Hartl and the author, and in the past years studied by many people. It generalizes the known construction for the usual affine Grassmannians, and makes sense for arbitrary symmetric Kac—Mody algebras. Abstract: The study of affine Deligne—Lusztig varieties originally arose from arithmetic geometry, but many problems on affine Deligne—Lusztig varieties are purely Lie-theoretic in nature. This survey deals with recent progress on several important problems on affine Deligne—Lusztig varieties.

The emphasis is on the Lie-theoretic aspect, while some connections and applications to arithmetic geometry will also be mentioned. Analysis and Operator Algebras Classical analysis. Real and Complex analysis in one and several variables, potential theory, quasiconformal mappings. Harmonic analysis.

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Linear and non-linear functional analysis, operator algebras, Banach algebras, Banach spaces. Non-commutative geometry, spectra of random matrices. Asymptotic geometric analysis. Metric geometry and applications. Geometric measure theory. Connections with sections 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, Abstract: The aim of this talk is to present some recent results on the structure of the singular part of measures satisfying a PDE constraint and to describe some applications in Geometric Measure Theory, in the Calculus of Variations and in real Analysis. Abstract: We present recent progress in theory of local conformal nets which is an operator algebraic approach to study chiral conformal field theory.

We emphasize representation theoretic aspects and relations to theory of vertex operator algebras which gives a different and algebraic formulation of chiral conformal field theory.

## Degree in Mathematics

Abstract: We survey some of the progress made recently in the classification of von Neumann algebras arising from countable groups and their measure preserving actions on probability spaces. We also present cocycle superrigidity theorems and some of their applications to orbit equivalence. Finally, we discuss several recent rigidity results for von Neumann algebras associated to groups.

Abstract: We shall begin by briefly reviewing a compactification of the Markov space for an infinite transition matrix, introduced by Marcelo Laca and the speaker roughly 20 years ago. Given a continuous potential we will then consider the problem of characterizing the conformal measures on that space. An example will be presented to show that conformal measures may live in the complement of the standard Markov space, hence being invisible to the standard theory.

In the context of the Markov shifts mentioned above we will then explore the connections between conformal and DLR measures.

We also introduce a new class of embeddings of general weighted planar graphs s-embeddings , which might, in particular, pave the way to true universality results for the planar Ising model. Abstract: The famous Banach—Tarski paradox and Hilbert's third problem are part of story of paradoxical equidecompositions and invariant finitely additive measures.

We review some of the classical results in this area including Laczkovich's solution to Tarski's circle-squaring problem: the disc of unit area can be cut into finitely many pieces that can be rearranged by translations to form the unit square. Abstract: The Hilbert transform is an average of dyadic shift operators.

### Course Descriptions for (MATH) 5xx and 7xxl

These can be seen as a coefficient shift and multiplier in a Haar wavelet expansion or as a time shifted operator on martingale differences. This observation and its generalisations are useful in deep characterisations of multi-parameter BMO spaces through commutators of multiplication by a BMO symbol with singular operators.

The connection between martingale transforms and Calderon—Zygmund operators had been understood for some time. It allows us to develop powerful tools with a probabilistic flavour to obtain deep results central to harmonic analysis. In the case of the Beurling—Ahlfors operator, a connection was particularly direct. Through the first sharp weighted estimate of a singular integral operator we showed that every weakly quasiregular map in the plane is quasiregular.

In other words, we established a borderline regularity result for the Beltrami equation. The central conjecture in sharp weighted theory was on the Hilbert transform though. Its first solution involved the precise model of the dyadic shift.

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Since then, weighted theory evolved through many outstanding contributions, giving deep insight into the nature of singular operators. Abstract: Positive closed currents, the analytic counterpart of effective cycles in algebraic geometry, are central objects in pluripotential theory. They were introduced in complex dynamics in the s and become now a powerful tool in the field. Challenging dynamical problems involve currents of any dimension. We will report recent developments on positive closed currents of arbitrary dimension, including the solutions to the regularization problem, the theory of super-potentials and the theory of densities.

Applications to dynamics such as properties of dynamical invariants e. It is an extrinsic geometric quantity that is invariant under global reparameterization of a surface and provide a natural extension of the classical mean curvature.

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We describe some properties of the NMC and the quasilinear differential operators that are involved when it acts on graphs. We also survey recent results on surfaces having constant NMC and describe their intimate link with some problems arising in the study of overdetermined boundary value problems. Abstract: Sofic entropy theory is a generalization of the classical Kolmogorov—Sinai entropy theory to actions of a large class of non-amenable groups called sofic groups. This is a short introduction with a guide to the literature.

Some are described herein. Abstract: We survey recent progress in the gap and type problems of Fourier analysis obtained via the use of Toeplitz operators in spaces of holomorphic functions. We discuss applications of such methods to spectral problems for differential operators. Abstract: This note describes the impact of disorder or irregularities in the ambient medium on the behavior of stationary solutions to elliptic partial differential equations and on spatial distribution of eigenfunctions, as well as the profound and somewhat surprising connections between these two topics which have been revealed in the past few years.

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## Geometry of Differential Equations: A Concise Introduction | SpringerLink

I will in particular focus on the role of quasidiagonality and amenability for classification, and on the regularity conjecture and its interplay with internal and external approximation properties. Geometric and qualitative theory of ordinary differential equations and smooth dynamical systems. Bifurcations and singularities.

Hamiltonian systems and dynamical systems of geometric origin.

## Glossary of areas of mathematics

One-dimensional and holomorphic dynamics. Strange attractors and chaotic dynamics. Lyapunov exponents. Multidimensional actions and rigidity in dynamics. Ergodic theory including applications to combinatorics and combinatorial number theory. Infinite dimensional dynamical systems and partial differential equations. Connections with sections 5, 7, 8, 10, 11, 12, 13, 15, We present some relations with Hausdorff dimension and measures with refined gauge functions of limit sets for geometric coding trees for rational functions on the Riemann sphere.

Abstract: We propose in these notes a list of some old and new questions related to quasi-periodic dynamics. A main aspect of quasi-periodic dynamics is the crucial influence of arithmetics on the dynamical features, with a strong duality in general between Diophantine and Liouville behavior. We will discuss rigidity and stability in Diophantine dynamics as well as their absence in Liouville ones. Beyond this classical dichotomy between the Diophantine and the Liouville worlds, we discuss some unified approaches and some phenomena that are valid in both worlds.

Our focus is mainly on low dimensional dynamics such as circle diffeomorphisms, disc dynamics, quasi-periodic cocycles, or surface flows, as well as finite dimensional Hamiltonian systems. In an opposite direction, the study of the dynamical properties of some diagonal and unipotent actions on the space of lattices can be applied to arithmetics, namely to the theory of Diophantine approximations.

We will mention in the last section some problems related to that topic. The field of quasi-periodic dynamics is very extensive and has a wide range of interactions with other mathematical domains. The list of questions we propose is naturally far from exhaustive and our choice was often motivated by our research involvements.

Abstract: I will explain a notion of arithmetic equidistribution that has found application in the study of complex dynamical systems. It was first introduced about 20 years ago, by Szpiro—Ullmo—Zhang, to analyze the geometry and arithmetic of abelian varieties.

In , Matt Baker and I used the theory to show when two complex rational functions have only finitely many pre- periodic points in common.

### Institut Teknologi Bandung

In this talk, I will explain how to obtain uniform bounds in families, and I will show how to use dynamical methods to return to the setting of abelian varieties. Abstract: The Ruelle resonances of a dynamical system are spectral characteristics of a system, describing the precise asymptotics of correlations.

While their existence can often be shown by abstract spectral analysis arguments, it is in general not possible to compute them exactly. After explaining the general background, illustrated by classical examples, I will focus on a specific example: in the case of linear pseudo-Anosov maps, Ruelle resonances can be completely described in terms of the action of the map on cohomology. Abstract: We report on recent results about the dimension and smoothness properties of self-similar sets and measures.