Faculty and scholars within the Graduate Center Mathematics program are prolific authors. Learn more about the books published by members of our community in the archive below.
Polynomial Methods and Incidence Theory
The past decade has seen numerous major mathematical breakthroughs for topics such as the finite field Kakeya conjecture, the cap set conjecture, Erdős's distinct distances problem, the joints problem, as well as others, thanks to the introduction of new polynomial methods. There has also been significant progress on a variety of problems from additive combinatorics, discrete geometry, and more. This book gives a detailed yet accessible introduction to these new polynomial methods and their applications, with a focus on incidence theory. Based on the author's own teaching experience, the text requires a minimal background, allowing graduate and advanced undergraduate students to get to grips with an active and exciting research front. The techniques are presented gradually and in detail, with many examples, warm-up proofs, and exercises included. An appendix provides a quick reminder of basic results and ideas.
Published March 2022
Cambridge University Press, 2022
A First Course in Stochastic Calculus
A First Course in Stochastic Calculus is a complete guide for advanced undergraduate students to take the next step in exploring probability theory and for master's students in mathematical finance who would like to build an intuitive and theoretical understanding of stochastic processes. This book is also an essential tool for finance professionals who wish to sharpen their knowledge and intuition about stochastic calculus.
Louis-Pierre Arguin offers an exceptionally clear introduction to Brownian motion and to random processes governed by the principles of stochastic calculus. The beauty and power of the subject are made accessible to readers with a basic knowledge of probability, linear algebra, and multivariable calculus. This is achieved by emphasizing numerical experiments using elementary Python coding to build intuition and adhering to a rigorous geometric point of view on the space of random variables. This unique approach is used to elucidate the properties of Gaussian processes, martingales, and diffusions. One of the book's highlights is a detailed and self-contained account of stochastic calculus applications to option pricing in finance.
Published November 2021
American Mathematical Society, 2021
A Course in Complex Analysis
A Course in Complex Analysis explores a central branch of mathematical analysis, with broad applications in mathematics and other fields such as physics and engineering. Ideally designed for a year-long graduate course on complex analysis and based on nearly twenty years of classroom lectures, this modern and comprehensive textbook is equally suited for independent study or as a reference for more experienced scholars.
Saeed Zakeri guides the reader through a journey that highlights the topological and geometric themes of complex analysis and provides a solid foundation for more advanced studies, particularly in Riemann surfaces, conformal geometry, and dynamics. He presents all the main topics of classical theory in great depth and blends them seamlessly with many elegant developments that are not commonly found in textbooks at this level. They include the dynamics of Möbius transformations, Schlicht functions and distortion theorems, boundary behavior of conformal and harmonic maps, analytic arcs and the general reflection principle, Hausdorff dimension and holomorphic removability, a multifaceted approach to the theorems of Picard and Montel, Zalcman’s rescaling theorem, conformal metrics and Ahlfors’s generalization of the Schwarz lemma, holomorphic branched coverings, geometry of the modular group, and the uniformization theorem for spherical domains.
Written with exceptional clarity and insightful style, A Course in Complex Analysis is accessible to beginning graduate students and advanced undergraduates with some background knowledge of analysis and topology. Zakeri includes more than 350 problems, with problem sets at the end of each chapter, along with numerous carefully selected examples. This well-organized and richly illustrated book is peppered throughout with marginal notes of historical and expository value.
Presenting a wealth of material in a single volume, A Course in Complex Analysis will be a valuable resource for students and working mathematicians.
Published November 2021
Princeton University Press 2021
Graphs and Discrete Dirichlet Spaces
The spectral geometry of infinite graphs deals with three major themes and their interplay: the spectral theory of the Laplacian, the geometry of the underlying graph, and the heat flow with its probabilistic aspects. In this book, all three themes are brought together coherently under the perspective of Dirichlet forms, providing a powerful and unified approach.
The book gives a complete account of key topics of infinite graphs, such as essential self-adjointness, Markov uniqueness, spectral estimates, recurrence, and stochastic completeness. A major feature of the book is the use of intrinsic metrics to capture the geometry of graphs. As for manifolds, Dirichlet forms in the graph setting offer a structural understanding of the interaction between spectral theory, geometry and probability. For graphs, however, the presentation is much more accessible and inviting thanks to the discreteness of the underlying space, laying bare the main concepts while preserving the deep insights of the manifold case.
Published January 2021
Research Trends in Graph Theory and Applications
The Workshop for Women in Graph Theory and Applications was held at the Institute for Mathematics and Its Applications (University of Minnesota, Minneapolis) on August 19-23, 2019. During this five-day workshop, 42 participants performed collaborative research, in six teams, each focused on open problems in different areas of graph theory and its applications. The research work of each team was led by two experts in the corresponding area, who prior to the workshop, carefully selected relevant and meaningful open problems that would yield high-quality research and results of strong impact. As a result, all six teams have made significant contributions to several open problems in their respective areas. The workshop led to the creation of the Women in Graph Theory and Applications Research Network, which provided the framework to continue collaborating and to produce this volume.
Published January 2021
Spring Nature, 2021
Simplicity: Ideals of Practice in Mathematics and the Arts
To find "criteria of simplicity" was the goal of David Hilbert's recently discovered twenty-fourth problem on his renowned list of open problems given at the 1900 International Congress of Mathematicians in Paris. At the same time, simplicity and economy of means are powerful impulses in the creation of artworks. This was an inspiration for a conference, titled the same as this volume, that took place at the Graduate Center of the City University of New York in April of 2013. This volume includes selected lectures presented at the conference, and additional contributions offering diverse perspectives from art and architecture, the philosophy and history of mathematics, and current mathematical practice.
Published January 2021
Topology: a Categorical Approach
A graduate-level textbook that presents basic topology from the perspective of category theory.
This graduate-level textbook on topology takes a unique approach: it reintroduces basic, point-set topology from a more modern, categorical perspective. Many graduate students are familiar with the ideas of point-set topology and they are ready to learn something new about them. Teaching the subject using category theory—a contemporary branch of mathematics that provides a way to represent abstract concepts—both deepens students' understanding of elementary topology and lays a solid foundation for future work in advanced topics.
After presenting the basics of both category theory and topology, the book covers the universal properties of familiar constructions and three main topological properties—connectedness, Hausdorff, and compactness. It presents a fine-grained approach to convergence of sequences and filters; explores categorical limits and colimits, with examples; looks in detail at adjunctions in topology, particularly in mapping spaces; and examines additional adjunctions, presenting ideas from homotopy theory, the fundamental groupoid, and the Seifert van Kampen theorem. End-of-chapter exercises allow students to apply what they have learned. The book expertly guides students of topology through the important transition from undergraduate student with a solid background in analysis or point-set topology to graduate student preparing to work on contemporary problems in mathematics.
Published August 2020
MIT Press, 2020
Justification Logic: Reasoning with Reasons
Cambridge University Press, 2019
Classical logic is concerned, loosely, with the behaviour of truths. Epistemic logic similarly is about the behaviour of known or believed truths. Justification logic is a theory of reasoning that enables the tracking of evidence for statements and therefore provides a logical framework for the reliability of assertions. This book, the first in the area, is a systematic account of the subject, progressing from modal logic through to the establishment of an arithmetic interpretation of intuitionistic logic. The presentation is mathematically rigorous but in a style that will appeal to readers from a wide variety of areas to which the theory applies. These include mathematical logic, artificial intelligence, computer science, philosophical logic and epistemology, linguistics, and game theory.
Published June 2019
Rotation Sets and Complex Dynamics
This monograph examines rotation sets under the multiplication by d (mod 1) map and their relation to degree d polynomial maps of the complex plane. These sets are higher-degree analogs of the corresponding sets under the angle-doubling map of the circle, which played a key role in Douady and Hubbard's work on the quadratic family and the Mandelbrot set. Presenting the first systematic study of rotation sets, treating both rational and irrational cases in a unified fashion, the text includes several new results on their structure, their gap dynamics, maximal and minimal sets, rigidity, and continuous dependence on parameters. This abstract material is supplemented by concrete examples which explain how rotation sets arise in the dynamical plane of complex polynomial maps and how suitable parameter spaces of such polynomials provide a complete catalog of all such sets of a given degree. As a main illustration, the link between rotation sets of degree 3 and one-dimensional families of cubic polynomials with a persistent indifferent fixed point is outlined.
The monograph will benefit graduate students as well as researchers in the area of holomorphic dynamics and related fields.
Published June 2018
A Graduate Course in Algebra
volume 1 & 2
This comprehensive two-volume book deals with algebra, broadly conceived. Volume 1 (Chapters 1–6) comprises material for a first year graduate course in algebra, offering the instructor a number of options in designing such a course. Volume 1, provides as well all essential material that students need to prepare for the qualifying exam in algebra at most American and European universities. Volume 2 (Chapters 7–13) forms the basis for a second year graduate course in topics in algebra. As the table of contents shows, that volume provides ample material accommodating a variety of topics that may be included in a second year course. To facilitate matters for the reader, there is a chart showing the interdependence of the chapters.
Published August 2017
World Scientific Publishing Company, 2017
Combinatorial and Additive Number Theory II
Based on talks from the 2015 and 2016 Combinatorial and Additive Number Theory (CANT) workshops at the City University of New York, these proceedings offer 19 peer-reviewed and edited papers on current topics in number theory. Held every year since 2003, the workshop series surveys state-of-the-art open problems in combinatorial and additive number theory and related parts of mathematics. Sumsets, partitions, convex polytopes and discrete geometry, Ramsey theory, primality testing, and cryptography are among the topics featured in this volume. Each contribution is dedicated to a specific topic that reflects the latest results by experts in the field. Researchers and graduate students interested in the current progress in number theory will find this selection of articles relevant and compelling.
Published January 2017
Lie groups and Geometric Aspects of Isometric Actions
This book provides quick access to the theory of Lie groups and isometric actions on smooth manifolds, using a concise geometric approach. After a gentle introduction to the subject, some of its recent applications to active research areas are explored, keeping a constant connection with the basic material. The topics discussed include polar actions, singular Riemannian foliations, cohomogeneity one actions, and positively curved manifolds with many symmetries. This book stems from the experience gathered by the authors in several lectures along the years, and was designed to be as self-contained as possible. It is intended for advanced undergraduates, graduate students, and young researchers in geometry, and can be used for a one-semester course or independent study.
Published May 2015
Lipman Bers, a Life in Mathematics
Lipman Bers (1914–1993) was a Latvian-born US mathematician who worked in a number of fields, especially complex analysis, minimal surfaces, and partial differential equations. This volume was created to commemorate the 100th anniversary of his birth. It is very well done and contains about 1/3 biographical materials and 2/3 survey papers of his work, with just a tiny amount of new results.
The main biographical item is a memoir that Bers worked on until the end of the life, though by then it had only reached about the year 1940 when he emigrated to the United States. Bers was a socialist and a Jew living in middle Europe, and judging by this memoir he spent most of his time dodging the police and the Nazis. He received his PhD in 1938, so the mathematical content of this portion is fairly small. The memoir is supplemented by some notes by his children with further details of their family life. There is also a lengthy obituary and a series of appreciations by other mathematicians. There are a large number of family and professional photographs bound in the center of the book.
Published January 2015
American Mathematical Society, 2015
Fixed Point Theorems and Their Applications
IOANNIS FARMAKIS AND MARTIN MOSKOWITZ
This is the only book that deals comprehensively with fixed point theorems throughout mathematics. Their importance is due to their wide applicability, which the book demonstrates. Beyond the first chapter, each one can be read independently, giving readers flexibility to follow their own interests. The book is written for graduate students and professional mathematicians and may also be of interest to physicists, economists, and engineers.
Published June 2013
World Scientific, 2013
Quasiconformal Mappings, Riemann Surfaces, and Teichmüller Spaces
This volume includes a wide range of papers on Teichmüller theory and related areas. It provides a broad survey of the present state of research and the applications of quasiconformal mappings, Riemann surfaces, complex dynamical systems, Teichmüller theory, and geometric function theory. The papers in this volume reflect the directions of research in different aspects of these fields and also give the reader an idea of how Teichmüller theory intersects with other areas of mathematics.
Published January 2012
American Mathematical Society, 2012
Non-Commutative Cryptography and Complexity of Group-Theoretic Problems
This book is about relations between three different areas of mathematics and theoretical computer science: combinatorial group theory, cryptography, and complexity theory. It explores how non-commutative (infinite) groups, which are typically studied in combinatorial group theory, can be used in public-key cryptography. It also shows that there is remarkable feedback from cryptography to combinatorial group theory because some of the problems motivated by cryptography appear to be new to group theory, and they open many interesting research avenues within group theory. In particular, a lot of emphasis in the book is put on studying search problems, as compared to decision problems traditionally studied in combinatorial group theory. Then, complexity theory, notably generic-case complexity of algorithms, is employed for cryptanalysis of various cryptographic protocols based on infinite groups, and the ideas and machinery from the theory of generic-case complexity are used to study asymptotically dominant properties of some infinite groups that have been applied in public-key cryptography so far. This book also describes new interesting developments in the algorithmic theory of solvable groups and another spectacular new development related to complexity of group-theoretic problems, which is based on the ideas of compressed words and straight-line programs coming from computer science.
Published November 2011
American Mathematical Society, 2011
Functions of Several Real Variables
This book begins with the basics of the geometry and topology of Euclidean space and continues with the main topics in the theory of functions of several real variables including limits, continuity, differentiation and integration. All topics and in particular, differentiation and integration, are treated in depth and with mathematical rigor. The classical theorems of differentiation and integration such as the Inverse and Implicit Function theorems, Lagrange's multiplier rule, Fubini's theorem, the change of variables formula, Green's, Stokes' and Gauss' theorems are proved in detail and many of them with novel proofs. The authors develop the theory in a logical sequence building one result upon the other, enriching the development with numerous explanatory remarks and historical footnotes. A number of well chosen illustrative examples and counter-examples clarify matters and teach the reader how to apply these results and solve problems in mathematics, the other sciences and economics.
Each of the chapters concludes with groups of exercises and problems, many of them with detailed solutions while others with hints or final answers. More advanced topics, such as Morse's lemma, Sard's theorem , the Weierstrass approximation theorem, the Fourier transform, Vector fields on spheres, Brouwer's fixed point theorem, Whitney's embedding theorem, Picard's theorem, and Hermite polynomials are discussed in stared sections.
Published April 2011
World Scientific Publishing Company, 2011
Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory
This book is based on a 10-day workshop given by leading experts in hyperbolic geometry, quantum topology and number theory, in June 2009 at Columbia University. Each speaker gave a minicourse consisting of three or four lectures aimed at graduate students and recent PhDs. The proceedings of this enormously successful workshop can serve as an introduction to this active research area in a way that is expository and broadly accessible to graduate students.
Published January 2011
American Mathematical Society, 2011
The Use of Ultraproducts in Commutative Algebra
In spite of some recent applications of ultraproducts in algebra, they remain largely unknown to commutative algebraists, in part because they do not preserve basic properties such as Noetherianity. This work wants to make a strong case against these prejudices. More precisely, it studies ultraproducts of Noetherian local rings from a purely algebraic perspective, as well as how they can be used to transfer results between the positive and zero characteristics, to derive uniform bounds, to define tight closure in characteristic zero, and to prove asymptotic versions of homological conjectures in mixed characteristic. Some of these results are obtained using variants called chromatic products, which are often even Noetherian. This book, neither assuming nor using any logical formalism, is intended for algebraists and geometers, in the hope of popularizing ultraproducts and their applications in algebra.
Published July 2010
Set Theory and the Continuum Problem
Revised and updated from the 1996 edition, this volume is a lucid, elegant, and complete survey of set theory drawn from the authors' substantial teaching experience. The book is intended to provide the reader with a complete foundation in modern set theory, explaining how the subject is axiomatized and the role of the axiom of choice, and elucidating ordinal and cardinal numbers, constructible sets, and forcing. The first of three parts focuses on axiomatic set theory, examining problems related to size comparisons between infinite sets, basics of class theory, and natural numbers, as well as Smullyan's double induction principle, super induction, ordinal numbers, order isomorphism and transfinite recursion, and the axiom of foundation and cardinals. The second part addresses Mostowski-Shepherdson mappings, reflection principles, constructible sets and constructibility, and the continuum hypothesis. The text concludes with an extensive exploration of forcing and independence results.
Published April 2010
Techmuller Theory and Moduli Problems
Ramanujan Lecture Notes Series, Vol. 10
Proceedings of the U.S.-India workshop in Techmuller theory and moduli problems, Harish-Chandra Research Institute, Allahabad, January 2006
Published April 2010
International Press of Boston, 2010
(Advanced Courses in Mathematics - CRM Barcelona)
This first monograph to fathom non-commutative cryptography permits the construction of cryptographic protocols, with rigorous mathematical justification. The coauthors explore relations between three areas of mathematics and theoretical computer science: complexity theory, cryptography, and combinatorial group theory. The book is intended for computer scientists and mathematicians, and for both graduate students and undergraduates.
Published July 2008
Basic Lie Theory
This volume provides a comprehensive account of basic Lie theory, a subject at the center of mathematics. It is designed to be a text for a year's graduate course in Lie theory, bringing the student to the point of being able to embark on research in this subject and can also serve as a reference work for professional mathematicians specializing in other subjects. It covers, among other things, the structure of Lie algebras, Lie groups and their homogeneous spaces, Haar measure, symmetric spaces of non-compact type, and lattices. Hossein Abbaspour, a recent alumnus of the Ph.D. program in mathematics at the Graduate Center is currently an assistant professor at the University of Nantes, France.
Published November 2007
World Scientific, Singapore, 2007
Incompleteness in the Land of Sets
This book gives a full presentation of the basic incompleteness and undecidability theorems of mathematical logic in the framework of set theory. Corresponding results for arithmetic follow easily, and are also given. Gödel numbering is generally avoided, except when an explicit connection is made between set theory and arithmetic. The book assumes little technical background from the reader. One needs mathematical ability, a general familiarity with formal logic, and an understanding of the completeness theorem, though not its proof. All else is developed and formally proved, from Tarski's Theorem to Gödel's Second Incompleteness Theorem. Exercises are scattered throughout.
Published February 2007
College Publications, 2007
Logic at the Crossroads: An Interdisciplinary View
The "First Indian Conference on Logic and its Relationship with Other Disciplines" took place in Mumbai at IIT Bombay, from January 8 to 12, 2005. Tutorials as well as advanced talks were given by Indian logicians and by visitors from Australia, the Czech Republic, Finland, Great Britain, Italy, Israel, Japan, the Netherlands, the USA, and other nations. Talks ranged from reflections on the range of mathematical proof and definability; to recent developments in computational logic; to new interfaces between logic, information dynamics, and games; to presentations on schools of Indian logic. One term used nowadays for this broad view of logic is "intelligent interaction." The Mumbai Conference took this term in the double sense of both information exchange and community formation, and indeed both processes were in evidence. The current volume contains a representative selection of material from that conference.
Published January 2007
Mumbai, Delhi, and other locations: Allied Publishers, 2007
The Structure of Models of Peano Arithmetic
These authors give an account of the present state of research on lattices of elementary substructures and automorphisms of nonstandard models of arithmetic. Major representation theorems are proved, and the important particular case of countable recursively saturated models is discussed in detail. All necessary technical tools are developed. The list includes: constructions of elementary simple extensions; a partial classification of arithmetic types, in particular Gaifman's theory of definable types; forcing in arithmetic; elements of the Kirby-Paris combinatorial theory of cuts; Lascar's generic automorphisms; and applications of Abramson and Harrington's generalization of Ramsey's theorem. There are also chapters discussing ?1-like models with interesting second order properties, and a chapter on order types of nonstandard models.
Published December 2006
Oxford University Press, 2006
Riemannian Geometry: A Modern Introduction
This book provides an introduction to Riemannian geometry, the geometry of curved spaces, for use in a graduate course. Requiring only an understanding of differentiable manifolds, the author covers the introductory ideas of Riemannian geometry followed by a selection of more specialized topics. Also featured are Notes and Exercises for each chapter, to develop and enrich the reader's appreciation of the subject. This second edition, first published in 2006, has a clearer treatment of many topics than the first edition, with new proofs of some theorems and a new chapter on the Riemannian geometry of surfaces. The main themes here are the effect of the curvature on the usual notions of classical Euclidean geometry, and the new notions and ideas motivated by curvature itself. Completely new themes created by curvature include the classical Rauch comparison theorem and its consequences in geometry and topology, and the interaction of microscopic behavior of the geometry with the macroscopic structure of the space.
Published January 2006
Cambridge University Press, 2006
Research Problems in Discrete Geometry
Written for advanced undergraduates, graduate students, and researchers in discrete geometry, this book provides an overview of the 150-year-old field and presents a collection of more than 500 attractive open problems in the field, some old, others new and never before published, and the most important partial results related to these problems. Intended as a source book for both professional mathematicians and graduate students who love beautiful mathematical questions, it provides a fine selection of research problems for graduate students looking for a dissertation topic. Moser is professor emeritus at McGill.
Published September 2005
Springer-Verlag, New York, 2005
A Course in Complex Analysis in One Variable
Complex analysis is a beautiful subject — perhaps the single most beautiful; and striking; in mathematics. It presents completely unforeseen results that are of a dramatic; even magical; nature. This invaluable book will convey to the student its excitement and extraordinary character. The exposition is organized in an especially efficient manner; presenting basic complex analysis in around 130 pages; with about 50 exercises. The material constantly relates to and contrasts with that of its sister subject; real analysis. An unusual feature of this book is a short final chapter containing applications of complex analysis to Lie theory.
Since much of the content originated in a one-semester course given at the CUNY Graduate Center; the text will be very suitable for first year graduate students in mathematics who want to learn the basics of this important subject. For advanced undergraduates; there is enough material for a year-long course or; by concentrating on the first three chapters; for one-semester course.
Published April 2002
World Scientific Publishing Company, 2002
Isoperimetric Inequalities: Differential Geometric and Analytic Perspectives
This introduction treats the classical isoperimetric inequality in Euclidean space and contrasting rough inequalities in noncompact Riemannian manifolds. In Euclidean space the emphasis is on a most general form of the inequality sufficiently precise to characterize the case of equality, and in Riemannian manifolds the emphasis is on those qualitiative features of the inequality that provide insight into the coarse geometry at infinity of Riemannian manifolds. The treatment in Euclidean space features a number of proofs of the classical inequality in increasing generality, providing in the process a transition from the methods of classical differential geometry to those of modern geometric measure theory; and the treatment in Riemannian manifolds features discretization techniques, and applications to upper bounds of large time heat diffusion in Riemannian manifolds. The result is an introduction to the rich tapestry of ideas and techniques of isoperimetric inequalities, a subject that has its beginnings in classical antiquity and which continues to inspire fresh ideas in geometry and analysis to this very day--and beyond!
Published July 2001
Cambridge University Press, 2001
Introduction to Model Theory
Model theory investigates mathematical structures by means of formal languages. So-called first-order languages have proved particularly useful in this respect.
This text introduces the model theory of first-order logic, avoiding syntactical issues not too relevant to model theory. In this spirit, the compactness theorem is proved via the algebraically useful ultrsproduct technique (rather than via the completeness theorem of first-order logic). This leads fairly quickly to algebraic applications, like Malcev's local theorems of group theory and, after a little more preparation, to Hilbert's Nullstellensatz of field theory.
Steinitz dimension theory for field extensions is obtained as a special case of a much more general model-theoretic treatment of strongly minimal theories. There is a final chapter on the models of the first-order theory of the integers as an abelian group. Both these topics appear here for the first time in a textbook at the introductory level, and are used to give hints to further reading and to recent developments in the field, such as stability (or classification) theory.
Published October 2000
Many problems in general relativity are essentially geometric in nature, in the sense that they can be understood in terms of Riemannian geometry and partial differential equations. This book is centered around the study of mass in general relativity using the techniques of geometric analysis. Specifically, it provides a comprehensive treatment of the positive mass theorem and closely related results, such as the Penrose inequality, drawing on a variety of tools used in this area of research, including minimal hypersurfaces, conformal geometry, inverse mean curvature flow, conformal flow, spinors and the Dirac operator, marginally outer trapped surfaces, and density theorems. This is the first time these topics have been gathered into a single place and presented with an advanced graduate student audience in mind; several dozen exercises are also included.
The main prerequisite for this book is a working understanding of Riemannian geometry and basic knowledge of elliptic linear partial differential equations, with only minimal prior knowledge of physics required. The second part of the book includes a short crash course on general relativity, which provides background for the study of asymptotically flat initial data sets satisfying the dominant energy condition.
Published January 1999
American Mathematical Society, 1999