Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
Not a member yet
2063 research outputs found
Sort by
Interfaces, Free Boundaries and Geometric Partial Differential Equations
No full textPartial differential equations arising in the context of interfaces and free boundaries encompass a flourishing area of research.
The workshop focused on new developments and emerging new themes.
At the same time also new interesting results on more traditional areas like, e.g. regularity theory and classical numerical approaches have been addressed.
By convening experts from various disciplines related to modeling, analysis, and numerical methods concerning interfaces and free boundaries, the workshop facilitated progress on longstanding open questions and paved the way for novel research directions
Arm Exponent for the Gaussian Free Field on Metric Graphs in Intermediate Dimensions
No full textWe investigate the bond percolation model on transient weighted graphs induced by the excursion sets of the Gaussian free field on the corresponding metric graph. We assume that balls in have polynomial volume growth with growth exponent and that the Green's function for the random walk on exhibits a power law decay with exponent , in the regime . In particular, this includes the cases of , for which , and , for which . For all such graphs, we determine the leading-order asymptotic behavior for the critical one-arm probability, which we prove decays with distance like . Our results are in fact more precise and yield logarithmic corrections when as well as corrections of order when . We further obtain very sharp upper bounds on truncated two-point functions close to criticality, which are new when and essentially optimal when . This extends previous results from [16].Acknowledgments: This research was supported through the programs ‘Oberwolfach Research Fellows’ and ‘Oberwolfach Leibniz Fellows’ by the Mathematisches Forschungsinstitut Oberwolfach in 2023. The research of AD has been supported by the Deutsche Forschungsgemeinschaft (DFG) grant DR 1096/2-1. AP has been supported by the Engineering and Physical
Sciences Research Council (EPSRC) grant EP/R022615/1, Isaac Newton Trust (INT) grant G101121, European Research Council (ERC) starting grant 804166 (SPRS), and the Swiss NSF. PFR thanks the Research Institute for Mathematical Sciences (RIMS), an International Joint Usage/Research Center located in Kyoto University, for its hospitality
Game-theoretic Statistical Inference: Optional Sampling, Universal Inference, and Multiple Testing Based on E-values
No full textThis half-size MFO workshop brings together researchers in mathematical statistics, probability theory, machine learning, medical sciences, and economics to discuss recent developments in sequential inference. New sequential inference methods that build on nonnegative martingale techniques allow us to elegantly solve prominent shortcomings of traditional statistical hypothesis tests. Instead of p-values, they are based on "e-values" which have the added benefit that their meaning is much easier to
communicate to applied researchers, due to their intuitive interpretation in terms of the wealth of a gambler playing a hypothetically fair game. Significant new contributions to this fast growing research area will be presented in order to stimulate collaborations, discuss and unify notation and concepts in the fields, and tackle a variety of open problems and address current major challenges
Free Boundary Problems in Fluid Dynamics
No full textOberwolfach Seminar 2243a: Free Boundary Problems in Fluid DynamicsThis book, originating from a seminar held at Oberwolfach in 2022, introduces to state-of-the-art methods and results in the study of free boundary problems which are arising from compressible as well as from incompressible Euler’s equations in general. A particular set of such problems is given by gaseous stars considered in a vacuum (modeled via the compressible Euler equations) as well as water waves in their full generality (seen as recasts of incompressible irrotational Euler equations). This is a broad research area which is highly relevant to many real life problems, and in which substantial progress has been made in the last decade
Analysis, Geometry and Topology of Positive Scalar Curvature Metrics
No full textRiemannian metrics with positive scalar curvature play an important role in differential geometry and general relativity.
To investigate these metrics, it is necessary to employ concepts and techniques from global analysis, geometric topology, metric geometry, index theory, and general relativity.
This workshop brought together researchers from a variety of backgrounds to combine their expertise and promote cross-disciplinary exchange
The ternary Goldbach problem
No full textLeonhard Euler (1707–1783), l’un des plus grands mathématiciens du XVIIIe siècle et de tous les temps, entretenait une correspondance régulière avec l’un de ses amis: Christian Goldbach (1690–1764), un amateur polymathe qui vivait et travaillait en Russie, tout comme Euler. Dans une lettre datée de juin 1742, Goldbach établit une conjecture (c’est-à-dire une hypothèse éclairée) sur les nombres premiers: Es scheinet wenigstens, dass eine jede Zahl, die größer ist als 2, ein aggregatum trium numerorum primorum sey. Il semble [ . . .] que tout nombre entier naturel supérieur à 2 puisse être écrit comme la somme de trois nombres premiers
On Overgroups of Distinguished Unipotent Elements in Reductive Groups and Finite Groups of Lie Type
No full textSuppose G is a simple algebraic group defined over an algebraically closed field of good characteristic p. In 2018 Korhonen showed that if H is a connected reductive subgroup of G which contains a distinguished unipotent element u of G of order p, then H is G-irreducible in the sense of Serre. We present a short and uniform proof of this result using so-called good A1 subgroups of G, introduced by Seitz. We also formulate a counterpart of Korhonen’s theorem for overgroups of u which are finite groups of Lie type. Moreover, we generalize both results above by removing the restriction on the order of u under a mild condition on p depending on the rank of G, and we present an analogue of Korhonen’s theorem for Lie algebras.We are grateful to M. Korhonen and D. Testerman for helpful comments on an earlier version of the manuscript. The research of this work was supported in part by the DFG (Grant #RO 1072/22-1 (project number: 498503969) to G. Röhrle). Some of this work was completed during a visit to the Mathematisches Forschungsinstitut Oberwolfach: we thank them for their support. 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: Permutation Patterns
No full textThe study of permutation patterns has recently seen several surprising results, and the purpose of this mini-workshop was to bring together researchers from across the field to focus on four hot topics related to these recent developments. The topics covered the nature of generating functions that enumerate permutation classes, the structure of permutation classes and the impact this has on their growth rates, and the study of permutons, which lies at the interface of permutation patterns and discrete probability. The workshop offered an opportunity for knowledge exchange, but also time and space to initiate group collaborations on open problems related to these topics
Randomness is Natural - an Introduction to Regularisation by Noise
No full textDifferential equations make predictions on the future state of a system given the present. In order to get a sensible prediction, sometimes it is necessary to include randomness in differential equations, taking microscopic effects into account. Surprisingly, despite the presence of randomness, our probabilistic prediction of future states is stable with respect to changes in the surrounding environment, even if the original prediction was unstable. This snapshot will unveil the core mathematical mechanism underlying this "regularisation by noise" phenomenon
Uncertainty as an Ingredient in Financial Modeling
No full textUncertainty - as opposed to risk - is used to describe events to which we are not able to assign a probability due to lack of information. Instead of assigning a probability to an uncertain event, we only assume that such an event is possible or that its probability is within some range. We illustrate the effects of the inclusion of uncertainty in modeling by looking at simple cases of an optimal investment problem