Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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Exact Rate of Convergence of k-Nearest-Neighbor Classification Rule
No full textResearch in Pairs 2017A binary classification problem is considered. The excess error probability of the k-nearest neighbor classification rule according to the error probability of the Bayes decision is revisited by a decomposition of the excess error probability into approximation and estimation error. Under a weak margin condition and under a modified Lipschitz condition, tight upper bounds are presented such that one avoids the condition that the feature vector is bounded
Set Theory
No full textThis workshop included selected talks on pure set theory and its applications, simultaneously showing diversity and coherence of the subject
Stochastic Analysis: Geometry of Random Processes
No full textA common feature shared by many natural objects arising in probability theory is that they tend to be very “rough”, as opposed to the “smooth” objects usually studied in other branches of mathematics. It is however still desirable to understand their geometric properties, be it from a metric, a topological, or a measure-theoretic perspective. In recent years, our understanding of such “random geometries” has seen spectacular advances on a number of fronts
Applications of Optimal Transportation in the Natural Sciences
No full textThe aim of this workshop was to gather a mixed group of experts and young researchers from different areas of applied mathematics in which optimal transport plays a central role. Applications in one of the classical areas of natural sciences, like physics, chemistry and (mathematical) biology were the main focus of the workshop
Proof Complexity and Beyond
No full textProof complexity is a multi-disciplinary intellectual endeavor that addresses questions of the general form “how difficult is it to prove certain mathematical facts?” The current workshop focused on recent advances in our understanding of logic-based proof systems and on connections to algorithms, geometry and combinatorics research, such as the analysis of approximation algorithms, or the size of linear or semidefinite programming formulations of combinatorial optimization problems, to name just two important examples
On Unipotent Radicals of Pseudo-Reductive Groups
No full textMSC: 20G15Research in Pairs 2015We establish some results on the structure of the geometric unipotent
radicals of pseudo-reductive k-groups. In particular, let be a purely
inseparable field extension of k of degree and let denote the Weil
restriction of scalars of a reductive -group . We prove that
the unipotent radical of the extension of scalars of to the
algebraic closure of has exponent . Our main theorem is to give
bounds on the nilpotency class of geometric unipotent radicals of standard
pseudo-reductive groups, which are sharp in many cases
The Varchenko Determinant of a Coxeter Arrangement
No full textOWLF 2017The Varchenko determinant is the determinant of a matrix defined from an arrangement of hyperplanes. Varchenko proved that this determinant has a beautiful factorization. It is, however, not possible to use this factorization to compute a Varchenko determinant from a certain level of complexity. Precisely at this point, we provide an explicit formula of this determinant for the hyperplane arrangements associated to the finite Coxeter groups. The intersections of hyperplanes with the chambers of such arrangements have nice properties which play a central role for the calculation of their relating determinants
Combinatorics
No full textCombinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization, Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions. This is a report on the meeting, containing abstracts of the presentations and a summary of the problem session
Composition of Irreducible Morphisms in Coils
No full textResearch in Pairs 2017We study the non-zero composition of n irreducible morphisms between modules lying in coils in relation with the powers of the radical of their module category
Z2-Thurston Norm and Complexity of 3-Manifolds, II
No full textResearch in Pairs 2017In this sequel to earlier papers by three of the authors, we obtain a new bound on the complexity of a closed 3-manifold, as well as a characterisation of manifolds realising our complexity bounds. As an application, we obtain the first infinite families of minimal triangulations of Seifert fibred spaces modelled on Thurston's geometry $\widetilde{\text{SL}_2(\mathbb{R})}.