Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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Resolutions in Local Algebra and Singularity Theory
No full textCommutative algebra is a vast subject, with connections to many different areas of mathematics, and beyond. The focus of this workshop was on three areas, all concerned with resolutions in various forms. One is the resolution of singularities of algebraic varieties, which remains a vibrant topic of research. The second is the theory of noncommutative resolution of singularities. Introduced two decades ago, this subject has witnessed remarkable growth developing connections to algebraic geometry, commutative algebra, cluster algebras, and the representation theory of algebras, both commutative and noncommutative, among others. The third intended meaning of the world "resolution" is as in free resolutions of algebras and modules in commutative algebra. There is another sense in which the title is appropriate: recently three long standing open problems in commutative algebra have been resolved. This workshop brought together experts and early career researchers in these various fields, to facilitate exchange of ideas and to explore potential collaborations
Flag-Accurate Arrangements
No full textIn [MR21], the first two authors introduced the notion of an accurate arrangement, a particular notion of freeness. In this paper, we consider a special subclass, where the property of accuracy stems from a flag of flats in the intersection lattice of the underlying arrangement. Members of this family are called flag-accurate. One relevance of this new notion is that it entails divisional freeness. There are a number of important natural classes which are flag-accurate, the most prominent one among them is the one consisting of Coxeter arrangements. This warrants a systematic study which is put forward in the present paper. More specifically, let be a free arrangement of rank . Suppose that for every , the first exponents of - when listed in increasing order - are realized as the exponents of a free restriction of to some intersection of reflecting hyperplanes of of dimension . Following [MR21], we call such an arrangement with this natural property accurate. If in addition the flats involved can be chosen to form a flag, we call flag-accurate. We investigate flag-accuracy among reflection arrangements, extended Shi and extended Catalan arrangements, and further for various families of graphic and digraphic arrangements. We pursue these both from theoretical and computational perspectives. Along the way we present examples of accurate arrangements that are not flag-accurate. The main result of [MR21] shows that MAT-free arrangements are accurate. We provide strong evidence for the conjecture that MAT-freeness actually entails flag-accuracy
Felder und Räume: Symmetrie und Lokalität in Mathematik und theoretischen Wissenschaften
No full textWir werden einige grundlegende Ideen der Eichtheorie und der dazugehörigen Differentialtopologie erkunden. Damit kann sich die Leserin ein Bild des Modulraums flacher Zusammenhänge machen und ihn mit den physikalisch motivierten Ideen dahinter in Beziehung bringen. Den Begriffen von Symmetrien und Feldern gehen wir gründlich nach. Außerdem werfen wir einen flüchtigen Blick auf unendliche Symmetrie in zwei Dimensionen und auf vor kurzem entdeckte Verallgemeinerungen
The Character Triple Conjecture for Maximal Defect Characters and the Prime 2
No full textWe prove that Späth’s Character Triple Conjecture holds for every finite group with respect to maximal defect characters at the prime 2. This is done by reducing the maximal defect case of the conjecture to the so-called inductive Alperin–McKay condition whose verification has recently been completed by Ruhstorfer for the prime 2. As a consequence we obtain the Character Triple Conjecture for all 2-blocks with abelian defect groups by applying Brauer’s Height Zero Conjecture, a proof of which is now available. We also obtain similar results for the block-free version of the Character Triple Conjecture at the prime 3.Large part of this work was carried out during a stay of the author at the Mathematisches Forschungsinstitut Oberwolfach
funded by a Leibniz Fellowship
Mini-Workshop: Multivariate Orthogonal Polynomials: New synergies with Numerical Analysis
No full textMultivariate polynomials and, in particular, multivariate orthogonal polynomials (MOPs) are research areas within the fields of special functions, Lie groups, quantum groups, computer algebra to name only some of them. However, there are many important areas in the field of numerical analysis where multivariate polynomials (of high order) play a crucial role: approximation by spectral methods and finite
elements, discrete calculus, polynomial trace liftings, exact sequence properties, sparsity, efficient and stable recursions, analysis of the geometry of the zeros. The miniworkshop brought together experts from the fields of MOPs and numerical analysis of partial differential equations
Algebras and Quantum Games
No full textEveryone loves a good game, but when the players can access the counterintuitive world of quantum mechanics, watch out
Geometric, Algebraic, and Topological Combinatorics
No full textThe 2023 Oberwolfach meeting "Geometric, Algebraic, and Topological
Combinatorics''
was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle),
Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered
a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics
with geometric flavor, and Topological Combinatorics. Some of the
highlights of the conference were (1) Federico Ardila and Tom Braden
discussed recent exciting developments in the intersection theory of matroids;
(2) Stavros Papadakis and Vasiliki Petrotou presented their proof of the
Lefschetz property for spheres, and, more generally, for pseudomanifolds and
cycles (this second part is joint with Karim Adiprasito); (3) Gaku Liu reported
on his joint work with Spencer Backman that establishes the existence of a
regular unimodular triangulation of an arbitrary matroid base polytope
Mini-Workshop: Standard Subspaces in Quantum Field Theory and Representation Theory
No full textReal standard subspaces of complex Hilbert spaces are long \linebreak known to provide the right language for Tomita-Takesaki modular theory of von Neumann algebras. In recent years they have also become an object of prominent interest in mathematical quantum field theory (QFT) and unitary representation theory of Lie groups. This workshop brought together mathematicians and physicists working with standard subspaces, particularly in QFT (construction of QFT models, characterization of entropy, information-theoretic aspects),
nets of standard subspaces on causal homogeneous spaces and aspects of reflection positivity and euclidean models related to standard subspaces and modular theory
Machine Learning for Science: Mathematics at the Interface of Data-driven and Mechanistic Modelling
No full textRapid progress in machine learning is enabling scientific advances across a range of disciplines. However, the utility of machine learning for science remains constrained by its current inability to translate insights from data about the dynamics of a system to new scientific knowledge about why those dynamics emerge, as traditionally represented by physical modelling. Mathematics is the interface that bridges data-driven and physical models of the world and can provide a foundation for delivering such knowledge. This workshop convened researchers working across domains with a shared interest in mathematics, machine learning, and their application in the sciences, to explore how tools of mathematics can help build machine learning tools for scientific discovery
Mathematical Analysis and Numerical Approximation
No full textThis book presents the notes from the seminar on wave phenomena given in 2019 at the Mathematical Research Center in Oberwolfach.
The research on wave-type problems is a fascinating and emerging field in mathematical research with many challenging applications in sciences and engineering. Profound investigations on waves require a strong interaction of several mathematical disciplines including functional analysis, partial differential equations, mathematical modeling, mathematical physics, numerical analysis, and scientific computing.
The goal of this book is to present a comprehensive introduction to the research on wave phenomena. Starting with basic models for acoustic, elastic, and electro-magnetic waves, topics such as the existence of solutions for linear and some nonlinear material laws, efficient discretizations and solution methods in space and time, and the application to inverse parameter identification problems are covered. The aim of this book is to intertwine analysis and numerical mathematics for wave-type problems promoting thus cooperative research projects in this field