of this class of stochastic processes which includes Brownian motion as well as estimation of parameters in the SDE or a nonparametric estimation of drift and diffusivity fields from observations. Observations can be in continuous time, in high frequency discrete time considering the limit of small inter-observation times or in discrete time with constant inter-obseration times. Application areas of SDEs where state spaces are naturally viewed as manifolds or stratified spaces include multivariate stochastic volatility models, stochastic evolution of shapes (e.g. of biological cells), time-varying image deformations for video analysis and phylogenetic trees.