http://publications.mfo.de/handle/mfo/3833 Oberwolfach Reports Volume 18 (2021) Tue, 07 Apr 2026 08:03:05 GMT 2026-04-07T08:03:05Z http://publications.mfo.de/handle/mfo/3926 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. Fri, 01 Jan 2021 00:00:00 GMT http://publications.mfo.de/handle/mfo/3926 2021-01-01T00:00:00Z 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. http://publications.mfo.de/handle/mfo/3922 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. Fri, 01 Jan 2021 00:00:00 GMT http://publications.mfo.de/handle/mfo/3922 2021-01-01T00:00:00Z 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. http://publications.mfo.de/handle/mfo/3919 Automorphic Forms, Geometry and Arithmetic (hybrid meeting) The workshop on automorphic forms, geometry and arithmetic focused on important recent developments within the research area, in particular, on the different recent approaches towards the Langlands functoriality principle and the Langlands correspondence, on their relative analogues, and on the relations between those advances and more arithmetic questions. Fri, 01 Jan 2021 00:00:00 GMT http://publications.mfo.de/handle/mfo/3919 2021-01-01T00:00:00Z The workshop on automorphic forms, geometry and arithmetic focused on important recent developments within the research area, in particular, on the different recent approaches towards the Langlands functoriality principle and the Langlands correspondence, on their relative analogues, and on the relations between those advances and more arithmetic questions. http://publications.mfo.de/handle/mfo/3918 Complexity Theory (hybrid meeting) Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including interactive proof systems, quantum information and computation, algorithmic coding theory, arithmetic complexity, expansion of hypergraphs and simplicial complexes, Markov chain Monte Carlo, and pseudorandomness. Many of the developments are related to diverse mathematical fields such as algebraic geometry, extremal combinatorics, combinatorial number theory, probability theory, representation theory, and operator algebras. Fri, 01 Jan 2021 00:00:00 GMT http://publications.mfo.de/handle/mfo/3918 2021-01-01T00:00:00Z Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including interactive proof systems, quantum information and computation, algorithmic coding theory, arithmetic complexity, expansion of hypergraphs and simplicial complexes, Markov chain Monte Carlo, and pseudorandomness. Many of the developments are related to diverse mathematical fields such as algebraic geometry, extremal combinatorics, combinatorial number theory, probability theory, representation theory, and operator algebras. http://publications.mfo.de/handle/mfo/3913 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. Fri, 01 Jan 2021 00:00:00 GMT http://publications.mfo.de/handle/mfo/3913 2021-01-01T00:00:00Z 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. http://publications.mfo.de/handle/mfo/3911 Convex Geometry and its Applications (hybrid meeting) The geometry of convex domains in Euclidean space plays a central role in several branches of mathematics: functional and harmonic analysis, the theory of PDE, linear programming and, increasingly, in the study of algorithms in computer science. The purpose of this meeting was to bring together researchers from the analytic, geometric and probabilistic groups who have contributed to these developments. Fri, 01 Jan 2021 00:00:00 GMT http://publications.mfo.de/handle/mfo/3911 2021-01-01T00:00:00Z The geometry of convex domains in Euclidean space plays a central role in several branches of mathematics: functional and harmonic analysis, the theory of PDE, linear programming and, increasingly, in the study of algorithms in computer science. The purpose of this meeting was to bring together researchers from the analytic, geometric and probabilistic groups who have contributed to these developments. http://publications.mfo.de/handle/mfo/3910 Mini-Workshop: Scattering Amplitudes, Cluster Algebras, and Positive Geometries (hybrid meeting) Cluster 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. Fri, 01 Jan 2021 00:00:00 GMT http://publications.mfo.de/handle/mfo/3910 2021-01-01T00:00:00Z Cluster 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. http://publications.mfo.de/handle/mfo/3909 Applied Harmonic Analysis and Data Science (hybrid meeting) Data science has become a field of major importance for science and technology nowadays and poses a large variety of challenging mathematical questions. The area of applied harmonic analysis has a significant impact on such problems by providing methodologies both for theoretical questions and for a wide range of applications in signal and image processing and machine learning. Building on the success of three previous workshops on applied harmonic analysis in 2012, 2015 and 2018, this workshop focused on several exciting novel directions such as mathematical theory of deep learning, but also reported progress on long-standing open problems in the field. Fri, 01 Jan 2021 00:00:00 GMT http://publications.mfo.de/handle/mfo/3909 2021-01-01T00:00:00Z Data science has become a field of major importance for science and technology nowadays and poses a large variety of challenging mathematical questions. The area of applied harmonic analysis has a significant impact on such problems by providing methodologies both for theoretical questions and for a wide range of applications in signal and image processing and machine learning. Building on the success of three previous workshops on applied harmonic analysis in 2012, 2015 and 2018, this workshop focused on several exciting novel directions such as mathematical theory of deep learning, but also reported progress on long-standing open problems in the field. http://publications.mfo.de/handle/mfo/3908 Enveloping Algebras and Geometric Representation Theory (hybrid meeting) The workshop brought together experts investigating algebraic Lie theory from the geometric and categorical viewpoints. Fri, 01 Jan 2021 00:00:00 GMT http://publications.mfo.de/handle/mfo/3908 2021-01-01T00:00:00Z The workshop brought together experts investigating algebraic Lie theory from the geometric and categorical viewpoints. http://publications.mfo.de/handle/mfo/3907 Arbeitsgemeinschaft: Derived Galois Deformation Rings and Cohomology of Arithmetic Groups (hybrid meeting) The purpose of the workshop was to study derived generalizations of Mazur's deformation ring of Galois representations, and the relationship of such a derived deformation ring to the homology of arithmetic groups. Fri, 01 Jan 2021 00:00:00 GMT http://publications.mfo.de/handle/mfo/3907 2021-01-01T00:00:00Z The purpose of the workshop was to study derived generalizations of Mazur's deformation ring of Galois representations, and the relationship of such a derived deformation ring to the homology of arithmetic groups.