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- ItemMFO-RIMS Tandem Workshop: Symmetries on Polynomial Ideals and Varieties (hybrid meeting)(Zürich : EMS Publ. House, 2021) Murai, Satoshi; Riener, Cordian; Yanagawa, KohjiThe study of symmetry as a structural property of algebraic objects is one of the fundamental pillows of the developments of modern mathematics, most prominently beginning with the work of Abel and Galois. The focus of the workshop was on permutation actions of the symmetric group on polynomial rings and algebraic and semi-algebraic sets. More concretely, it was centered around recent developments in the asymptotic setup of symmetric ideals in the polynomial ring in infinitely many variables.
- ItemHistory of Mathematics: A Global Cultural Approach (online meeting)(Zürich : EMS Publ. House, 2020) Montelle, Clemency; Remmert, Volker; Rowe, David E.The primary purpose of this workshop was to take account of progress on an ongoing six-volume cultural history of mathematics from antiquity to the present. This project is led by nine editors working with a large team of authors. Since the workshop had to be held remotely, it took the form of various group meetings held throughout the week. The final session involved assessments by editors of the six volumes with an eye toward completing the project by the end of 2021. The abstracts below summarize the contents of the individual chapters in the entire project, which will be published in Bloomsbury's cultural history series.
- ItemReal Algebraic Geometry with a View Toward Hyperbolic Programming and Free Probability(Zürich : EMS Publ. House, 2020) Kuhlmann, Salma; Speicher, Roland; Vinnikov, VictorContinuing the tradition initiated in the MFO workshops held in 2014 and 2017, this workshop was dedicated to the newest developments in real algebraic geometry and polynomial optimization, with a particular emphasis on free non-commutative real algebraic geometry and hyperbolic programming. A particular effort was invested in exploring the interrelations with free probability. This established an interesting dialogue between researchers working in real algebraic geometry and those working in free probability, from which emerged new exciting and promising synergies.
- ItemHomotopic and Geometric Galois Theory (online meeting)(Zürich : EMS Publ. House, 2021) Dèbes, Pierre; Nakamura, Hiroaki; Stix, JakobIn his "Letter to Faltings'', Grothendieck lays the foundation of what will become part of his multi-faceted legacy to arithmetic geometry. This includes the following three branches discussed in the workshop: the arithmetic of Galois covers, the theory of motives and the theory of anabelian Galois representations. Their geometrical paradigms endow similar but complementary arithmetic insights for the study of the absolute Galois group $\mathrm{G}_{\mathbb{Q}}$ of the field of rational numbers that initially crystallized into a functorially group-theoretic unifying approach. Recent years have seen some new enrichments based on modern geometrical constructions - e.g. simplicial homotopy, Tannaka perversity, automorphic forms - that endow some higher considerations and outline new geometric principles. This workshop brought together an international panel of young and senior experts of arithmetic geometry who sketched the future desire paths of homotopic and geometric Galois theory.
- ItemApplications of Optimal Transportation in the Natural Sciences (online meeting)(Zürich : EMS Publ. House, 2021) Ehrlacher, Virginie; Matthes, DanielConcepts 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.
- 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.