Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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2063 research outputs found
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Hyperbolic Balance Laws: Interplay between Scales and Randomness
No full textHyperbolic 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
Arbeitsgemeinschaft: Geometry and Representation Theory around the P=W Conjecture
No full textGiven a smooth projective curve , nonabelian Hodge theory gives a diffeomorphism between two different moduli spaces associated to . The first is the moduli space of Higgs bundles on of rank , which is equipped with the structure of an algebraic completely integrable Hamiltonian system. The second is the character variety of representations of the fundamental group of into . In 2012, de Cataldo, Hausel, and Migliorini proposed the conjecture which identifies the perverse filtration on the cohomology of the Higgs moduli space with the weight filtration on the cohomology of the character variety.
Recently, in 2022, two independent proofs of the Conjecture appeared, in work of Maulik & Shen and Hausel, Mellit, Minets & Schiffmann. The aim of the Arbeitsgemeinschaft was to understand the Conjecture and these two recent proofs
The Subgroup Structure of Pseudo-Reductive Groups
No full textLet be a field. We investigate the relationship between subgroups of a pseudo-reductive -group and its maximal reductive quotient , with applications to the subgroup structure of . Let be the minimal field of definition for the geometric unipotent radical of , and let be the quotient map. We first characterise those smooth subgroups of for which . We next consider the following questions: given a subgroup of , does there exist a subgroup of such that , and if is smooth can we find such a that is smooth? We find sufficient conditions for a positive answer to these questions. In general there are various obstructions to the existence of such a subgroup , which we illustrate with several examples. Finally, we apply these results to relate the maximal smooth subgroups of with those of .Work on this paper began during a visit to the Mathematisches Forschungsinstitut Oberwolfach under the Oberwolfach Research Fellows Programme; we thank them for their support. The fourth author was supported by a postdoctoral fellowship of the Alexander von Humboldt Foundation. We’re grateful to David Stewart for comments and for pointing out a mistake in an earlier draft. For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission
Mini-Workshop: Growth and Expansion in Groups
No full textThe aim of the workshop was to give a panoramic view of the main lines of research on various aspects of growth and expansion in finite groups. The main topics included: diameter bounds for Cayley graphs of alternating and classical groups, with results for generating sets that can be arbitrary, random, or containing special elements; constructions of expander families of finite simple groups, with the use of property (T); and character bounds for finite simple groups, yielding applications to word map problems and random walks on Cayley graphs. A series of smaller topics explored connections and similarities to different problems in group theory
Mini-Workshop: New Horizons in Linear Dynamics, Universality, and the Invariant Subspace Problem
No full textThe mini-workshop "New Horizons in Linear Dynamics, Universality, and the Invariant Subspace Problem" discussed recent advances in the study of dynamical properties of continuous linear operators, and in the related study of universal properties of holomorphic functions. Ideas from topological dynamics, ergodic theory, and very recent advances on the invariant subspace problem were also considered
Proof Mining and the Convex Feasibility Problem : the Curious Case of Dykstra's Algorithm
No full textIn a recent proof mining application, the proof-theoretical analysis of Dykstra's cyclic projections algorithm resulted in quantitative information expressed via primitive recursive functionals in the sense of Gödel. This was surprising as the proof relies on several compactness principles and its quantitative analysis would require the functional interpretation of arithmetical comprehension. Therefore, a priori one would expect the need of Spector’s bar-recursive functionals. In this paper, we explain how the use of bounded collection principles allows for a modified intermediate proof justifying the finitary results obtained, and discuss the approach in the context of previous eliminations of weak compactness arguments in proof mining
A Gentle Introduction to Interpolation on the Grassmann Manifold
No full textThis work was supported through the program "Oberwolfach Research Fellows" by the Mathematisches Forschungsinstitut Oberwolfach in 2023
Applied Harmonic Analysis and Data Science
No full textData science is a field of major importance for science and technology
nowadays and poses a large variety of
challenging mathematical questions. The area
of applied harmonic analysis has a significant impact on such problems by providing methodologies
both for theoretical questions and for a wide range of applications in machine learning, as well as in
in signal and image processing.
Building on the success of four previous workshops on applied harmonic analysis in 2012, 2015, 2018, 2021,
this workshop focused on several exciting directions, such as mathematical theory of
deep learning, phase-retrieval time-frequency analysis, and sampling on t-design curves,
and discussed open problems in the field
Voronoi Cells: Or How to Find the Nearest Bakery
No full textDeciding which mall, hospital or school is closest to us is a problem we face everyday. It even comes on holidays with us, when we optimize our plans to make sure that we have enough time to visit all the attractions we want to see. In this article, we show how concepts from metric algebraic geometry help us to rise to this task while planning a weekend trip to the Black Forest
On the Halpern Method with Adaptive Anchoring Parameters
No full textWe establish the convergence of a speed-up version of the Halpern iteration with adaptive anchoring parameters in the general geodesic setting of Hadamard spaces, generalizing a recent result by He, Xu, Dong and Mei from a linear to a nonlinear setting. In particular, our results extend the fast rates of asymptotic regularity obtained by these authors for the first time to a nonlinear setting. Our approach relies on a quantitative study of these previous results in the linear setting, combined with certain optimizations and an elimination of the weak compactness arguments employed crucially in the linear setting, which not only allows for the lift of the result to a nonlinear setting but also streamlines the previous convergence analysis considerably. This work is set in the context of recent developments in proof mining, and as byproduct of our approach, we further obtain quantitative information in the form of highly uniform rates of metastability of low complexity, which are new already in the context of Hilbert spaces.This research was supported through the program "Oberwolfach Leibniz Fellows" by the Mathematisches Forschungsinstitut Oberwolfach in 2024. The fist author was supported by the DFG Project PI 2070/1-1. The second author was supported by the DFG Project KO 1737/6-2