Michael Lusztig's Books

Representations of Finite Dimensional Algebras and Related Topics in Lie Theory and Geometry

By: Vlastimil Dlab,Claus Michael Ringel

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Representations of Finite Dimensional Algebras and Related Topics in Lie Theory and Geometry

By: Vlastimil Dlab,Claus Michael Ringel

These proceedings are from the Tenth International Conference on Representations of Algebras and Related Topics (ICRA X) held at The Fields Institute. In addition to the traditional ``instructional'' workshop preceding the conference, there were also workshops on ``Commutative Algebra, Algebraic Geometry and Representation Theory'', ``Finite Dimensional Algebras, Algebraic Groups and Lie Theory'', and ``Quantum Groups and Hall Algebras''. These workshops reflect the latest developments and the increasing interest in areas that are closely related to the representation theory of finite dimensional associative algebras. Although these workshops were organized separately, their topics are strongly interrelated. The workshop on Commutative Algebra, Algebraic Geometry and Representation Theory surveyed various recently established connections, such as those pertaining to the classification of vector bundles or Cohen-Macaulay modules over Noetherian rings, coherent sheaves on curves, or ideals in Weyl algebras. In addition, methods from algebraic geometry or commutative algebra relating to quiver representations and varieties of modules were presented. The workshop on Finite Dimensional Algebras, Algebraic Groups and Lie Theory surveyed developments in finite dimensional algebras and infinite dimensional Lie theory, especially as the two areas interact and may have future interactions. The workshop on Quantum Groups and Hall Algebras dealt with the different approaches of using the representation theory of quivers (and species) in order to construct quantum groups, working either over finite fields or over the complex numbers. In particular, these proceedings contain a quite detailed outline of the use of perverse sheaves in order to obtain canonical bases. The book is recommended for graduate students and researchers in algebra and geometry.

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502

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American Mathematical Soc.

Published Date

ISBN 10

0821834169

ISBN 13

9780821834169

Perspectives on Solvable Models

Dedicated to Vladimir Rittenberg on the Occasion of His 60th Birthday

By: Uwe Grimm,Michael Baake

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Perspectives on Solvable Models

Dedicated to Vladimir Rittenberg on the Occasion of His 60th Birthday

By: Uwe Grimm,Michael Baake

This volume consists of a collection of recent research articles dedicated to Vladimir Rittenberg on the occasion of his 60th birthday. Various aspects of solvable models in different areas of theoretical and mathematical physics are covered. Particular topics include diffusion, self-organized criticality, classical and quantum spin chains, two-dimensional lattice models, quantum algebras, and conformal field theory. The list of contributing authors contains altogether 34 names, including among others, Baxter, Cardy, Itzykson, Martin, McCoy, Nahm, Pearce and de Vega.

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314

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World Scientific

Published Date

ISBN 10

981022107X

ISBN 13

9789810221072

Noncommutative Algebraic Geometry

By: Gwyn Bellamy,Daniel Rogalski,Travis Schedler,J. Toby Stafford,Michael Wemyss

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Noncommutative Algebraic Geometry

By: Gwyn Bellamy,Daniel Rogalski,Travis Schedler,J. Toby Stafford,Michael Wemyss

This book provides a comprehensive introduction to the interactions between noncommutative algebra and classical algebraic geometry.

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367

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Cambridge University Press

Published Date

ISBN 10

1107129540

ISBN 13

9781107129542

Representation Theory and Complex Analysis

Lectures Given at the C.I.M.E. Summer School Held in Venice, Italy, June 10-17, 2004

By: Michael Cowling

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Representation Theory and Complex Analysis

Lectures Given at the C.I.M.E. Summer School Held in Venice, Italy, June 10-17, 2004

By: Michael Cowling

Six leading experts lecture on a wide spectrum of recent results on the subject of the title. They present a survey of various interactions between representation theory and harmonic analysis on semisimple groups and symmetric spaces, and recall the concept of amenability. They further illustrate how representation theory is related to quantum computing; and much more. Taken together, this volume provides both a solid reference and deep insights on current research activity.

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400

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Springer Science & Business Media

Published Date

ISBN 10

3540768912

ISBN 13

9783540768913

Computational Methods for Representations of Groups and Algebras

Euroconference in Essen (Germany), April 1-5, 1997

By: P. Dräxler,Gerhard Michler,Claus Michael Ringel

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Computational Methods for Representations of Groups and Algebras

Euroconference in Essen (Germany), April 1-5, 1997

By: P. Dräxler,Gerhard Michler,Claus Michael Ringel

I Introductory Articles.- 1 Classification Problems in the Representation Theory of Finite-Dimensional Algebras.- 2 Noncommutative Gröbner Bases, and Projective Resolutions.- 3 Construction of Finite Matrix Groups.- II Keynote Articles.- 4 Derived Tubularity: a Computational Approach.- 5 Problems in the Calculation of Group Cohomology.- 6 On a Tensor Category for the Exceptional Lie Groups.- 7 Non-Commutative Gröbner Bases and Anick's Resolution.- 8 A new Existence Proof of Janko's Simple Group J4.- 9 The Normalization: a new Algorithm, Implementation and Comparisons.- 10 A Computer Algebra Approach to sheaves over Weighted Projective Lines.- 11 Open Problems in the Theory of Kazhdan-Lusztig polynomials.- 12 Relative Trace Ideals and Cohen Macaulay Quotients.- 13 On Sims' Presentation for Lyons' Simple Group.- 14 A Presentation for the Lyons Simple Group.- 15 Reduction of Weakly Definite Unit Forms.- 16 Decision Problems in Finitely Presented Groups.- 17 Some Algorithms in Invariant Theory of Finite Groups.- 18 Coxeter Transformations associated with Finite Dimensional Algebras.- 19 The 2-Modular Decomposition Numbers of Co2.- 20 Bimodule and Matrix Problems.

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378

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Springer Science & Business Media

Published Date

ISBN 10

3764360631

ISBN 13

9783764360634

Mathematics without Apologies

Portrait of a Problematic Vocation

By: Michael Harris

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Mathematics without Apologies

Portrait of a Problematic Vocation

By: Michael Harris

An insightful reflection on the mathematical soul What do pure mathematicians do, and why do they do it? Looking beyond the conventional answers—for the sake of truth, beauty, and practical applications—this book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twenty-first century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources. Drawing on his personal experiences and obsessions as well as the thoughts and opinions of mathematicians from Archimedes and Omar Khayyám to such contemporary giants as Alexander Grothendieck and Robert Langlands, Michael Harris reveals the charisma and romance of mathematics as well as its darker side. In this portrait of mathematics as a community united around a set of common intellectual, ethical, and existential challenges, he touches on a wide variety of questions, such as: Are mathematicians to blame for the 2008 financial crisis? How can we talk about the ideas we were born too soon to understand? And how should you react if you are asked to explain number theory at a dinner party? Disarmingly candid, relentlessly intelligent, and richly entertaining, Mathematics without Apologies takes readers on an unapologetic guided tour of the mathematical life, from the philosophy and sociology of mathematics to its reflections in film and popular music, with detours through the mathematical and mystical traditions of Russia, India, medieval Islam, the Bronx, and beyond.

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467

Publisher

Princeton University Press

Published Date

ISBN 10

1400885523

ISBN 13

9781400885527

Period Domains over Finite and p-adic Fields

By: Jean-François Dat,Sascha Orlik,Michael Rapoport

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Period Domains over Finite and p-adic Fields

By: Jean-François Dat,Sascha Orlik,Michael Rapoport

This book is, on the one hand, a pedagogical introduction to the formalism of slopes, of semi-stability and of related concepts in the simplest possible context. It is therefore accessible to any graduate student with a basic knowledge in algebraic geometry and algebraic groups. On the other hand, the book also provides a thorough introduction to the basics of period domains, as they appear in the geometric approach to local Langlands correspondences and in the recent conjectural p-adic local Langlands program. The authors provide numerous worked examples and establish many connections to topics in the general area of algebraic groups over finite and local fields. In addition, the end of each section includes remarks on open questions, historical context and references to the literature.

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395

Publisher

Cambridge University Press

Published Date

ISBN 10

1139488341

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9781139488341

Applications of Curves over Finite Fields

1997 AMS-IMS-SIAM Joint Summer Research Conference on Applications of Curves Over Finite Fields, July 27-31, 1997, University of Washington, Seattle

By: Michael D. Fried

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Applications of Curves over Finite Fields

1997 AMS-IMS-SIAM Joint Summer Research Conference on Applications of Curves Over Finite Fields, July 27-31, 1997, University of Washington, Seattle

By: Michael D. Fried

This volume presents the results of the AMS-IMS-SIAM Joint Summer Research Conference held at the University of Washington (Seattle). The talks were devoted to various aspects of the theory of algebraic curves over finite fields and its numerous applications. The three basic themes are the following: 1. Curves with many rational points. Several articles describe main approaches to the construction of such curves: the Drinfeld modules and fiber product methods, the moduli space approach, and the constructions using classical curves. 2. Monodromy groups of characteristic $p$ covers. A number of authors presented the results and conjectures related to the study of the monodromy groups of curves over finite fields. In particular, they study the monodromy groups from genus 0 covers, reductions of covers, and explicit computation of monodromy groups over finite fields. 3. Zeta functions and trace formulas. To a large extent, papers devoted to this topic reflect the contributions of Professor Bernard Dwork and his students. This conference was the last attended by Professor Dwork before his death, and several papers inspired by his presence include commentaries about the applications of trace formulas and L-function. The volume also contains a detailed introduction paper by Professor Michael Fried, which helps the reader to navigate the material presented in the book.

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254

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821809253

ISBN 13

9780821809259

Sub-State Nationalism

A Comparative Analysis of Institutional Design

By: Helen Catt,Michael Murphy

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Sub-State Nationalism

A Comparative Analysis of Institutional Design

By: Helen Catt,Michael Murphy

A major feature of the political development of Western democracies is the growth of indigenous, ethnic and national groups striving for political self-determination. This book analyses the institutional responses individual governments have made to these demands. Sub-State Nationalism provides a much needed categorization and genuinely comparative analysis of the political voice gained by sub-state national groups in multinational democratic communities. The book includes international case-studies drawn from Canada, New Zealand, Australia and the USA. It covers the empirical question of what voice these groups have, and how its institutions are structured, and the analytical question of how such knowledge contributes to our theoretical understanding of the politics of group rights and representation.

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266

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Routledge

Published Date

ISBN 10

1134533926

ISBN 13

9781134533923

The Penrose Transform

Its Interaction with Representation Theory

By: Robert J. Baston,Michael G. Eastwood

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The Penrose Transform

Its Interaction with Representation Theory

By: Robert J. Baston,Michael G. Eastwood

Brings to the reader a huge amount of information, well organized and condensed into less than two hundred pages." — Mathematical Reviews In recent decades twistor theory has become an important focus for students of mathematical physics. Central to twistor theory is the geometrical transform known as the Penrose transform, named for its groundbreaking developer. Geared toward students of physics and mathematics, this advanced text explores the Penrose transform and presupposes no background in twistor theory and a minimal familiarity with representation theory. An introductory chapter sketches the development of the Penrose transform, followed by reviews of Lie algebras and flag manifolds, representation theory and homogeneous vector bundles, and the Weyl group and the Bott-Borel-Weil theorem. Succeeding chapters explore the Penrose transform in terms of the Bernstein-Gelfand-Gelfand resolution, followed by worked examples, constructions of unitary representations, and module structures on cohomology. The treatment concludes with a review of constructions and suggests further avenues for research.

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257

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Courier Dover Publications

Published Date

ISBN 10

0486797295

ISBN 13

9780486797298

The Penrose Transform and Analytic Cohomology in Representation Theory

AMS-IMS-SIAM Summer Research Conference, June 27 to July 3, 1992, Mount Holyoke College, South Hadley, Massachusetts

By: Michael G. Eastwood

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The Penrose Transform and Analytic Cohomology in Representation Theory

AMS-IMS-SIAM Summer Research Conference, June 27 to July 3, 1992, Mount Holyoke College, South Hadley, Massachusetts

By: Michael G. Eastwood

This book contains refereed papers presented at the AMS-IMS-SIAM Summer Research Conference on the Penrose Transform and Analytic Cohomology in Representation Theory held in the summer of 1992 at Mount Holyoke College. The conference brought together some of the top experts in representation theory and differential geometry. One of the issues explored at the conference was the fact that various integral transforms from representation theory, complex integral geometry, and mathematical physics appear to be instances of the same general construction, which is sometimes called the ``Penrose transform''. There is considerable scope for further research in this area, and this book would serve as an excellent introduction.

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274

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821851764

ISBN 13

9780821851760

Subgroup Decomposition in Out(Fn)

By: Michael Handel,Lee Mosher

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Subgroup Decomposition in Out(Fn)

By: Michael Handel,Lee Mosher

In this work the authors develop a decomposition theory for subgroups of Out(Fn) which generalizes the decomposition theory for individual elements of Out(Fn) found in the work of Bestvina, Feighn, and Handel, and which is analogous to the decomposition theory for subgroups of mapping class groups found in the work of Ivanov.

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290

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American Mathematical Soc.

Published Date

ISBN 10

1470441136

ISBN 13

9781470441135

Factorizable Sheaves and Quantum Groups

By: Roman Bezrukavnikov,Michael Finkelberg,Vadim Schechtman

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Factorizable Sheaves and Quantum Groups

By: Roman Bezrukavnikov,Michael Finkelberg,Vadim Schechtman

The book is devoted to the geometrical construction of the representations of Lusztig's small quantum groups at roots of unity. These representations are realized as some spaces of vanishing cycles of perverse sheaves over configuration spaces. As an application, the bundles of conformal blocks over the moduli spaces of curves are studied. The book is intended for specialists in group representations and algebraic geometry.

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300

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Springer

Published Date

ISBN 10

3540692312

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9783540692317

Fields Medallists' Lectures, 2nd Edition

By: Michael Atiyah,Daniel Iagolnitzer

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Fields Medallists' Lectures, 2nd Edition

By: Michael Atiyah,Daniel Iagolnitzer

Although the Fields Medal does not have the same public recognition as the Nobel Prizes, they share a similar intellectual standing. It is restricted to one field — that of mathematics — and an age limit of 40 has become an accepted tradition. Mathematics has in the main been interpreted as pure mathematics, and this is not so unreasonable since major contributions in some applied areas can be (and have been) recognized with Nobel Prizes.A list of Fields Medallists and their contributions provides a bird's-eye view of mathematics over the past 60 years. It highlights the areas in which, at various times, greatest progress has been made. This volume does not pretend to be comprehensive, nor is it a historical document. On the other hand, it presents contributions from Fields Medallists and so provides a highly interesting and varied picture.The second edition of Fields Medallists' Lectures features additional contributions from the following Medallists: Kunihiko Kodaira (1954), Richard E Borcherds (1998), William T Gowers (1998), Maxim Kontsevich (1998), Curtis T McMullen (1998) and Vladimir Voevodsky (2002).

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824

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World Scientific

Published Date

ISBN 10

9814486809

ISBN 13

9789814486804

The Generalized Triangle Inequalities in Symmetric Spaces and Buildings with Applications to Algebra

By: Michael Kapovich,Bernhard Leeb,John James Millson

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The Generalized Triangle Inequalities in Symmetric Spaces and Buildings with Applications to Algebra

By: Michael Kapovich,Bernhard Leeb,John James Millson

In this paper the authors apply their results on the geometry of polygons in infinitesimal symmetric spaces and symmetric spaces and buildings to four problems in algebraic group theory. Two of these problems are generalizations of the problems of finding the constraints on the eigenvalues (resp. singular values) of a sum (resp. product) when the eigenvalues (singular values) of each summand (factor) are fixed. The other two problems are related to the nonvanishing of the structure constants of the (spherical) Hecke and representation rings associated with a split reductive algebraic group over $\mathbb{Q}$ and its complex Langlands' dual. The authors give a new proof of the Saturation Conjecture for $GL(\ell)$ as a consequence of their solution of the corresponding saturation problem for the Hecke structure constants for all split reductive algebraic groups over $\mathbb{Q}$.

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American Mathematical Soc.

Published Date

ISBN 10

0821840541

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9780821840542

The Geometry and Topology of Coxeter Groups. (LMS-32)

By: Michael W. Davis

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The Geometry and Topology of Coxeter Groups. (LMS-32)

By: Michael W. Davis

The Geometry and Topology of Coxeter Groups is a comprehensive and authoritative treatment of Coxeter groups from the viewpoint of geometric group theory. Groups generated by reflections are ubiquitous in mathematics, and there are classical examples of reflection groups in spherical, Euclidean, and hyperbolic geometry. Any Coxeter group can be realized as a group generated by reflection on a certain contractible cell complex, and this complex is the principal subject of this book. The book explains a theorem of Moussong that demonstrates that a polyhedral metric on this cell complex is nonpositively curved, meaning that Coxeter groups are "CAT(0) groups." The book describes the reflection group trick, one of the most potent sources of examples of aspherical manifolds. And the book discusses many important topics in geometric group theory and topology, including Hopf's theory of ends; contractible manifolds and homology spheres; the Poincaré Conjecture; and Gromov's theory of CAT(0) spaces and groups. Finally, the book examines connections between Coxeter groups and some of topology's most famous open problems concerning aspherical manifolds, such as the Euler Characteristic Conjecture and the Borel and Singer conjectures.

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601

Publisher

Princeton University Press

Published Date

ISBN 10

1400845947

ISBN 13

9781400845941

Michael Atiyah Collected Works

Volume 2: K-Theory

By: Michael Atiyah

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Michael Atiyah Collected Works

Volume 2: K-Theory

By: Michael Atiyah

One of the greatest mathematicians in the world, Michael Atiyah has earned numerous honors, including a Fields Medal, the mathematical equivalent of the Nobel Prize. While the focus of his work has been in the areas of algebraic geometry and topology, he has also participated in research with theoretical physicists. For the first time, these volumes bring together Atiyah's collected papers--both monographs and collaborative works-- including those dealing with mathematical education and current topics of research such as K-theory and gauge theory. The volumes are organized thematically. They will be of great interest to research mathematicians, theoretical physicists, and graduate students in these areas.

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876

Publisher

Oxford University Press

Published Date

ISBN 10

0198532768

ISBN 13

9780198532767

Anomalies in Net Present Value, Returns and Polynomials, and Regret Theory in Decision-Making

By: Michael C. I. Nwogugu

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Anomalies in Net Present Value, Returns and Polynomials, and Regret Theory in Decision-Making

By: Michael C. I. Nwogugu

This book explores why Modified Internal Rate of Return (MIRR) and Net Present Value (NPV) are not necessarily accurate or efficient tools for valuation and decision-making. The author specifically addresses the biases and framing effects inherent in the NPV/MIRR/IRR model and in related approaches such as Adjusted Present Value (APV), Net Future Value (NFV), and by extension, Polynomials. In doing so, the book presents new ways of solving higher order polynomials using invariants and homomorphisms and explains why the “Fundamental Theorem of Algebra”, the Binomial Theorem and the “Descartes Sign Rule” are unreliable. Chapters also discuss how International Asset Pricing Theory (IAPT) and Intertemporal Capital Asset Pricing Models (ICAPM) can produce inaccurate results in certain circumstances. The conditions under which ICAPM and IAPT may be accurate are described; as well as why those conditions cannot, or are unlikely to, exist. The conditions under which negative interest rates may exist or are justified are also outlined. Moreover, the author explains why traditional Consumption-Savings-Investment-Production models of allocation can be inefficient, and then introduces a new model of allocation that can be applied to individuals, households and companies. Finally, the book explains why the Elasticity of Intertemporal Substitution is a flawed concept and introduces the Marginal Rate of Intertemporal Joint Substitution as a solution.

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336

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Springer

Published Date

ISBN 10

1137446986

ISBN 13

9781137446985

Language Politics of Regional Integration

Cases from the Americas

By: Michael A. Morris

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Language Politics of Regional Integration

Cases from the Americas

By: Michael A. Morris

Language policies impact language choice, language prestige, and language spread. Rising regional integration, both formal and informal, adds to the sensitivity and complexity of language politics, whether in North America, South America or Europe. This book shows how language politics vary across the Americas and contrast with Europe.

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222

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Springer

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ISBN 10

1137561475

ISBN 13

9781137561473

Multiculturalism

A Critical Introduction

By: Michael Murphy

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Multiculturalism

A Critical Introduction

By: Michael Murphy

What is multiculturalism and what are the different theories used to justify it? Are multicultural policies a threat to liberty and equality? Can liberal democracies accommodate minority groups without sacrificing peace and stability? In this clear introduction to the subject, Michael Murphy explores these questions and critically assesses multiculturalism from the standpoint of political philosophy and political practice. The book explores the origins and contemporary usage of the concept of multiculturalism in the context of debates about citizenship, egalitarian justice and conflicts between individual and collective rights. The ideas of some of the most influential champions and critics of multiculturalism, including Will Kymlicka, Chandran Kukathas, Susan Okin and Brian Barry, are also clearly explained and evaluated. Key themes include the tension between multiculturalism and gender equality, cultural relativism and the limits of liberal toleration, and the impact of multicultural policies on social cohesion ethnic conflict. Murphy also surveys the legal practices and policies enacted to accommodate multiculturalism, drawing on examples from the Americas, Australasia, Europe, Asia and the Middle East. Multiculturalism: A Critical Introduction is an ideal starting point for anyone coming to the topic for the first time as well as those already familiar with some of the key issues.

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205

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Routledge

Published Date

ISBN 10

1136520112

ISBN 13

9781136520112

Algebraic Geometry Codes: Advanced Chapters

By: Michael Tsfasman,Serge Vlǎduţ,Dmitry Nogin

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Algebraic Geometry Codes: Advanced Chapters

By: Michael Tsfasman,Serge Vlǎduţ,Dmitry Nogin

Algebraic Geometry Codes: Advanced Chapters is devoted to the theory of algebraic geometry codes, a subject related to local_libraryBook Catalogseveral domains of mathematics. On one hand, it involves such classical areas as algebraic geometry and number theory; on the other, it is connected to information transmission theory, combinatorics, finite geometries, dense packings, and so on. The book gives a unique perspective on the subject. Whereas most books on coding theory start with elementary concepts and then develop them in the framework of coding theory itself within, this book systematically presents meaningful and important connections of coding theory with algebraic geometry and number theory. Among many topics treated in the book, the following should be mentioned: curves with many points over finite fields, class field theory, asymptotic theory of global fields, decoding, sphere packing, codes from multi-dimensional varieties, and applications of algebraic geometry codes. The book is the natural continuation of Algebraic Geometric Codes: Basic Notions by the same authors. The concise exposition of the first volume is included as an appendix.

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466

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American Mathematical Soc.

Published Date

ISBN 10

1470448653

ISBN 13

9781470448653

The Independence of Scotland

Self-government and the Shifting Politics of Union

By: Michael Keating

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The Independence of Scotland

Self-government and the Shifting Politics of Union

By: Michael Keating

In a thought-provoking analysis Keating reviews the political, constitutional, and legal issues around Scottish independence and the political economy of independence, surveying the options for a social and economic project for an independent Scotland.

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225

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Oxford University Press

Published Date

ISBN 10

0199545952

ISBN 13

9780199545957

Fields Medallists' Lectures

By: Michael Francis Atiyah,Daniel Iagolnitzer

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Fields Medallists' Lectures

By: Michael Francis Atiyah,Daniel Iagolnitzer

Although not as publicly well-known as the Nobel Prizes, the Fields Medal shares the same intellectual standing and is the equivalent award in the field of mathematics. This volume presents a selected list of 22 Fields Medallists and their contributions to give a highly interesting and varied bird's eye view of mathematics over the past 60 years. The contributions relate directly to the work for which the Medals were awarded or to the medallists' more current interests. In most cases, they are preceded by the introductory speech given by another leading mathematician during the prize ceremony, a photograph and up-to-date biographical notice.

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652

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World Scientific

Published Date

ISBN 10

9810231172

ISBN 13

9789810231170

Localization for $THH(ku)$ and the Topological Hochschild and Cyclic Homology of Waldhausen Categories

By: Andrew J. Blumberg,Michael A. Mandell

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Localization for $THH(ku)$ and the Topological Hochschild and Cyclic Homology of Waldhausen Categories

By: Andrew J. Blumberg,Michael A. Mandell

The authors resolve the longstanding confusion about localization sequences in $THH$ and $TC$ and establish a specialized devissage theorem.

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100

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American Mathematical Soc.

Published Date

ISBN 10

1470441780

ISBN 13

9781470441784

Vector Fields on Manifolds

By: Michael Francis Atiyah

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Vector Fields on Manifolds

By: Michael Francis Atiyah

This paper is a contribution to the topological study of vector fields on manifolds. In particular we shall be concerned with the problems of exist ence of r linearly independent vector fields. For r = 1 the classical result of H. Hopf asserts that the vanishing of the Euler characteristic is the necessary and sufficient condition, and our results will give partial extens ions of Hopf's theorem to the case r > 1. Arecent article by E. Thomas [10] gives a good survey of work in this general area. Our approach to these problems is based on the index theory of elliptic differential operators and is therefore rather different from the standard topological approach. Briefly speaking, what we do is to observe that certain invariants of a manifold (Euler characteristic, signature, etc. ) are indices of elliptic operators (see [5]) and the existence of a certain number of vector fields implies certain symmetry conditions for these operators and hence corresponding results for their indices. In this way we obtain certain necessary conditions for the existence of vector fields and, more generally , for the existence of fields of tangent planes. For example, one of our results is the following THEOREM (1. 1). Let X be a compact oriented smooth manifold 0/ dimension 4 q, and assume that X possesses a tangent fteld of oriented 2-planes (that is, an oriented 2-dimensional sub-bundle 0/ the tangent vector bundle).

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Springer

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ISBN 10

3322985032

ISBN 13

9783322985033

Algebraic Geometric Codes: Basic Notions

By: Michael Tsfasman,Serge Vlǎduţ,Dmitry Nogin

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Algebraic Geometric Codes: Basic Notions

By: Michael Tsfasman,Serge Vlǎduţ,Dmitry Nogin

The book is devoted to the theory of algebraic geometric codes, a subject formed on the border of several domains of mathematics. On one side there are such classical areas as algebraic geometry and number theory; on the other, information transmission theory, combinatorics, finite geometries, dense packings, etc. The authors give a unique perspective on the subject. Whereas most books on coding theory build up coding theory from within, starting from elementary concepts and almost always finishing without reaching a certain depth, this book constantly looks for interpretations that connect coding theory to algebraic geometry and number theory. There are no prerequisites other than a standard algebra graduate course. The first two chapters of the book can serve as an introduction to coding theory and algebraic geometry respectively. Special attention is given to the geometry of curves over finite fields in the third chapter. Finally, in the last chapter the authors explain relations between all of these: the theory of algebraic geometric codes.

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338

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American Mathematical Society

Published Date

ISBN 10

1470470071

ISBN 13

9781470470074

Weighted Littlewood-Paley Theory and Exponential-Square Integrability

By: Michael Wilson

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Weighted Littlewood-Paley Theory and Exponential-Square Integrability

By: Michael Wilson

Littlewood-Paley theory is an essential tool of Fourier analysis, with applications and connections to PDEs, signal processing, and probability. It extends some of the benefits of orthogonality to situations where orthogonality doesn’t really make sense. It does so by letting us control certain oscillatory infinite series of functions in terms of infinite series of non-negative functions. Beginning in the 1980s, it was discovered that this control could be made much sharper than was previously suspected. The present book tries to give a gentle, well-motivated introduction to those discoveries, the methods behind them, their consequences, and some of their applications.

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233

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Springer Science & Business Media

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ISBN 10

3540745823

ISBN 13

9783540745822

Pseudo-Differential Operators

Quantization and Signals

By: Hans G. Feichtinger,Bernard Helffer,Michael Lamoureux,Nicolas Lerner,Joachim Toft

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Pseudo-Differential Operators

Quantization and Signals

By: Hans G. Feichtinger,Bernard Helffer,Michael Lamoureux,Nicolas Lerner,Joachim Toft

Pseudo-differential operators were initiated by Kohn, Nirenberg and Hörmander in the sixties of the last century. Beside applications in the general theory of partial differential equations, they have their roots also in the study of quantization first envisaged by Hermann Weyl thirty years earlier. Thanks to the understanding of the connections of wavelets with other branches of mathematical analysis, quantum physics and engineering, such operators have been used under different names as mathematical models in signal analysis since the last decade of the last century. The volume investigates the mathematics of quantization and signals in the context of pseudo-differential operators, Weyl transforms, Daubechies operators, Wick quantization and time-frequency localization operators. Applications to quantization, signal analysis and the modern theory of PDE are highlighted.

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235

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Springer

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ISBN 10

3540682686

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9783540682684

Enumerative Invariants in Algebraic Geometry and String Theory

Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 6-11, 2005

By: Marcos Marino,Michael Thaddeus,Ravi Vakil

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Enumerative Invariants in Algebraic Geometry and String Theory

Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 6-11, 2005

By: Marcos Marino,Michael Thaddeus,Ravi Vakil

Starting in the middle of the 80s, there has been a growing and fruitful interaction between algebraic geometry and certain areas of theoretical high-energy physics, especially the various versions of string theory. Physical heuristics have provided inspiration for new mathematical definitions (such as that of Gromov-Witten invariants) leading in turn to the solution of problems in enumerative geometry. Conversely, the availability of mathematically rigorous definitions and theorems has benefited the physics research by providing the required evidence in fields where experimental testing seems problematic. The aim of this volume, a result of the CIME Summer School held in Cetraro, Italy, in 2005, is to cover part of the most recent and interesting findings in this subject.

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219

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Springer

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ISBN 10

3540798145

ISBN 13

9783540798149

A Concise Course on Stochastic Partial Differential Equations

By: Claudia Prévôt,Michael Röckner

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A Concise Course on Stochastic Partial Differential Equations

By: Claudia Prévôt,Michael Röckner

These lectures concentrate on (nonlinear) stochastic partial differential equations (SPDE) of evolutionary type. There are three approaches to analyze SPDE: the "martingale measure approach", the "mild solution approach" and the "variational approach". The purpose of these notes is to give a concise and as self-contained as possible an introduction to the "variational approach". A large part of necessary background material is included in appendices.

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149

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Springer Science & Business Media

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ISBN 10

3540707808

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9783540707806

Differential Equations Driven by Rough Paths

Ecole d’Eté de Probabilités de Saint-Flour XXXIV-2004

By: Terry J. Lyons,Michael J. Caruana,Thierry Lévy

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Differential Equations Driven by Rough Paths

Ecole d’Eté de Probabilités de Saint-Flour XXXIV-2004

By: Terry J. Lyons,Michael J. Caruana,Thierry Lévy

Each year young mathematicians congregate in Saint Flour, France, and listen to extended lecture courses on new topics in Probability Theory. The goal of these notes, representing a course given by Terry Lyons in 2004, is to provide a straightforward and self supporting but minimalist account of the key results forming the foundation of the theory of rough paths.

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126

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Springer

Published Date

ISBN 10

3540712852

ISBN 13

9783540712855

Cluster Algebras and Poisson Geometry

By: Michael Gekhtman,Michael Shapiro,Alek Vainshtein

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Cluster Algebras and Poisson Geometry

By: Michael Gekhtman,Michael Shapiro,Alek Vainshtein

The first book devoted to cluster algebras, this work contains chapters on Poisson geometry and Schubert varieties; an introduction to cluster algebras and their main properties; and geometric aspects of the cluster algebra theory, in particular on its relations to Poisson geometry and to the theory of integrable systems.

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264

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American Mathematical Soc.

Published Date

ISBN 10

0821849727

ISBN 13

9780821849729

The Republican Hero

From Homer to Batman

By: Michael Lusztig

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The Republican Hero

From Homer to Batman

By: Michael Lusztig

Politically speaking, do heroes matter? Are we living in a post-heroic age? The Republican Hero addresses both these questions. The general tenor of modern thinking is that heroes do matter but that the modern age is characterized by a narrowing of moral horizons once illuminated by heroes, secular and spiritual. Michael Lusztig argues that the modern world is not post-heroic. He makes the case that the modern age is the most heroic age, if measured in terms of the Aristotelian currency of balance and completeness. To this end, he identifies four main hero-types—the epic, magnanimous, Romantic, and common. Each can rightfully be called a republican hero: each contributes to the promotion or protection or provision of republican values. Each exemplifies the heroic virtues of their age. However, taken conjunctively, each contributes to what Lusztig conceives as the complete republican hero of the modern age.

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379

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State University of New York Press

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ISBN 10

1438495382

ISBN 13

9781438495385

Geometric Representation Theory and Gauge Theory

Cetraro, Italy 2018

By: Alexander Braverman,Michael Finkelberg,Andrei Negut,Alexei Oblomkov

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Geometric Representation Theory and Gauge Theory

Cetraro, Italy 2018

By: Alexander Braverman,Michael Finkelberg,Andrei Negut,Alexei Oblomkov

This book offers a review of the vibrant areas of geometric representation theory and gauge theory, which are characterized by a merging of traditional techniques in representation theory with the use of powerful tools from algebraic geometry, and with strong inputs from physics. The notes are based on lectures delivered at the CIME school "Geometric Representation Theory and Gauge Theory" held in Cetraro, Italy, in June 2018. They comprise three contributions, due to Alexander Braverman and Michael Finkelberg, Andrei Negut, and Alexei Oblomkov, respectively. Braverman and Finkelberg’s notes review the mathematical theory of the Coulomb branch of 3D N=4 quantum gauge theories. The purpose of Negut’s notes is to study moduli spaces of sheaves on a surface, as well as Hecke correspondences between them. Oblomkov's notes concern matrix factorizations and knot homology. This book will appeal to both mathematicians and theoretical physicists and will be a source of inspiration for PhD students and researchers.

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137

Publisher

Springer Nature

Published Date

ISBN 10

303026856X

ISBN 13

9783030268565

Book's by Michael Lusztig

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