Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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2063 research outputs found
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On Dykstra’s Algorithm with Bregman Projections
No full textWe provide quantitative results on the asymptotic behavior of Dykstra’s algorithm with Bregman projections, a combination of the well-known Dykstra’s algorithm and the method of cyclic Bregman projections, designed to find best approximations and solve the convex feasibility problem in a non-Hilbertian setting. The result we provide arise through the lens of proof mining, a program in mathematical logic which extracts computational information from non-effective proofs. Concretely, we provide a highly uniform and computable rate of metastability of low complexity and, moreover, we also specify general circumstances in which one can obtain full and effective rates of convergence. As a byproduct of our quantitative analysis, we also for the first time establish the strong convergence of Dykstra’s method with Bregman projections in infinite dimensional (reflexive) Banach spaces.This research was supported through the program “Oberwolfach Leibniz Fellows” by the Mathematisches Forschungsinstitut Oberwolfach in 2024. Further, the authors were supported by the DFG Project KO 1737/6-2
Mechanics of Materials: Multiscale Design of Advanced Materials and Structures
No full textMaterials 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.)
Cluster Algebras and Its Applications
No full textThis 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
Metric Algebraic Geometry
No full textOberwolfach Seminar: Metric Algebraic Geometry 2322bOberwolfach Seminar: Metric Algebraic Geometry 2322bMetric algebraic geometry combines concepts from algebraic geometry and differential geometry. Building on classical foundations, it offers practical tools for the 21st century. Many applied problems center around metric questions, such as optimization with respect to distances.
After a short dive into 19th-century geometry of plane curves, we turn to problems expressed by polynomial equations over the real numbers. The solution sets are real algebraic varieties. Many of our metric problems arise in data science, optimization and statistics. These include minimizing Wasserstein distances in machine learning, maximum likelihood estimation, computing curvature, or minimizing the Euclidean distance to a variety.
This book addresses a wide audience of researchers and students and can be used for a one-semester course at the graduate level. The key prerequisite is a solid foundation in undergraduate mathematics, especially in algebra and geometry.
This is an openaccess book.[Open Access
Mini-Workshop: Bridging Number Theory and Nichols Algebras via Deformations
No full textNichols 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
Ky Fan Theorem for Sphere Bundles
No full textThe classic Ky Fan theorem is a combinatorial equivalent of Borsuk-Ulam theorem. It is a generalization and extension of Tucker’s lemma and, just like its predecessor, it pinpoints important properties of antipodal colorings of vertices of a triangulated sphere Sn. Here we describe generalizations of Ky Fan theorem for the case when the sphere is replaced by the total space of a triangulated sphere bundle.It is our pleasure to acknowledge the hospitality of the Byurakan Astrophysical observatory, where this research was initiated, and also the hospitality of Mathematisches Forschungsinstitut Oberwolfach, where in the spring of 2024 this paper was completed as a part of ’research in pairs’ project. R. ˇZivaljevi´c was supported by the Science Fund of the Republic of Serbia, Grant No. 7744592, Integrability and Extremal Problems in Mechanics, Geometry and Combinatorics - MEGI
Real and Logarithmic Enumerative Geometry
No full textKey topics of the workshop included enumerative geometry, Gromov-Witten invariants, and their extensions to general ground fields and tropical counting. Significant contributions were made in refined invariants, topological recursion, and Givental reconstruction. Discussions also covered the log double ramification cycle, Brill-Noether theory, Hilbert schemes of points, moduli spaces, log smooth degenerations, mirror symmetry, and the topology of real algebraic varieties
A CFSG-Free Explicit Jordan’s Theorem over Arbitrary Fields
No full textWe prove a version of Jordan's classification theorem for finite subgroups of that is at the same time quantitatively explicit, CFSG-free, and valid for arbitrary . This is the first proof to satisfy all three properties at once. Our overall strategy follows Larsen and Pink [24], with explicit computations based on techniques developed by the authors and Helfgott [2, 3], particularly in relation to dimensional estimates.The second author was funded by a Young Researcher Fellowship from the HUN-REN Alfréd Rényi Institute of Mathematics, and by the Leibniz Fellowship 2405p from the Mathematisches Forschungsinstitut Oberwolfach (MFO).
The authors would like to thank the MFO for providing a wonderful environment for the development of this article
Statistics and Learning Theory in the Era of Artificial Intelligence
No full textThe workshop highlighted recent theoretical advances on inference in high-dimensional statistical models based on the interplay of techniques from mathematical statistics, machine learning, theoretical computer
science and related areas. The workshop brought together about 50 researchers in order to present new results, exchange ideas and explore open
problems
Applications of Optimal Transportation
No full textThe 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