Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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2063 research outputs found
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Cauchy Completeness, Lax Epimorphisms and Effective Descent for Split Fibrations
No full textFor any suitable base category , we find that -fully faithful lax epimorphisms in - are precisely those -functors whose induced -functors between the Cauchy completions are equivalences. For the case , this is equivalent to requiring that the induced functor between the categories of split (op)fibrations is an equivalence.
By reducing the study of effective descent functors with respect to the indexed category of split (op)fibrations to the study of the codescent factorization, we find that these observations on fully faithful lax epimorphisms provide us with a characterization of (effective) -descent morphisms in the category of small categories ; namely, we find that they are precisely the (effective) descent morphisms with respect to the indexed categories of discrete opfibrations -- previously studied by Sobral. We include some comments on the Beck-Chevalley condition and future work
MFO-RIMS Tandem Workshop 2023: Arithmetic Homotopy and Galois Theory
No full textThis report presents a general panorama of recent progress in the arithmetic-geometry theory of Galois and homotopy groups and its ramifications. While still relying on Grothendieck's original pillars, the present program has now evolved beyond the classical group-theoretic legacy to result in an autonomous project that exploits a new geometrization of the original insight and sketches new frontiers between homotopy geometry, homology geometry, and diophantine geometry. This panorama "closes the loop'' by including the last twenty-year progress of the Japanese arithmetic-geometry school via Ihara's program and Nakamura-Tamagawa-Mochizuki's anabelian approach, which brings its expertise in terms of algorithmic, combinatoric, and absolute reconstructions. These methods supplement and interact with those from the classical arithmetic of covers and Hurwitz spaces and the motivic and geometric Galois representations. This workshop has brought together the next generation of arithmetic homotopic Galois geometers, who, with the support of senior experts, are developing new techniques and principles for the exploration of the next research frontiers
Design and Analysis of Infectious Disease Studies
No full textThis was the sixth workshop on mathematical and statistical methods for the transmission of infectious diseases. Building on epidemiologic models which were the subject of earlier workshops, this workshop concentrated on disentangling who infected whom by analysing high-resolution genomic data of pathogens which are routinely collected during outbreaks. Following the trail of the small mutations which continuously occur in different places of pathogens' genomes, mathematical tools and computational algorithms were used to reconstruct transmission trees and contact networks. In the past three years these methods were developed and used particularly in the context of the SARS-Cov-2 (Covid-19) pandemic
Mini-Workshop: Poisson and Poisson-type algebras
No full textThe first historical encounter with Poisson-type algebras is with Hamiltonian mechanics. With the abstraction of many notions in Physics, Hamiltonian systems were geometrized into manifolds that model the set of all possible configurations of the system, and the cotangent bundle of this manifold describes its phase space, which is endowed with a Poisson structure. Poisson brackets led to other algebraic structures, and the notion of Poisson-type algebra arose, including transposed Poisson algebras, Novikov-Poisson algebras, or commutative pre-Lie algebras, for example. These types of algebras have long gained popularity in the scientific world and are not only of their own interest to study, but are also an important tool for researching other mathematical and physical objects
Variational Methods for Evolution
No full textVariational principles for evolutionary systems arise in many settings, both in those describing
the physical world and in man-made algorithms for data science and optimization tasks.
Variational principles are available for Hamiltonian systems
in classical mechanics, gradient flows for dissipative systems, as well as in time-incremental minimization techniques for more general
evolutionary problems. Additional challenges arise via the interplay of two or more functionals (e.g. a free energy and a dissipation potential), thus encompassing a large variety of applications in the modeling of materials and fluids, in biology, and in multi-agent systems.
Variational principles and associated evolutions are also at the core of the modern approaches to machine learning tasks, since many of them are posed as minimizing an objective functional that models the problem. The discrete and random nature of these problems and the need for accurate computation in high dimension present a set of challenges that require new mathematical insights.
Variational methods for evolution allow for the
usage of the rich toolbox provided by the calculus of
variations, metric-space geometry, partial differential equations, and other branches of applied analysis.
The variational methods for evolution have seen a rapid growth over
the last two decades. This workshop continued the
successful line of meetings (2011, 2014, 2017, and 2020), while evolving and
branching into new directions. We have brought together a wide scope of
mathematical researchers from
calculus of variations, partial differential equations, numerical analysis, and
stochastics, as well as researchers from data science and machine learning, to exchange ideas, foster interaction, develop new avenues, and generally bring these communities closer together
Skew Braces and the Yang-Baxter Equation
No full textThe workshop was focused on the interplay between set-theoretic solutions to the Yang-Baxter equation and several
algebraic structures
used to construct and understand new solutions.
In this vein, the YBE and properties of these algebraic
structures are used as a source of inspiration
to study other mathematical problems not directly
related to the YBE
Homotopy Theory
No full textThe workshop brought together experts in homotopy theory from many areas, including chromatic homotopy theory, algebraic K-theory, derived algebra and equivariant homotopy theory. We had a lecture series on the recent disproof of the telescope conjecture. In addition to a total of 24 research talks we had two gong-shows where participants presented their research in
10-minute talks
Morphisms in Low Dimensions
No full textThis workshop brought together experts on interrelated topics in low-dimensional topology, centred around the common theme of 'morphisms'. Our goal was to improve community understanding of recent developments in the field and to promote new advances in the study of global properties of 4-manifolds
MATRIX-MFO Tandem Workshop: Stochastic Reinforcement Processes and Graphs
No full textStochastic processes with reinforcement are the central theme of the present tandem workshop. We assembled a diverse group of international experts that worked on reinforcement dynamics from several different perspectives. We discussed progress and future strategies around a number of key open problems in the area of interacting urns with graph based interaction, preferential attachment, and reinforced random walks
Bochner-Riesz Means at the Critical Index: Weighted and Sparse Bounds
No full textWe consider Bochner-Riesz means on weighted spaces, at the critical index . For every -weight we obtain an extension of Vargas' weak type inequality in some range of . To prove this result we establish new endpoint results for sparse domination. These are almost optimal in dimension ; partial results as well as conditional results are proved in higher dimensions. For the means of index we prove fully optimal sparse bounds.This research was supported through the program Oberwolfach Research Fellows by Mathematisches Forschungsinstitut Oberwolfach in 2023.
The authors were supported in part by National Science Foundation grants DMS-1954479 (D.B.), DMS-2154835 (J.R.), DMS-2054220 (A.S.), and by the AEI grants RYC2020-029151-I and PID2022-140977NA-I00 (D.B.)