Search Results
Mini-Workshop: (Anosov)$^3$ (hybrid meeting)
Three different active fields are subsumed under the keyword Anosov theory: Spectral theory of Anosov flows, dynamical rigidity of Anosov actions, and Anosov representations. In all three fields there have been dynamic developments and substantial breakthroughs in recent years. The mini-workshop brought together researchers from the three different communities and sparked a joint discussion of current ideas, common interests, and open problems.
Mini-Workshop: Variable Curvature Bounds, Analysis and Topology on Dirichlet Spaces (hybrid meeting)
A 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.
Algebraic Groups (hybrid meeting)
Linear algebraic groups is an active research area in contempo- rary mathematics. It has rich connections to algebraic geometry, representa- tion theory, algebraic combinatorics, number theory, algebraic topology, and differential equations. The foundations of this theory were laid by A. Borel, C. Chevalley, J.-P. Serre, T. A. Springer and J. Tits in the second half of the 20th century. The Oberwolfach workshops on algebraic groups, led by Springer and Tits, played an important role in this effort as a forum for re- searchers, meeting at approximately 3 year intervals since the 1960s. The present workshop continued this tradition, covering a range of topics, with an emphasis on recent developments in the subject.
Mini-Workshop: Three Facets of R-Matrices (hybrid meeting)
By definition, an $R$-matrix with spectral parameter is a solution to the Yang-Baxter equation, introduced in the 1970's by C.N. Yang and R.J. Baxter. Such a matrix encodes the Boltzmann weights of a lattice model of statistical mechanics, and the Yang-Baxter equation appears naturally as a sufficient condition for its solvability. In the last decade, several mathematical and physical theories have led to seemingly different constructions of $R$-matrices. The theme of this workshop was the interaction of three such approaches, each of which has independently proven to be valuable: the geometric, analytic and gauge-theoretic constructions of $R$-matrices. Its aim was to bring together leading experts and researchers from each school of thought, whose recent works have given novel interpretations to this nearly classical topic.
Mini-Workshop: Analysis of Data-driven Optimal Control (hybrid meeting)
This hybrid mini-workshop discussed recent mathematical methods for analyzing the opportunities and limitations of data-driven and machine-learning approaches to optimal feedback control. The analysis concerned all aspects of such approaches, ranging from approximation theory particularly for high-dimensional problems via complexity analysis of algorithms to robustness issues.
Combinatorial Optimization (hybrid meeting)
Combinatorial 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.
Applications of Optimal Transportation in the Natural Sciences (online meeting)
Concepts and methods from the mathematical theory of optimal transportation have reached significant importance in various fields of the natural sciences. The view on classical problems from a "transport perspective'' has lead to the development of powerful problem-adapted mathematical tools, and sometimes to a novel geometric understanding of the matter. The natural sciences, in turn, are the most important source of ideas for the further development of the optimal transport theory, and are a driving force for the design of efficient and reliable numerical methods to approximate Wasserstein distances and the like. The presentations and discussions in this workshop have been centered around recent analytical results and numerical methods in the field of optimal transportation that have been motivated by specific applications in statistical physics, quantum mechanics, and chemistry.
Analysis, Geometry and Topology of Positive Scalar Curvature Metrics (hybrid meeting)
The investigation of Riemannian metrics with lower scalar curvature bounds has been a central topic in differential geometry for decades. It addresses foundational problems, combining ideas and methods from global analysis, geometric topology, metric geometry and general relativity. Seminal contributions by Gromov during the last years have led to a significant increase of activities in the area which have produced a number of impressive results. Our workshop reflected the state of the art of this thriving field of research.
Computation and Learning in High Dimensions (hybrid meeting)
The most challenging problems in science often involve the learning and accurate computation of high dimensional functions. High-dimensionality is a typical feature for a multitude of problems in various areas of science. The so-called curse of dimensionality typically negates the use of traditional numerical techniques for the solution of high-dimensional problems. Instead, novel theoretical and computational approaches need to be developed to make them tractable and to capture fine resolutions and relevant features. Paradoxically, increasing computational power may even serve to heighten this demand, since the wealth of new computational data itself becomes a major obstruction. Extracting essential information from complex problem-inherent structures and developing rigorous models to quantify the quality of information in a high-dimensional setting pose challenging tasks from both theoretical and numerical perspective. This has led to the emergence of several new computational methodologies, accounting for the fact that by now well understood methods drawing on spatial localization and mesh-refinement are in their original form no longer viable. Common to these approaches is the nonlinearity of the solution method. For certain problem classes, these methods have drastically advanced the frontiers of computability. The most visible of these new methods is deep learning. Although the use of deep neural networks has been extremely successful in certain application areas, their mathematical understanding is far from complete. This workshop proposed to deepen the understanding of the underlying mathematical concepts that drive this new evolution of computational methods and to promote the exchange of ideas emerging in various disciplines about how to treat multiscale and high-dimensional problems.
Analysis, Geometry and Topology of Singular PDE (hybrid meeting)
This is a report on the Oberwolfach conference "Analysis, geometry and topology of singular PDE'', June 2-12, 2021. This workshop, held in an hybrid format, focused on the topology, geometry and geometric analysis of certain spaces with singularities: stratified spaces, compactifications of moduli spaces, spaces carrying a (singular) foliation, etc., and on the microlocal techniques, either in their classical forms or in more recent versions developed to handle singular PDE, or via the groupoid approach.