Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
Not a member yet
2063 research outputs found
Sort by
Combinatorial *-algebras
No full textThis workshop aimed to strengthen ties and foster collaborations between different communities working on combinatorial -algebras,
including -and pure algebraists
K-Stability, Birational Geometry and Mirror Symmetry
No full textThe 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)
Alternating Snake Modules and a Determinantal Formula
No full textWe introduce a family of modules for the quantum affine algebra which include as very special cases both the snake modules and the modules arising from a monoidal categorification of cluster algebras. We give necessary and sufficient conditions for these modules to be prime and prove a unique factorization result. We also give an explicit formula expressing the module as an alternating sum of Weyl modules. Finally, we give an application of our results to classical questions in the category . Specifically we apply our results to show that there are a large family of non-regular, non-dominant weights for which the non-zero Kazhdan-Lusztig coefficients are .The authors thank David Hernandez and Bernard Leclerc for many helpful discussions and insightful questions. They thank Ryo Fujita for drawing their attention to the connection with the Arakawa–Suzuki functor and for pertinent references. A substantial portion of this work was carried out during two visits to Oberwolfach as part of the OWRF program in 2023; the authors are deeply appreciative of the excellent environment at the Mathematisches Forschungsinstitut. M.B. is grateful to the Department of Mathematics, UCR, for their hospitality during a visit when part of this research was carried out
Cutoff-Phänomen: überraschendes Verhalten beim Kartenmischen und bei weiteren Markovketten
No full textDieser Schnappschuss vergleicht zwei Arten des Kartenmischens und untersucht, wie lange es dauert einen "gut gemischten" Kartenstapel zu erhalten. Überraschenderweise kann das Mischverhalten auch für sehr ähnlich ausschauende Kartenmischtechniken sehr unterschiedlich sein.[Also available in English
Higgs bundles without geometry
No full textLes fibrés de Higgs sont apparus il y a quelques décennies comme solutions de certaines équations en physique, et ils ont attiré beaucoup d’attention en géométrie comme dans d’autres domaines des mathématiques et de la physique. Ici, nous donnons un aperçu très informel de quelques aspects d’algèbre linéaire qui anticipent la structure profonde de l’espace de modules des fibrés de Higgs
The Congruence Properties of Romik’s Sequence of Taylor Coefficients of Jacobi’s Theta Function
No full textIn [Ramanujan J. 52 (2020), 275-290], Romik considered the Taylor expansion of Jacobi's theta function at and encoded it in an integer sequence for which he provided a recursive procedure to compute the terms of the sequence. He observed intriguing behaviour of modulo primes and prime powers. Here we prove (1) that eventually vanishes modulo any prime power with (mod 4), (2) that is eventually periodic modulo any prime power with (mod 4), and (3) that is purely periodic modulo any 2-power . Our results also provide more detailed information on period length, respectively from when on the sequence vanishes or becomes periodic. The corresponding bounds may not be optimal though, as computer data suggest. Our approach shows that the above congruence properties hold at a much finer, polynomial level.We thank Tanay Wakhare for helpful correspondence. The authors also thank the Mathematische Forschungsinstitut Oberwolfach for the opportunity of an Oberwolfach Research Fellowship in August/September 2024, during which they succeeded to improve the divisibility results for primes p ≡ 3 (mod 4) significantly
Fracture as an Emergent Phenomenon
No full textThe 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
Arrangements, Matroids and Logarithmic Vector Fields
No full textThe focus of this workshop was on the ongoing interaction between
geometric aspects of matroid theory with various directions in the
study of hyperplane arrangements. A hyperplane arrangement
is exactly a linear realization of a (loop-free, simple) matroid.
While a matroid is a purely combinatorial object, though, an arrangement
is associated with a range of algebraic and geometric constructions
that connect closely with the combinatorics of matroids.
The meeting brought together researchers involved with complementary
angles on the subject, many of whom had not met before, so an important
underlying objective was to make introductions between groups with
overlapping interests in order to facilitate new collaborations and
advances in the subject
Nonlinear Optics: Physics, Analysis, and Numerics
No full textWhen 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
Waves and Incidences
No full textThe wave equation in Euclidean spaces describes many natural phenomena such as sound, light, or water waves. We explore how its solutions are related to the geometric problem of how long thin cylinders can intersect each other and discuss a related open problem