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- ItemMini-Workshop: Gibbs Measures for Nonlinear Dispersive Equations(Zürich : EMS Publ. House, 2018) Schlein, Benjamin; Sohinger, VedranIn this mini-workshop we brought together leading experts working on the application of Gibbs measures to the study of nonlinear PDEs. This framework is a powerful tool in the probabilistic study of solutions to nonlinear dispersive PDEs, in many ways alternative or complementary to deterministic methods. Among the special topics discussed were the construction of the measures, applications to dynamics, as well as the microscopic derivation of Gibbs measures from many-body quantum mechanics.
- ItemMini-Workshop: Self-adjoint Extensions in New Settings(Zürich : EMS Publ. House, 2019) Kostenko, Aleksey; Pankrashkin, KonstantinThe main focus of the workshop is on the analysis of boundary value problems for differential and difference operators in some non-classical geometric settings, such as fractal graphs, sub-Riemannian manifolds or non-elliptic transmission problems. Taking into account their importance in modern mathematical analysis, we aim at developing suitable tools in the operator theory to deal with the new problem settings.
- ItemStatistics for Data with Geometric Structure(Zürich : EMS Publ. House, 2018) Hotz, Thomas; Huckemann, Stephan; Miller, EzraStatistics for data with geometric structure is an active and diverse topic of research. Applications include manifold spaces in directional data or symmetric positive definite matrices and some shape representations. But in some cases, more involved metric spaces like stratified spaces play a crucial role in different ways. On the one hand, phylogenetic trees are represented as points in a stratified data space, whereas branching trees, for example of veins, are data objects, whose stratified structure is of essential importance. For the latter case, one important tool is persistent homology, which is currently a very active area of research. As data sets become not only larger but also more complex, the need for theoretical and methodological progress in dealing with data on non-Euclidean spaces or data objects with nontrivial geometric structure is growing. A number of fundamental results have been achieved recently and the development of new methods for refined, more informative data representation is ongoing. Two complimentary approaches are pursued: on the one hand developing sophisticated new parameters to describe the data, like persistent homology, and on the other hand achieving simpler representations in terms of given parameters, like dimension reduction. Some foundational works in stochastic process theory on manifolds open the doors to this field and stochastic analysis on manifolds, thus enabling a well-founded treatment of non-Euclidean dynamic data. The results presented in the workshop by leading experts in the field are great accomplishments of collaboration between mathematicians from statistics, geometry and topology and the open problems which were discussed show the need for an expansion of this interdisciplinary effort, which could also tie in more closely with computer science.
- ItemMini-Workshop: Mathematical and Numerical Analysis of Maxwell's Equations(Zürich : EMS Publ. House, 2018) Langer, Ulrich; Monk, Peter; Pauly, DirkIn this mini-workshop 17 leading mathematicians from Europe and United States met at the MFO to discuss and present new developments in the mathematical and numerical analysis of Maxwell’s equations and related systems of partial differential equations. The report at hand offers the extended abstracts of their talks.
- ItemNon-Archimedean Geometry and Applications(Zürich : EMS Publ. House, 2019) Gubler, Walter; Schneider, Peter; Werner, AnnetteThe workshop focused on recent developments in non-Archimedean analytic geometry with various applications to other fields. The topics of the talks included applications to complex geometry, mirror symmetry, p-adic Hodge theory, tropical geometry, resolution of singularities, p-adic dynamical systems and diophantine geometry.
- ItemRandom Matrices(Zürich : EMS Publ. House, 2019) Götze, Friedrich; Guionnet, AliceLarge complex systems tend to develop universal patterns that often represent their essential characteristics. For example, the cumulative effects of independent or weakly dependent random variables often yield the Gaussian universality class via the central limit theorem. For non-commutative random variables, e.g. matrices, the Gaussian behavior is often replaced by another universality class, commonly called random matrix statistics. Nearby eigenvalues are strongly correlated, and, remarkably, their correlation structure is universal, depending only on the symmetry type of the matrix. Even more surprisingly, this feature is not restricted to matrices; in fact Eugene Wigner, the pioneer of the field, discovered in the 1950s that distributions of the gaps between energy levels of complicated quantum systems universally follow the same random matrix statistics. This claim has never been rigorously proved for any realistic physical system but experimental data and extensive numerics leave no doubt as to its correctness. Since then random matrices have proved to be extremely useful phenomenological models in a wide range of applications beyond quantum physics that include number theory, statistics, neuroscience, population dynamics, wireless communication and mathematical finance. The ubiquity of random matrices in natural sciences is still a mystery, but recent years have witnessed a breakthrough in the mathematical description of the statistical structure of their spectrum. Random matrices and closely related areas such as log-gases have become an extremely active research area in probability theory. This workshop brought together outstanding researchers from a variety of mathematical backgrounds whose areas of research are linked to random matrices. While there are strong links between their motivations, the techniques used by these researchers span a large swath of mathematics, ranging from purely algebraic techniques to stochastic analysis, classical probability theory, operator algebra, supersymmetry, orthogonal polynomials, etc.
- ItemMathematical Methods in Quantum Chemistry(Zürich : EMS Publ. House, 2018) Friesecke, Gero; Helgaker, Trygve Ulf; Lin, LinThe field of quantum chemistry is concerned with the modelling and simulation of the behaviour of molecular systems on the basis of the fundamental equations of quantum mechanics. Since these equations exhibit an extreme case of the curse of dimensionality (the Schrödinger equation for N electrons being a partial differential equation on R3N ), the quantum-chemical simulation of even moderate-size molecules already requires highly sophisticated model-reduction, approximation, and simulation techniques. The workshop brought together selected quantum chemists and physicists, and the growing community of mathematicians working in the area, to report and discuss recent advances on topics such as coupled-cluster theory, direct approximation schemes in full configuration-interaction (FCI) theory, interacting Green’s functions, foundations and computational aspects of densityfunctional theory (DFT), low-rank tensor methods, quantum chemistry in the presence of a strong magnetic field, and multiscale coupling of quantum simulations.
- ItemApplied Harmonic Analysis and Data Processing(Zürich : EMS Publ. House, 2018) Kutyniok, Gitta; Rauhut, Holger; Strohmer, ThomasMassive data sets have their own architecture. Each data source has an inherent structure, which we should attempt to detect in order to utilize it for applications, such as denoising, clustering, anomaly detection, knowledge extraction, or classification. Harmonic analysis revolves around creating new structures for decomposition, rearrangement and reconstruction of operators and functions—in other words inventing and exploring new architectures for information and inference. Two previous very successful workshops on applied harmonic analysis and sparse approximation have taken place in 2012 and in 2015. This workshop was the an evolution and continuation of these workshops and intended to bring together world leading experts in applied harmonic analysis, data analysis, optimization, statistics, and machine learning to report on recent developments, and to foster new developments and collaborations.
- ItemNonlinear Evolution Equations: Analysis and Numerics(Zürich : EMS Publ. House, 2019) Koch, Herbert; Oh, Sung-Jin; Ostermann, AlexanderThe qualitative theory of nonlinear evolution equations is an important tool for studying the dynamical behavior of systems in science and technology. A thorough understanding of the complex behavior of such systems requires detailed analytical and numerical investigations of the underlying partial differential equations.
- ItemRepresentations of Finite Groups(Zürich : EMS Publ. House, 2019) Geck, Meinolf; Kessar, Radha; Navarro, GabrielThe workshop Representations of Finite Groups was organised by Joseph Chuang (London), Meinolf Geck (Stuttgart), Radha Kessar (London) and Gabriel Navarro (Valencia). It covered a wide variety of aspects of representation theory of finite groups and its relations to other areas of mathematics.