Lectures On The Arthur Selberg Trace Formula

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Lectures on the Arthur-Selberg Trace Formula

Author : Stephen S. Gelbart
Publisher : American Mathematical Soc.
ISBN 13 : 0821805711
Total Pages : 112 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Lectures on the Arthur-Selberg Trace Formula by : Stephen S. Gelbart

Download or read book Lectures on the Arthur-Selberg Trace Formula written by Stephen S. Gelbart and published by American Mathematical Soc.. This book was released on 1996 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Arthur-Selberg trace formula is an equality between two kinds of traces: the geometric terms given by the conjugacy classes of a group and the spectral terms given by the induced representations. In general, these terms require a truncation in order to converge, which leads to an equality of truncated kernels. The formulas are difficult in general and even the case of $GL$(2) is nontrivial. The book gives proof of Arthur's trace formula of the 1970s and 1980s, with special attention given to $GL$(2). The problem is that when the truncated terms converge, they are also shown to be polynomial in the truncation variable and expressed as ``weighted'' orbital and ``weighted'' characters. In some important cases the trace formula takes on a simple form over $G$. The author gives some examples of this, and also some examples of Jacquet's relative trace formula. This work offers for the first time a simultaneous treatment of a general group with the case of $GL$(2). It also treats the trace formula with the example of Jacquet's relative formula. Features: Discusses why the terms of the geometric and spectral type must be truncated, and why the resulting truncations are polynomials in the truncation of value $T$. Brings into play the significant tool of ($G, M$) families and how the theory of Paley-Weiner is applied. Explains why the truncation formula reduces to a simple formula involving only the elliptic terms on the geometric sides with the representations appearing cuspidally on the spectral side (applies to Tamagawa numbers). Outlines Jacquet's trace formula and shows how it works for $GL$(2).

The Selberg-Arthur Trace Formula

Author : Salahoddin Shokranian
Publisher : Springer
ISBN 13 : 3540466592
Total Pages : 104 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis The Selberg-Arthur Trace Formula by : Salahoddin Shokranian

Download or read book The Selberg-Arthur Trace Formula written by Salahoddin Shokranian and published by Springer. This book was released on 2006-11-14 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book based on lectures given by James Arthur discusses the trace formula of Selberg and Arthur. The emphasis is laid on Arthur's trace formula for GL(r), with several examples in order to illustrate the basic concepts. The book will be useful and stimulating reading for graduate students in automorphic forms, analytic number theory, and non-commutative harmonic analysis, as well as researchers in these fields. Contents: I. Number Theory and Automorphic Representations.1.1. Some problems in classical number theory, 1.2. Modular forms and automorphic representations; II. Selberg's Trace Formula 2.1. Historical Remarks, 2.2. Orbital integrals and Selberg's trace formula, 2.3.Three examples, 2.4. A necessary condition, 2.5. Generalizations and applications; III. Kernel Functions and the Convergence Theorem, 3.1. Preliminaries on GL(r), 3.2. Combinatorics and reduction theory, 3.3. The convergence theorem; IV. The Ad lic Theory, 4.1. Basic facts; V. The Geometric Theory, 5.1. The JTO(f) and JT(f) distributions, 5.2. A geometric I-function, 5.3. The weight functions; VI. The Geometric Expansionof the Trace Formula, 6.1. Weighted orbital integrals, 6.2. The unipotent distribution; VII. The Spectral Theory, 7.1. A review of the Eisenstein series, 7.2. Cusp forms, truncation, the trace formula; VIII.The Invariant Trace Formula and its Applications, 8.1. The invariant trace formula for GL(r), 8.2. Applications and remarks

Harmonic Analysis, the Trace Formula, and Shimura Varieties

Author : Clay Mathematics Institute. Summer School
Publisher : American Mathematical Soc.
ISBN 13 : 9780821838440
Total Pages : 708 pages
Book Rating : 4.8/5 (384 download)

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Book Synopsis Harmonic Analysis, the Trace Formula, and Shimura Varieties by : Clay Mathematics Institute. Summer School

Download or read book Harmonic Analysis, the Trace Formula, and Shimura Varieties written by Clay Mathematics Institute. Summer School and published by American Mathematical Soc.. This book was released on 2005 with total page 708 pages. Available in PDF, EPUB and Kindle. Book excerpt: Langlands program proposes fundamental relations that tie arithmetic information from number theory and algebraic geometry with analytic information from harmonic analysis and group representations. This title intends to provide an entry point into this exciting and challenging field.

An Introduction to the Langlands Program

Author : Joseph Bernstein
Publisher : Springer Science & Business Media
ISBN 13 : 0817682260
Total Pages : 283 pages
Book Rating : 4.8/5 (176 download)

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Book Synopsis An Introduction to the Langlands Program by : Joseph Bernstein

Download or read book An Introduction to the Langlands Program written by Joseph Bernstein and published by Springer Science & Business Media. This book was released on 2013-12-11 with total page 283 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a broad, user-friendly introduction to the Langlands program, that is, the theory of automorphic forms and its connection with the theory of L-functions and other fields of mathematics. Each of the twelve chapters focuses on a particular topic devoted to special cases of the program. The book is suitable for graduate students and researchers.

Lectures on Harmonic Analysis

Author : Thomas H. Wolff
Publisher : American Mathematical Soc.
ISBN 13 : 0821834495
Total Pages : 154 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Lectures on Harmonic Analysis by : Thomas H. Wolff

Download or read book Lectures on Harmonic Analysis written by Thomas H. Wolff and published by American Mathematical Soc.. This book was released on 2003-09-17 with total page 154 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book demonstrates how harmonic analysis can provide penetrating insights into deep aspects of modern analysis. It is both an introduction to the subject as a whole and an overview of those branches of harmonic analysis that are relevant to the Kakeya conjecture. The usual background material is covered in the first few chapters: the Fourier transform, convolution, the inversion theorem, the uncertainty principle and the method of stationary phase. However, the choice of topics is highly selective, with emphasis on those frequently used in research inspired by the problems discussed in the later chapters. These include questions related to the restriction conjecture and the Kakeya conjecture, distance sets, and Fourier transforms of singular measures. These problems are diverse, but often interconnected; they all combine sophisticated Fourier analysis with intriguing links to other areas of mathematics and they continue to stimulate first-rate work. The book focuses on laying out a solid foundation for further reading and research. Technicalities are kept to a minimum, and simpler but more basic methods are often favored over the most recent methods. The clear style of the exposition and the quick progression from fundamentals to advanced topics ensures that both graduate students and research mathematicians will benefit from the book.

On The Langlands Program: Endoscopy And Beyond

Author : Wee Teck Gan
Publisher : World Scientific
ISBN 13 : 9811285837
Total Pages : 449 pages
Book Rating : 4.8/5 (112 download)

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Book Synopsis On The Langlands Program: Endoscopy And Beyond by : Wee Teck Gan

Download or read book On The Langlands Program: Endoscopy And Beyond written by Wee Teck Gan and published by World Scientific. This book was released on 2024-04-15 with total page 449 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a collection of lecture notes from the minicourses in the December 2018 Langlands Workshop: Endoscopy and Beyond. The volume combines seven introductory chapters on trace formulas, local Arthur packets, and beyond endoscopy. It aims to introduce the endoscopy classification via a basic example of the trace formula for SL(2), explore the more refined questions on the structure of Arthur packets, and look beyond endoscopy following the suggestions of Langlands, Braverman-Kazhdan, Ngo, and Altuğ. The book is a helpful reference for undergraduates, postgraduates, and researchers.

Lectures on Tensor Categories and Modular Functors

Author : Bojko Bakalov
Publisher : American Mathematical Soc.
ISBN 13 : 0821826867
Total Pages : 232 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Lectures on Tensor Categories and Modular Functors by : Bojko Bakalov

Download or read book Lectures on Tensor Categories and Modular Functors written by Bojko Bakalov and published by American Mathematical Soc.. This book was released on 2001 with total page 232 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives an exposition of the relations among the following three topics: monoidal tensor categories (such as a category of representations of a quantum group), 3-dimensional topological quantum field theory, and 2-dimensional modular functors (which naturally arise in 2-dimensional conformal field theory). The following examples are discussed in detail: the category of representations of a quantum group at a root of unity and the Wess-Zumino-Witten modular functor. The idea that these topics are related first appeared in the physics literature in the study of quantum field theory. Pioneering works of Witten and Moore-Seiberg triggered an avalanche of papers, both physical and mathematical, exploring various aspects of these relations. Upon preparing to lecture on the topic at MIT, however, the authors discovered that the existing literature was difficult and that there were gaps to fill. The text is wholly expository and finely succinct. It gathers results, fills existing gaps, and simplifies some proofs. The book makes an important addition to the existing literature on the topic. It would be suitable as a course text at the advanced-graduate level.

Lectures on Hilbert Schemes of Points on Surfaces

Author : Hiraku Nakajima
Publisher : American Mathematical Soc.
ISBN 13 : 0821819569
Total Pages : 146 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Lectures on Hilbert Schemes of Points on Surfaces by : Hiraku Nakajima

Download or read book Lectures on Hilbert Schemes of Points on Surfaces written by Hiraku Nakajima and published by American Mathematical Soc.. This book was released on 1999 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt: It has been realized that Hilbert schemes originally studied in algebraic geometry are closely related to several branches of mathematics, such as singularities, symplectic geometry, representation theory - even theoretical physics. This book reflects this feature of Hilbert schemes.

Three Lectures on Commutative Algebra

Author : Holger Brenner
Publisher : American Mathematical Soc.
ISBN 13 : 0821844342
Total Pages : 202 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Three Lectures on Commutative Algebra by : Holger Brenner

Download or read book Three Lectures on Commutative Algebra written by Holger Brenner and published by American Mathematical Soc.. This book was released on 2008 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt: These lectures provides detailed introductions to some of the latest advances in three significant areas of rapid development in commutative algebra and its applications: tight closure and vector bundles; combinatorics and commutative algebra; constructive desingularization."

Introduction To Non-abelian Class Field Theory, An: Automorphic Forms Of Weight 1 And 2-dimensional Galois Representations

Author : Toyokazu Hiramatsu
Publisher : World Scientific
ISBN 13 : 9813142286
Total Pages : 188 pages
Book Rating : 4.8/5 (131 download)

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Book Synopsis Introduction To Non-abelian Class Field Theory, An: Automorphic Forms Of Weight 1 And 2-dimensional Galois Representations by : Toyokazu Hiramatsu

Download or read book Introduction To Non-abelian Class Field Theory, An: Automorphic Forms Of Weight 1 And 2-dimensional Galois Representations written by Toyokazu Hiramatsu and published by World Scientific. This book was released on 2016-09-13 with total page 188 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph provides a brief exposition of automorphic forms of weight 1 and their applications to arithmetic, especially to Galois representations. One of the outstanding problems in arithmetic is a generalization of class field theory to non-abelian Galois extension of number fields. In this volume, we discuss some relations between this problem and cusp forms of weight 1.

Representation Theory and Automorphic Forms

Author : T. N. Bailey
Publisher : American Mathematical Soc.
ISBN 13 : 0821806092
Total Pages : 490 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Representation Theory and Automorphic Forms by : T. N. Bailey

Download or read book Representation Theory and Automorphic Forms written by T. N. Bailey and published by American Mathematical Soc.. This book was released on 1997 with total page 490 pages. Available in PDF, EPUB and Kindle. Book excerpt: The lectures from a course in the representation theory of semi- simple groups, automorphic forms, and the relations between them. The purpose is to help analysts make systematic use of Lie groups in work on harmonic analysis, differential equations, and mathematical physics; and to provide number theorists with the representation-theoretic input to Wiles's proof of Fermat's Last Theorem. Begins with an introductory treatment of structure theory and ends with the current status of functionality. Annotation copyrighted by Book News, Inc., Portland, OR

Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations

Author : Evgeniĭ Borisovich Dynkin
Publisher : American Mathematical Soc.
ISBN 13 : 082183682X
Total Pages : 130 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations by : Evgeniĭ Borisovich Dynkin

Download or read book Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations written by Evgeniĭ Borisovich Dynkin and published by American Mathematical Soc.. This book was released on 2004 with total page 130 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to the applications of probability theory to the theory of nonlinear partial differential equations. More precisely, it is shown that all positive solutions for a class of nonlinear elliptic equations in a domain are described in terms of their traces on the boundary of the domain. The main probabilistic tool is the theory of superdiffusions, which describes a random evolution of a cloud of particles. A substantial enhancement of this theory is presented that will be of interest to anyone who works on applications of probabilistic methods to mathematical analysis. The book is suitable for graduate students and research mathematicians interested in probability theory and its applications to differential equations. Also of interest by this author is Diffusions, Superdiffusions and Partial Differential Equations in the AMS series, Colloquium Publications.

Oscillating Patterns in Image Processing and Nonlinear Evolution Equations

Author : Yves Meyer
Publisher : American Mathematical Soc.
ISBN 13 : 9780821829202
Total Pages : 138 pages
Book Rating : 4.8/5 (292 download)

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Book Synopsis Oscillating Patterns in Image Processing and Nonlinear Evolution Equations by : Yves Meyer

Download or read book Oscillating Patterns in Image Processing and Nonlinear Evolution Equations written by Yves Meyer and published by American Mathematical Soc.. This book was released on 2001 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: Image compression, the Navier-Stokes equations, and detection of gravitational waves are three seemingly unrelated scientific problems that, remarkably, can be studied from one perspective. The notion that unifies the three problems is that of ``oscillating patterns'', which are present in many natural images, help to explain nonlinear equations, and are pivotal in studying chirps and frequency-modulated signals. The first chapter of this book considers image processing, moreprecisely algorithms of image compression and denoising. This research is motivated in particular by the new standard for compression of still images known as JPEG-2000. The second chapter has new results on the Navier-Stokes and other nonlinear evolution equations. Frequency-modulated signals and theiruse in the detection of gravitational waves are covered in the final chapter. In the book, the author describes both what the oscillating patterns are and the mathematics necessary for their analysis. It turns out that this mathematics involves new properties of various Besov-type function spaces and leads to many deep results, including new generalizations of famous Gagliardo-Nirenberg and Poincare inequalities. This book is based on the ``Dean Jacqueline B. Lewis Memorial Lectures'' given bythe author at Rutgers University. It can be used either as a textbook in studying applications of wavelets to image processing or as a supplementary resource for studying nonlinear evolution equations or frequency-modulated signals. Most of the material in the book did not appear previously inmonograph literature.

Traces of Hecke Operators

Author : Andrew Knightly
Publisher : American Mathematical Soc.
ISBN 13 : 0821837397
Total Pages : 392 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Traces of Hecke Operators by : Andrew Knightly

Download or read book Traces of Hecke Operators written by Andrew Knightly and published by American Mathematical Soc.. This book was released on 2006 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Fourier coefficients of modular forms are of widespread interest as an important source of arithmetic information. In many cases, these coefficients can be recovered from explicit knowledge of the traces of Hecke operators. The original trace formula for Hecke operators was given by Selberg in 1956. Many improvements were made in subsequent years, notably by Eichler and Hijikata. This book provides a comprehensive modern treatment of the Eichler-Selberg/Hijikata trace formulafor the traces of Hecke operators on spaces of holomorphic cusp forms of weight $\mathtt{k >2$ for congruence subgroups of $\operatorname{SL 2(\mathbf{Z )$. The first half of the text brings together the background from number theory and representation theory required for the computation. Thisincludes detailed discussions of modular forms, Hecke operators, adeles and ideles, structure theory for $\operatorname{GL 2(\mathbf{A )$, strong approximation, integration on locally compact groups, the Poisson summation formula, adelic zeta functions, basic representation theory for locally compact groups, the unitary representations of $\operatorname{GL 2(\mathbf{R )$, and the connection between classical cusp forms and their adelic counterparts on $\operatorname{GL 2(\mathbf{A )$. Thesecond half begins with a full development of the geometric side of the Arthur-Selberg trace formula for the group $\operatorname{GL 2(\mathbf{A )$. This leads to an expression for the trace of a Hecke operator, which is then computed explicitly. The exposition is virtually self-contained, withcomplete references for the occasional use of auxiliary results. The book concludes with several applications of the final formula.

Arithmetic Noncommutative Geometry

Author : Matilde Marcolli
Publisher : American Mathematical Soc.
ISBN 13 : 0821838334
Total Pages : 152 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Arithmetic Noncommutative Geometry by : Matilde Marcolli

Download or read book Arithmetic Noncommutative Geometry written by Matilde Marcolli and published by American Mathematical Soc.. This book was released on 2005 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: Arithmetic Noncommutative Geometry uses ideas and tools from noncommutative geometry to address questions in a new way and to reinterpret results and constructions from number theory and arithmetic algebraic geometry. This general philosophy is applied to the geometry and arithmetic of modular curves and to the fibers at Archimedean places of arithmetic surfaces and varieties. Noncommutative geometry can be expected to say something about topics of arithmetic interest because it provides the right framework for which the tools of geometry continue to make sense on spaces that are very singular and apparently very far from the world of algebraic varieties. This provides a way of refining the boundary structure of certain classes of spaces that arise in the context of arithmetic geometry. With a foreword written by Yuri Manin and a brief introduction to noncommutative geometry, this book offers a comprehensive account of the cross fertilization between two important areas, noncommutative geometry and number theory. It is suitable for graduate students and researchers interested in these areas.

Quadratic Algebras

Author : Alexander Polishchuk
Publisher : American Mathematical Soc.
ISBN 13 : 0821838342
Total Pages : 176 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Quadratic Algebras by : Alexander Polishchuk

Download or read book Quadratic Algebras written by Alexander Polishchuk and published by American Mathematical Soc.. This book was released on 2005 with total page 176 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces recent developments in the study of algebras defined by quadratic relations. One of the main problems in the study of these (and similarly defined) algebras is how to control their size. A central notion in solving this problem is the notion of a Koszul algebra, which was introduced in 1970 by S. Priddy and then appeared in many areas of mathematics, such as algebraic geometry, representation theory, non commutative geometry, $K$-theory, number theory, and non commutative linear algebra.The authors give a coherent exposition of the theory of quadratic and Koszul algebras, including various definitions of Koszulness, duality theory, Poincare-Birkhoff-Witt-type theorems for Koszul algebras, and the Koszul deformation principle. In the concluding chapter of the book, they explain a surprising connection between Koszul algebras and one-dependent discrete-time stochastic processes. The book can be used by graduate students and researchers working in algebra and any of the above-mentioned areas of mathematics.

Automorphic Forms and $L$-functions II

Author : David Ginzburg
Publisher : American Mathematical Soc.
ISBN 13 : 0821847082
Total Pages : 339 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Automorphic Forms and $L$-functions II by : David Ginzburg

Download or read book Automorphic Forms and $L$-functions II written by David Ginzburg and published by American Mathematical Soc.. This book was released on 2009 with total page 339 pages. Available in PDF, EPUB and Kindle. Book excerpt: Includes articles that represent global aspects of automorphic forms. This book covers topics such as: the trace formula; functoriality; representations of reductive groups over local fields; the relative trace formula and periods of automorphic forms; Rankin - Selberg convolutions and L-functions; and, p-adic L-functions.