Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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Quadratically Enriched Plane Curve Counting via Tropical Geometry
No full textWe thank Jesse Pajwani for pointing out that identities in the Grothendieck-Witt ring can be checked on multiquadratic finite ´etale algebras and for his help with computations. We thank Erwan Brugall´e, Andreas Gross, Marc Levine, Dhruv Ranganathan and Kirsten Wickelgren for useful discussions. The first, second and fourth author acknowledge support by DFG-grant MA 4797/9-1. The third author acknowledges support by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the Collaborative Research Centre TRR 326 Geometry and Arithmetic of Uniformized Structures, project number 444845124. The first author thanks the Universität Duisburg-Essen and the Università degli Studi di Napoli Federico II for support. Part of this work was completed while the authors stayed as Research Fellows at the Mathematisches Forschungsinstitut Oberwolfach in March 2024. We thank the institute for hosting us and for providing ideal working conditions.We prove that the quadratically enriched count of rational curves in a smooth toric del Pezzo surface passing through -rational points and pairs of conjugate points in quadratic field extensions can be determined by counting certain tropical stable maps through vertically stretched point conditions with a suitable multiplicity. Building on the floor diagram technique in tropical geometry, we provide an algorithm to compute these numbers.
Our tropical algorithm computes not only these new quadratically enriched enumerative invariants, but simultaneously also the complex Gromov-Witten invariant, the real Welschinger invariant counting curves satisfying real point conditions only, the real Welschinger invariant of curves satisfying pairs of complex conjugate and real point conditions, and the quadratically enriched count of curves satisfying -rational point conditions
Revised Version
No full textThis work was supported through the program "Oberwolfach Research Fellows" by the Mathematisches Forschungsinstitut Oberwolfach in 2023.Acknowledgements. The authors gratefully acknowledge the financial support of the Mathematisches Forschungsinstitut Oberwolfach for a two-week stay at the Institute under the "Oberwolfach Research Fellows" program. Further, the authors acknowledge two reviewers, and Gunther Dirr, Knut Hüper and Federico Stra for clarifying discussions, for pointing out several references, and for a careful reading of
the revised version of this manuscript. The first and third authors are members of the INdAM-GNCS group.Revised Versio
Truncated Fusion Rules for Supergroups
No full textIn the '70s, physicists introduced a new type of symmetry – supersymmetry – to address some unresolved issues in particle physics models. Its mathematical foundations involve the representation theory of the associated symmetry groups, called supergroups. Our aim is to understand fusion rules, which describe how a combination of two physical systems can be broken down into more fundamental building blocks. Although the answer is largely unknown, we can get approximate answers in some cases
Convex Polytopes and Linear Programs
No full textConvex polytopes are geometric objects that look deceptively simple. They occur everywhere in mathematics and have practical applications in everyday life – like organizing your grocery shopping list. In this snapshot, you get into contact with a long-standing, unsolved question in mathematics, which you can explore interactively
On the Complexity of Epimorphism Testing with Virtually Abelian Targets
No full textFriedl and Löh (2021, Confl. Math.) prove that testing whether or not there is an epimorphism from a finitely presented group to the direct product of an abelian and a finite group, or to a virtually cyclic group, is decidable. Here we prove that these problems are NP-complete. In addition we show that testing epimorphism is NP-complete when the target is a restricted type of semi-direct product of a finitely generated free abelian group and a finite group, thus extending the class of virtually abelian target groups for which decidability of epimorphism is known. We also consider epimorphism from a finitely presented group to a fixed finite group. We show the epimorphism problem is NP-complete when the target is a dihedral group of order that is not a power of 2, complementing the work on Kuperberg and Samperton (2018, Geom. Topol.) who showed the same result when the target is non-Abelian finite simple.This research was supported by Australian Research Council grant DP210100271. The second author was supported by an Australian Government Research Training Program Stipend. The third author was partially supported by DFG grant WE 6835/1–2. The first and third author were supported through the program “Oberwolfach Research Fellows” at the Mathematisches Forschungsinstitut Oberwolfach in 2023.
We wish to thank Michal Ferov, Robert Tang, Alexander Thumm and Kane Townsend for fruitful discussions about this project
Domain-Scaled Regular Variation: Mathematical Foundations for a New Tail Process
No full textThreshold exceedances of stochastic processes in space and time often appear to be more localized the more extreme they are. While classical regularly varying stochastic processes cannot model this effect, we introduce an adapted version of regular variation, where a suitable domain-scaling can be incorporated to accommodate this behaviour. Our theory is inspired by the triangular array convergence of domain-scaled maxima of Gaussian processes to a Brown-Resnick process and turns out to be natural in this context. We study key properties of the resulting tail process and demonstrate its ability to approximate conditional exceedance probabilities of Gaussian processes. Mathematical convenience arises from the recently rediscovered concept of vague convergence based on boundedness.The authors would like to thank the Mathematisches Forschungsinstitut Oberwolfach for the kind hospitality during our two weeks at the institute in March 2024 supported through the program "Oberwolfach Research Fellows", which has allowed us to kickstart this research. KS wishes to thank her home institution, Cardiff University, for granting her research leave in autumn 2024 to complete the research and writing of this manuscript, including financial support for visiting MO at the University of Stuttgart during this time
Proof Complexity and Beyond
No full textProof 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
Mini-Workshop: Mathematics of Entropic AI in the Natural Sciences
No full textThe mathematical framework of approximate entropic learning introduced very recently promises to provide robust, cheap and efficient ways of machine learning in the so called "small data" regime, when the underlying learning task is highly-underdetermined, due to a large problem dimension and relatively small data statistics size. Such "small data" learning challenges are particularly common in the natural sciences (e.g., in geosciences, in climate research, in economics, and in biomedicine), imposing considerable difficulties for the numerics of common "data-hungry" Artificial Intelligence (AI) tools like Deep Learning (DL). The aim of this workshop will be to bring together experts in the emergent fields of entropic and DL mathematics/numerics, with some lead experts applying AI in the domain disciplines. The goal will be to detect and to discuss the commonalities in the challenges and in their mathematical solutions, as well as to discuss and fine-tune common mathematical problem formulations that are motivated by the AI applications in natural sciences. The establishment of a common mathematical framework for such small-data machine learning tasks would not only bolster future methodological developments but would also lay solid foundations to further in-depth rigorous analysis and theoretically founded interpretation of these methods and their results
The Alternating Halpern-Mann Iteration for Families of Maps
No full textWe generalize the alternating Halpern-Mann iteration to countably infinite families of nonexpansive maps and prove its strong convergence towards a common fixed point in the general nonlinear setting of Hadamard spaces. Our approach is based on a quantitative perspective which allowed to circumvent prevalent troublesome arguments and in the end provide a simple convergence proof. In that sense, discussing both the asymptotic regularity and the strong convergence of the iteration in quantitative terms, we furthermore provide low complexity uniform rates of convergence and of metastability (in the sense of T. Tao). In CAT(0) spaces, we obtain linear and quadratic uniform rates of convergence. Our results are made possible by proof-theoretical insights of the research program proof mining and extend several previous theorems in the literature