Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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On a Cheeger Type Inequality in Cayley Graphs of Finite Groups
No full textLet be a finite group. It was remarked by Breuillard-Green-Guralnick-Tao that if the Cayley graph is an expander graph and is non-bipartite then the spectrum of the adjacency operator is bounded away from . In this article we are interested in explicit bounds for the spectrum of these graphs. Specifically, we show that the non-trivial spectrum of the adjacency operator lies in the interval , where denotes the (vertex) Cheeger constant of the regular graph with respect to a symmetric set of generators and
Counting self-avoiding walks on the hexagonal lattice
No full textIn how many ways can you go for a walk along a
lattice grid in such a way that you never meet your
own trail? In this snapshot, we describe some combinatorial
and statistical aspects of these so-called
self-avoiding walks. In particular, we discuss a recent
result concerning the number of self-avoiding walks
on the hexagonal (“honeycomb”) lattice. In the last
part, we briefly hint at the connection to the geometry
of long random self-avoiding walks
Arbeitsgemeinschaft: Zimmer's Conjecture
No full textThe aim of this Arbeitsgemeinschaft was to understand
the recent progress on Zimmer's conjecture in [1,2]. The week focuses
on the cocompact case from [1]
Large Scale Stochastic Dynamics
No full textThe goal of this workshop was to explore the recent advances in the
mathematical understanding of the macroscopic properties which emerge on large space-time scales from interacting microscopic particle systems. There were 55 participants,
including postdocs and graduate students, working in diverse
intertwining areas of probability and statistical mechanics. During
the meeting, 29 talks of 45 minutes were scheduled and an evening
session was organised with 10 more short talks of 10 minutes, mostly by younger participants.
These talks addressed the following topics :
randomness emerging from deterministic dynamics,
hydrodynamic limits, interface growth models and slow convergence to
equilibrium in kinetically
constrained dynamics
On the Lie Algebra Structure of of a Finite-Dimensional Algebra A
No full textLet be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the Ext-quiver of is a simple directed graph, then is a solvable Lie algebra. The second main result shows that if the Ext-quiver of has no loops and at most two parallel arrows in any direction, and if is a simple Lie algebra, then char(k) is not equal to and . The third result investigates symmetric algebras with a quiver which has a vertex with a single loop
Mini-Workshop: Self-adjoint Extensions in New Settings
No full textThe main focus of the workshop is on the analysis of boundary value problems for differential and difference operators
in some non-classical geometric settings, such as fractal graphs, sub-Riemannian manifolds or non-elliptic transmission problems.
Taking into account their importance in modern mathematical analysis, we aim at developing suitable tools in the operator theory to deal with the new problem settings
Nonlinear Acoustics
No full textNonlinear acoustics has been a topic of research for
more than 250 years. Driven by a wide range and a
large number of highly relevant industrial and medical
applications, this area has expanded enormously
in the last few decades. Here, we would like to give
a glimpse of the mathematical modeling techniques
that are commonly employed to tackle problems in
this area of research, with a selection of references
for the interested reader to further their knowledge
into this mathematically interesting field
A Quantitative Analysis of the “Lion-Man” Game
No full textIn this paper we analyze, based on an interplay between ideas and techniques from logic and geometric analysis, a pursuit-evasion game. More precisely, we focus on a discrete lion and man game with an -capture criterion. We prove that in uniformly convex bounded domains the lion always wins and, using ideas stemming from proof mining, we extract a uniform rate of
convergence for the successive distances between the lion and the man. As a byproduct of our analysis, we study the relation among different convexity properties in the setting of geodesic spaces
Nonlinear Evolution Equations: Analysis and Numerics
No full textThe qualitative theory of nonlinear evolution equations is an
important tool for studying the dynamical behavior of systems in
science and technology. A thorough understanding of the complex
behavior of such systems requires detailed analytical and numerical
investigations of the underlying partial differential equations
Singularities and Homological Aspects of Commutative Algebra
No full textCommutative algebra has recently witnessed a number of spectacular developments, resulting in the resolution of long-standing problems. The new techniques and perspectives, such as methods from the theory of perfectoid spaces, are leading to an extraordinary transformation in the field. There is also remarkable progress on the study of singularities in positive characteristics, and in particular on the problem of resolution of singularities. This workshop brought together researchers driving these developments with a broader group of young researchers in commutative algebra and allied fields, with the aim of spurring new collaborations and progress