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Mini-Workshop: Superpotentials in Algebra and Geometry
Mirror symmetry has been at the epicenter of many mathematical discoveries in the past twenty years. It was discovered by physicists in the setting of super conformal field theories (SCFTs) associated to closed string theory, mathematically described by $\sigma$-models. These $\sigma$-models turn out in two different ways: the A-model and the B-model. Physical considerations predict that deformations of the SCFT of either $\sigma$-model should be isomorphic. Thus the mirror symmetry conjecture states that the A-model of a particular Calabi-Yau space $X$ must be isomorphic to the B-model of its mirror $\check{X}$. Mirror symmetry has been extended beyond the Calabi-Yau setting, in particular to Fano varieties, using the so called Landau-Ginzburg models. That is a non-compact manifold equipped with a complex valued function called the \emph{superpotential}. In general, there is no clear recipe to construct the mirror for a given variety which demonstrates the need of joining mathematical forces from a wide range. The main aim of this Mini-Workshop was to bring together experts from the different communities (such as symplectic geometry and topology, the theory of cluster varieties, Lie theory and algebraic combinatorics) and to share the state of the art on superpotentials and explore connections between different constructions.
Mini-Workshop: Dimers, Ising and Spanning Trees beyond the Critical Isoradial Case (online meeting)
The goal of this mini-workshop is to gather specialists of the dimer, Ising and spanning tree models around recent and ongoing progress in two directions. One is understanding the connection to the spectral curve of these models in the cases when the curve has positive genus. The other is the introduction of universal embeddings associated to these models. We aim to use these new tools to progress in the study of scaling limits.
Higgs bundles without geometry
Higgs bundles appeared a few decades ago as solutions to certain equations from physics and have attracted much attention in geometry as well as other areas of mathematics and physics. Here, we take a very informal stroll through some aspects of linear algebra that anticipate the deeper structure in the moduli space of Higgs bundles.
Discrete Geometry (hybrid meeting)
A number of important recent developments in various branches of discrete geometry were presented at the workshop, which took place in hybrid format due to a pandemic situation. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics, algebraic geometry or functional analysis. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry.
Algebraic Geometry: Moduli Spaces, Birational Geometry and Derived Aspects (hybrid meeting)
The talks at the workshop and the research done during the week focused on aspects of algebraic geometry in the broad sense. Special emphasis was put on hyperkähler manifolds and derived categories.
Combinatorics
Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization, Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions. This is a report on the meeting, containing extended abstracts of the presentations and a summary of the problem session.
From Betti numbers to ℓ²-Betti numbers
We provide a leisurely introduction to ℓ²-Betti numbers, which are topological invariants, by relating them to their much older cousins, Betti numbers. In the end we present an open research problem about ℓ²-Betti numbers.
Mini-Workshop: Computational Optimization on Manifolds (online meeting)
The goal of the mini-workshop was to study the geometry, algorithms and applications of unconstrained and constrained optimization problems posed on Riemannian manifolds. Focus topics included the geometry of particular manifolds, the formulation and analysis of a number of application problems, as well as novel algorithms and their implementation.
Classical and Quantum Mechanical Models of Many-Particle Systems (online meeting)
The collective behaviour of many-particle systems is a common denominator in the challenges of a highly diverse range of applications: from classical problems in Physics (gas dynamics e.g. Boltzmann's equation, plas\-ma dynamics e.g. various Vlasov equations, semiconductors, quantum mechanics) to current models in biology (kinetic models for collective interaction e.g. swarming, evolution of trait-structured species) to rising topics in social sciences (opinion formation, crowding phenomena) and economics (wealth distribution, mean-field games).\\ Key mathematical questions concern the analysis (global-in-time wellposedness, regularity), rigorous scaling resp. macroscospic limits (model reduction from many-particle models to mean-field/mesoscopic descriptions to macroscopic evolutions), efficient and asymptotic preserving numerical methods and qualitative results (e.g. large-time equilibration).
Cohomology of Finite Groups: Interactions and Applications (hybrid meeting)
The cohomology of finite groups is an important tool in many subjects including representation theory and algebraic topology. This meeting was the fifth in a series that has emphasized the interactions of group cohomology with other areas. In spite of the Covid-19 epidemic, this hybrid meeting ran smoothly with about half the participants physically present and the other half participating via Zoom.