Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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2063 research outputs found
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Mini-Workshop: Mathematical Aspects of Nonlinear Wave Propagation in Solid Mechanics
No full textNonlinear elastodynamics sets a plethora of challenging mathematical problems such as those concerning wave propagation in solids.
Elastic vibrations and acoustic waves have been widely studied because of their applications in nondestructive tests of materials and structures, and, in recent times, several novel aspects of the theory of wave propagation in solids have blossomed thanks to the introduction of metamaterials and new technological devices.
The goal of this workshop was to bring together researchers with different backgrounds to discuss recent advances, and to stimulate future work
Modular Forms
No full textThe theory of Modular Forms has been central in mathematics with a
rich history and connections to many other areas of mathematics. The
workshop explored recent developments and future directions with a
particular focus on connections to the theory of periods
Curvatura escalar positiva y aplicaciones
No full textWe introduce the idea of curvature, including how it developed historically, and focus on the scalar curvature of a manifold. A major current research topic involves understanding positive scalar curvature. We discuss why this is interesting and how it relates to general relativity.
[also available in Spanish][Also available in Spanish
Algebraic K-theory
No full textAlgebraic -theory has seen a fruitful development during the last three
years. Part of this recent progress was driven by the use of -categories and
related techniques originally developed in algebraic topology. On the other hand we have
seen continuing progress based on motivic homotopy theory which has been an important
theme in relation to
algebraic -theory for twenty years
On Residuals of Finite Groups
No full textA theorem of Dolfi, Herzog, Kaplan, and Lev [DHKL07, Thm. C] asserts that in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order, and that the inequality is sharp. Inspired by this result and some of the arguments in [DHKL07], we establish the following generalisation: if is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and is the extension-closure of , then there exists an (optimal) constant depending only on such that, for all non-trivial finite groups G with trivial -radical, , where is the -residual of . When , the class of finite nilpotent groups, it follows that , the class of finite soluble groups, thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J. G. Thompson's classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations X of full characteristic such that , thus providing applications of our main result beyond the reach of [DHKL07, Thm. C
Mini-Workshop: Lorentz Gas Dynamics: particle systems and scaling limits
No full textThis workshop had the aim to bring together leading experts from different fields, i.e. interacting particle systems, dynamical systems and kinetic theory, to consider
questions related to the dynamics of the Lorentz gas and to promote the exchange of information concerning the techniques that have been developed in different contexts and communities
The Mathematics of Fluids and Solids
No full textFluid-structure interaction is a rich and active field
of mathematics that studies the interaction between
fluids and solid objects. In this short article, we give
a glimpse into this exciting field, as well as a sample
of the most significant questions that mathematicians
try to answer
Random Matrices
No full textLarge complex systems tend to develop universal patterns that often represent their essential characteristics. For example, the cumulative effects of independent or weakly dependent random variables often yield the Gaussian universality class via the central limit theorem.
For non-commutative random variables, e.g. matrices, the Gaussian behavior is often replaced by another universality class, commonly called random matrix statistics. Nearby eigenvalues are strongly correlated, and, remarkably, their correlation structure is universal, depending only on the symmetry type of the matrix. Even more surprisingly, this feature is not restricted to matrices; in fact Eugene Wigner, the pioneer of the field, discovered in the 1950s that distributions of the gaps between energy levels of complicated quantum systems universally follow the same random matrix statistics. This
claim has never been rigorously proved for any realistic physical system but experimental data and extensive numerics leave no doubt as to its correctness. Since then random matrices have proved to be extremely useful phenomenological models in a wide range of applications beyond quantum physics that include number theory, statistics, neuroscience, population dynamics, wireless communication and mathematical finance.
The ubiquity of random matrices in natural sciences is still a mystery, but recent years have witnessed a breakthrough in the mathematical description of the statistical structure of their spectrum. Random matrices and closely related areas such as log-gases have become an extremely active research area in probability theory.
This workshop brought together outstanding researchers from a variety of mathematical
backgrounds whose areas of research are linked to random matrices. While there are strong links between their motivations, the techniques
used by these researchers span a large swath of mathematics, ranging from purely algebraic techniques to
stochastic analysis, classical probability theory, operator algebra, supersymmetry,
orthogonal polynomials, etc
Arithmetic of Shimura Varieties
No full textArithmetic properties of Shimura varieties are an exciting topic which has roots in classical
topics of algebraic geometry and of number theory such as modular curves and modular forms. This very active research field has contributed to some of the most spectacular developments in
number theory and arithmetic geometry in the last twenty years. Shimura varieties and their equal characteristic analogue, moduli spaces of shtukas, are closely related to the Langlands program (classical as
well as -adic). A particular case is given by moduli spaces of abelian
varieties, a classical object of study in algebraic geometry
Some Results Related to Schiffer's Problem
No full textWe consider the following semilinear overdetermined problem on a two dimensional bounded or unbounded domain with analytic boundary having at least one bounded connected component \begin{eqnarray*} \left\{ \begin{array}{l} - \Delta u = g(u) \quad \hbox{in } \Omega,\\ \frac{\partial u}{\partial \nu} =0 \, \hbox{ and } \, u = c \hbox{ on } \partial \Omega, \end{array} \right. \end{eqnarray*} where is a constant. When the constant solution is the unique solution. For , we show that the boundary is a circle if and only if the problem admits a solution that has constant third or fourth normal derivative along the boundary. A similar result involving the fifth normal derivative is proved