Oberwolfach Reports (OWR)

Permanent URI for this collectionhttps://oa.tib.eu/renate/handle/123456789/14627

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Mini-Workshop: Bridging Number Theory and Nichols Algebras via Deformations
(Zürich : EMS Publ. House, 2024) Carnovale, Giovanna; Heckenberger, István; Vendramin, Leandro
Nichols algebras are graded Hopf algebra objects in braided tensor categories. They appeared first in a paper by Nichols in 1978 in the search for new examples of Hopf algebras. Rediscovered later several times, they also provide a conceptual explanation of the construction of quantum groups. The aim of the workshop is to review recent developments in the field, initiate collaborations, and discuss new approaches to open problems.
Interfaces, Free Boundaries and Geometric Partial Differential Equations
(Zürich : EMS Publ. House, 2024) Elliott, Charles M.; Garcke, Harald; Niethammer, Barbara; Simonett, Gieri
Partial differential equations arising in the context of interfaces and free boundaries encompass a flourishing area of research. The workshop focused on new developments and emerging new themes. At the same time also new interesting results on more traditional areas like, e.g. regularity theory and classical numerical approaches have been addressed. By convening experts from various disciplines related to modeling, analysis, and numerical methods concerning interfaces and free boundaries, the workshop facilitated progress on longstanding open questions and paved the way for novel research directions.
Mini-Workshop: Permutation Patterns
(Zürich : EMS Publ. House, 2024) Bóna, Miklós; Bouvel, Mathilde; Brignall, Robert; Pantone, Jay
The study of permutation patterns has recently seen several surprising results, and the purpose of this mini-workshop was to bring together researchers from across the field to focus on four hot topics related to these recent developments. The topics covered the nature of generating functions that enumerate permutation classes, the structure of permutation classes and the impact this has on their growth rates, and the study of permutons, which lies at the interface of permutation patterns and discrete probability. The workshop offered an opportunity for knowledge exchange, but also time and space to initiate group collaborations on open problems related to these topics.
Mini-Worskhop: Artin Groups meet Triangulated Categories
(Zürich : EMS Publ. House, 2024) Boyd, Rachael; Heng, Edmund; Ozornova, Viktoriya
Artin and Coxeter groups are naturally occurring generalisations of the braid and symmetric groups respectively. However, unlike for Coxeter groups, many basic group theoretic questions remain unanswered for general Artin groups – most notably the K(π,1)-conjecture for Artin groups remains open except for certain special families of Artin groups. Recently, Artin groups have also appeared as groups acting on triangulated categories, where the associated spaces of Bridgeland's stability conditions provide new realisations of the corresponding K(π,1) spaces. The aim of the workshop is to bring together experts and early career researchers from two seemingly different areas of research: (i) geometric and combinatorial group theory and topology, and (ii) triangulated categories and stability conditions, to explore their intersection via the K(π,1)-conjecture.
Hyperbolic Balance Laws: Interplay between Scales and Randomness
(Zürich : EMS Publ. House, 2024) Abgrall, Rémi; Garavello, Mauro; Lukáčová-Medvid'ová, Mária; Trivisa, Konstantina
Hyperbolic balance laws are fundamental in the mathematical modeling of transport-dominated processes in natural, socio-economic and engineering sciences. The aim of the workshop was to discuss open questions in the area of nonlinear hyperbolic conservation and balance laws. We have focused on a delicate interplay between scale hierarchies and random/stochastic effects and discuss them from analytical, numerical and modeling point of view. This leads to questions of admissibility criteria connecting to ill-posedness of weak entropy solutions, hyperbolic problems with non-local terms, mean field theory, multiscale and structure preserving numerical methods, random solutions and uncertainty quantification methods, as well as data-based methods.
Applications of Optimal Transportation
(Zürich : EMS Publ. House, 2024) Carlier, Guillaume; Colombo, Maria; Ehrlacher, Virginie; Matthes, Daniel
The mathematical theory of optimal transportation is constantly expanding its range of application, while applications give impulses for new research directions in the field. This workshop was specifically devoted to applications of optimal transportation in the natural sciences, and to the recent developments of the theory that have been motivated by these. The bouquet of current applications includes mathematical models for large-scale air motion, dynamics of plasmas, material design, pattern formation in fluids, collective behaviour in biology, and many more. Related theoretical developments are in the analysis of the Hellinger-Kantorovich metric for modeling reaction–diffusion processes, and in efficient numerical methods for multi-marginal optimal transport, to name two of many examples encountered in this workshop.
Discrete Geometry
(Zürich : EMS Publ. House, 2024) Adiprasito, Karim; Goaoc, Xavier; Patáková, Zuzana
A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as convex geometry, combinatorics, or topology. Two open problem sessions highlighted the abundance of open questions and many of the results presented were obtained by young researchers, confirming the vitality of the field.
Analysis, Geometry and Topology of Positive Scalar Curvature Metrics
(Zürich : EMS Publ. House, 2024) Ammann, Bernd; Hanke, Bernhard; Sakovich, Anna
Riemannian metrics with positive scalar curvature play an important role in differential geometry and general relativity. To investigate these metrics, it is necessary to employ concepts and techniques from global analysis, geometric topology, metric geometry, index theory, and general relativity. This workshop brought together researchers from a variety of backgrounds to combine their expertise and promote cross-disciplinary exchange.
K-Stability, Birational Geometry and Mirror Symmetry
(Zürich : EMS Publ. House, 2024) Delcroix, Thibaut; Heuberger, Liana; Zimmermann, Susanna
The workshop K-stability, Birational Geometry and Mirror Symmetry presented recent advances in all three topics, in the form of research level mini-courses, research talks and lightning talk sessions. Deep interactions between the three topics were highlighted, together with applications (e.g. to K-moduli or subgroups of the Cremona groups) and new directions (such as non-Archimedean methods in K-stability and Mirror Symmetry).
Nonlinear Optics: Physics, Analysis, and Numerics
(Zürich : EMS Publ. House, 2024) Busch, Kurt; El-Ganainy, Ramy; Hochbruck, Marlis
When high-intensity electromagnetic waves at optical frequencies interact with solids and/or nanostructures the materials' response cannot anymore be described via simple linear relations. The resulting science of nonlinear optics has recently witnessed exiting developments that have brought to the fore numerous mathematical challenges that need to be addressed in order to fully exploit the opportunities that result from these developments. The mathematical modeling involves a system of partial differential equations where the Maxwell equations are coupled to evolution equations of the materials and their response to electromagnetic fields. Typically, the full coupled systems are quite complicated or even intractable so that the derivation, the analysis, and the numerical treatment of simplified effective models is often indispensable. In turn, this requires the close cooperation between researchers from theoretical physics and analysis/numerics in order to push forward the field on nonlinear optics.
Proof Complexity and Beyond
(Zürich : EMS Publ. House, 2024) Atserias, Albert; Mahajan, Meena; Nordström, Jakob; Razborov, Alexander
Proof complexity is a multi-disciplinary research area that addresses questions of the general form “how difficult is it to prove certain mathematical facts?” The current workshop focussed on recent advances in our understanding that the analysis of an appropriately tailored concept of “proof” underlies many of the arguments in algorithms, geometry or combinatorics research that make the core of modern theoretical computer science. These include the analysis of practical Boolean satisfiability (SAT) solving algorithms, the size of linear or semidefinite programming formulations of combinatorial optimization problems, the complexity of solving total NP search problems by local methods, and the complexity of describing winning strategies in two-player round-based games, to name just a few important examples.
Mechanics of Materials: Multiscale Design of Advanced Materials and Structures
(Zürich : EMS Publ. House, 2024) Forest, Samuel; McDowell, David; Müller, Stefan; Werner, Ewald
Materials can now be designed and architectured like structural components for targeted mechanical and physical properties. Structures and microstructures should not be studied independently and their design will benefit from a multiscale approach combining nonlinear continuum mechanics approaches and physical descriptions of elasticity, viscoplasticity, phase transformations and damage of microstructures, at various scales. The aim of the workshop was to gather outstanding junior and senior researchers in the various branches of mathematics, physics and engineering sciences suited to address the question of design of materials and structures by means of multiscale discrete and continuum approaches to their constitutive behavior. Examples include atomic or macroscopic lattices, random or periodic cellular materials, smart materials like shape memory alloys, 3D woven composites, acoustic and electromagnetic metamaterials, etc. Modern continuum mechanics relies on sophisticated constitutive laws for anisotropic materials exhibiting elastoviscoplastic behavior, still a field of intense research with new mathematical concepts. In particular size-dependent properties are addressed by resorting to generalized continua such as gradient or micromorphic and phase field models. The latter are attractive for the simulation of microstructure evolution coupled with mechanics, due to thermodynamic and metallurgical processes and damage. Scale transition and homogenization methods for continuous and discrete systems are required for the determination of effective material and structural behavior. Metamaterials are architectured materials specifically designed to achieve certain propagation and dispersion properties of elastic and plastic waves. Optimization strategies for the design of optimal architectures are involved in the design process. Target functions for optimization are now based on multicriteria (stiffness, strength, thermal expansion, transport properties, anisotropy etc.).
Fracture as an Emergent Phenomenon
(Zürich : EMS Publ. House, 2024) Diehl, Patrick; Lipton, Robert; Pandolfi, Anna; Wick, Thomas
The mechanics of fracture propagation provides essential knowledge for the risk tolerance design of devices, structures, and vehicles. Techniques of free energy minimization provide guidance, but have limited applicability to material systems evolving away from equilibrium. Experimental evidence shows that the material response depends on driving forces arising from mechanical fields. Recent years have witnessed the development of new methods for modeling complex dynamic and quasistatic fracture. New approaches may differ remarkably from previous ones, as they involve implicit coupling between damaged and undamaged states, allowing fracture to be modeled as emergent phenomena. The focus of this workshop is on the most advanced techniques for modeling fracture, represented by eigenerosion methods, variational approaches, phase field fracture models, and non-local approaches. Technical progress is contingent on the further development of the mathematical framework underlying these techniques. This is necessary for accurate and reliable computational modeling of fracture for multiple freely propagating cracks. The objective of this workshop is to mathematically identify and discuss open issues related to fracture modeling and to highlight recent advances. Addressing fundamental issues will foster exchange between the different communities, essential for advancing the field.
Combinatorial ∗-algebras
(Zürich : EMS Publ. House, 2024) Cortiñas, Guillermo; Eilers, Søren; Gillaspy, Elizabeth; Hazrat, Roozbeh
This workshop aimed to strengthen ties and foster collaborations between different communities working on combinatorial ∗-algebras, including C∗-and pure algebraists.
Cluster Algebras and Its Applications
(Zürich : EMS Publ. House, 2024) Baur, Karin; Marsh, Bethany; Schiffler, Ralf; Schroll, Sibylle
This workshop focused on recent developments in cluster algebras and their applications as well as interactions with other areas of mathematics. In addition to new advances in the theory of cluster algebras themselves, it included applications to knot theory and geometry as well as interactions with representation theory and categorification, Grassmannians, combinatorics, geometric surfaces models and Lie theory.
Low-dimensional Topology
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Friedl, Stefan; Moriah, Yoav; Purcell, Jessica; Schleimer, Saul
The workshop brought together experts from across all areas of low-dimensional topology, including knot theory, computational topology, three-manifolds and four-manifolds. In addition to the standard research talks we had two survey talks by Marc Lackenby and Joel Hass, leading to discussions of open problems. Furthermore we had three sessions of five-minute talks by a total of roughly thirty participants.
Combinatorics
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Keevash, Peter; Samotij, Wojciech; Sudakov, Benny
Combinatorics is an area of mathematics primarily concerned with counting and studying properties of discrete objects such as graphs, set systems, partial orders, polyhedra, etc. Combinatorial problems naturally arise in many areas of mathematics, such as algebra, geometry, probability theory, and topology, and in theoretical computer science. Historically, such questions were often studied using ad hoc arguments. However, over the last few decades, the development of general and powerful methods have elevated combinatorics to a thriving branch of mathematics with many connections to other subjects. The workshop brought together the established leading experts and the brightest young talents from different parts of this very broad area in order to discuss the most exciting recent developments, current themes and trends, and the most promising new directions for future research.
Morphisms in Low Dimensions
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Lobb, Andrew; Miller, Maggie; Ray, Arunima
This workshop brought together experts on interrelated topics in low-dimensional topology, centred around the common theme of 'morphisms'. Our goal was to improve community understanding of recent developments in the field and to promote new advances in the study of global properties of 4-manifolds.
Arithmetic of Shimura Varieties
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Fargues, Laurent; Görtz, Ulrich; Viehmann, Eva; Wedhorn, Torsten
The aim of this workshop was to discuss recent developments on the arithmetic of Shimura varieties and on related topics within the Langlands program, and to initiate and support further research in this direction. The talks presented new methods and results covering topics ranging from geometric questions on the reduction of Shimura varieties to representations in their cohomology, automorphic forms, and questions on the geometry and arithmetic of moduli spaces of bundles on the Fargues-Fontaine curve.
Representation Theory of Quivers and Finite-Dimensional Algebras
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Amiot, Claire; Crawley-Boevey, William; Iyama, Osamu; Schröer, Jan
This workshop was about the representation theory of quivers and finite-dimensional (associative) algebras, and links to other areas of mathematics, including other areas of representation theory, homological algebra, cluster algebras, algebraic geometry and singularity theory. Particularly active topics included $\tau$-tilting theory, algebras arising from surface triangulations and the study of exact categories and their generalizations.

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