$n$-Harmonic Mappings between Annuli

by Tadeusz Iwaniec

Author	: Tadeusz Iwaniec
Publisher	: American Mathematical Soc.
Total Pages	: 120
Release	: 2012
Genre	: Mathematics
ISBN	: 0821853570

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Iwaniec and Onninen (both mathematics, Syracuse U., US) address concrete questions regarding energy minimal deformations of annuli in Rn. One novelty of their approach is that they allow the mappings to slip freely along the boundaries of the domains, where it is most difficult to establish the existence, uniqueness, and invertibility properties of the extremal mappings. At the core of the matter, they say, is the underlying concept of free Lagrangians. After an introduction, they cover in turn principal radial n-harmonics, and the n-harmonic energy. There is no index. Annotation ©2012 Book News, Inc., Portland, OR (booknews.com).

An Introduction to the Theory of Higher-Dimensional Quasiconformal Mappings

by Frederick W. Gehring

Author	: Frederick W. Gehring
Publisher	: American Mathematical Soc.
Total Pages	: 442
Release	: 2017-05-03
Genre	: Mathematics
ISBN	: 0821843605

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This book offers a modern, up-to-date introduction to quasiconformal mappings from an explicitly geometric perspective, emphasizing both the extensive developments in mapping theory during the past few decades and the remarkable applications of geometric function theory to other fields, including dynamical systems, Kleinian groups, geometric topology, differential geometry, and geometric group theory. It is a careful and detailed introduction to the higher-dimensional theory of quasiconformal mappings from the geometric viewpoint, based primarily on the technique of the conformal modulus of a curve family. Notably, the final chapter describes the application of quasiconformal mapping theory to Mostow's celebrated rigidity theorem in its original context with all the necessary background. This book will be suitable as a textbook for graduate students and researchers interested in beginning to work on mapping theory problems or learning the basics of the geometric approach to quasiconformal mappings. Only a basic background in multidimensional real analysis is assumed.

Complex Analysis and Dynamical Systems III

by Mark Lʹvovich Agranovskiĭ

Author	: Mark Lʹvovich Agranovskiĭ
Publisher	: American Mathematical Soc.
Total Pages	: 482
Release	: 2008
Genre	: Mathematics
ISBN	: 0821841505

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The papers in this volume cover a wide variety of topics in the geometric theory of functions of one and several complex variables, including univalent functions, conformal and quasiconformal mappings, minimal surfaces, and dynamics in infinite-dimensional spaces. In addition, there are several articles dealing with various aspects of approximation theory and partial differential equations. Taken together, the articles collected here provide the reader with a panorama of activity in complex analysis, drawn by a number of leading figures in the field.

Modular Branching Rules for Projective Representations of Symmetric Groups and Lowering Operators for the Supergroup $Q(n)$

by Aleksandr Sergeevich Kleshchëv

Author	: Aleksandr Sergeevich Kleshchëv
Publisher	: American Mathematical Soc.
Total Pages	: 148
Release	: 2012
Genre	: Mathematics
ISBN	: 0821874314

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There are two approaches to projective representation theory of symmetric and alternating groups, which are powerful enough to work for modular representations. One is based on Sergeev duality, which connects projective representation theory of the symmetric group and representation theory of the algebraic supergroup $Q(n)$ via appropriate Schur (super)algebras and Schur functors. The second approach follows the work of Grojnowski for classical affine and cyclotomic Hecke algebras and connects projective representation theory of symmetric groups in characteristic $p$ to the crystal graph of the basic module of the twisted affine Kac-Moody algebra of type $A_{p-1}^{(2)}$. The goal of this work is to connect the two approaches mentioned above and to obtain new branching results for projective representations of symmetric groups.

The Kohn-Sham Equation for Deformed Crystals

by Weinan E

Author	: Weinan E
Publisher	: American Mathematical Soc.
Total Pages	: 109
Release	: 2013-01-25
Genre	: Mathematics
ISBN	: 0821875604

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The solution to the Kohn-Sham equation in the density functional theory of the quantum many-body problem is studied in the context of the electronic structure of smoothly deformed macroscopic crystals. An analog of the classical Cauchy-Born rule for crystal lattices is established for the electronic structure of the deformed crystal under the following physical conditions: (1) the band structure of the undeformed crystal has a gap, i.e. the crystal is an insulator, (2) the charge density waves are stable, and (3) the macroscopic dielectric tensor is positive definite. The effective equation governing the piezoelectric effect of a material is rigorously derived. Along the way, the authors also establish a number of fundamental properties of the Kohn-Sham map.

Pseudo-Differential Operators with Discontinuous Symbols: Widom's Conjecture

by Aleksandr Vladimirovich Sobolev

Author	: Aleksandr Vladimirovich Sobolev
Publisher	: American Mathematical Soc.
Total Pages	: 116
Release	: 2013-02-26
Genre	: Mathematics
ISBN	: 0821884875

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Relying on the known two-term quasiclassical asymptotic formula for the trace of the function $f(A)$ of a Wiener-Hopf type operator $A$ in dimension one, in 1982 H. Widom conjectured a multi-dimensional generalization of that formula for a pseudo-differential operator $A$ with a symbol $a(\mathbf{x}, \boldsymbol{\xi})$ having jump discontinuities in both variables. In 1990 he proved the conjecture for the special case when the jump in any of the two variables occurs on a hyperplane. The present paper provides a proof of Widom's Conjecture under the assumption that the symbol has jumps in both variables on arbitrary smooth bounded surfaces.

Potential Wadge Classes

by Dominique Lecomte

Author	: Dominique Lecomte
Publisher	: American Mathematical Soc.
Total Pages	: 95
Release	: 2013-01-25
Genre	: Mathematics
ISBN	: 0821875574

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Let $\bf\Gamma$ be a Borel class, or a Wadge class of Borel sets, and $2\!\leq\! d\!\leq\!\omega$ be a cardinal. A Borel subset $B$ of ${\mathbb R}^d$ is potentially in $\bf\Gamma$ if there is a finer Polish topology on $\mathbb R$ such that $B$ is in $\bf\Gamma$ when ${\mathbb R}^d$ is equipped with the new product topology. The author provides a way to recognize the sets potentially in $\bf\Gamma$ and applies this to the classes of graphs (oriented or not), quasi-orders and partial orders.

Characterization and Topological Rigidity of Nobeling Manifolds

by Andrzej Nagórko

Author	: Andrzej Nagórko
Publisher	: American Mathematical Soc.
Total Pages	: 106
Release	: 2013-04-22
Genre	: Mathematics
ISBN	: 082185366X

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The author develops a theory of Nobeling manifolds similar to the theory of Hilbert space manifolds. He shows that it reflects the theory of Menger manifolds developed by M. Bestvina and is its counterpart in the realm of complete spaces. In particular the author proves the Nobeling manifold characterization conjecture.

Character Identities in the Twisted Endoscopy of Real Reductive Groups

by Paul Mezo

Author	: Paul Mezo
Publisher	: American Mathematical Soc.
Total Pages	: 106
Release	: 2013-02-26
Genre	: Mathematics
ISBN	: 0821875655

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Suppose $G$ is a real reductive algebraic group, $\theta$ is an automorphism of $G$, and $\omega$ is a quasicharacter of the group of real points $G(\mathbf{R})$. Under some additional assumptions, the theory of twisted endoscopy associates to this triple real reductive groups $H$. The Local Langlands Correspondence partitions the admissible representations of $H(\mathbf{R})$ and $G(\mathbf{R})$ into $L$-packets. The author proves twisted character identities between $L$-packets of $H(\mathbf{R})$ and $G(\mathbf{R})$ comprised of essential discrete series or limits of discrete series.