Softcover ISBN: 978-1-4704-8433-0
Product Code: SURV/295
Expected availability date: June 10, 2026
Mathematical Surveys and Monographs Volume: 295;
2026; Estimated: 430 pp
MSC: Primary 57; 53; 14; 32; 20
This book is a compendium of problems in low-dimensional topology, each presented with background and references. It is aimed at graduate students and more experienced researchers alike, highlighting the problem-driven nature of the field. The problems are intended to stimulate research and point to new directions in an area that has been extremely active and has broadened tremendously over the past 50 years.
The problem list is the outcome of a collaborative, community-driven effort led by R. İnanç Bakur, R. C. Kirby, and D. Ruberman. It began at a week-long workshop at the American Institute of Mathematics in fall 2023 and grew substantially through the involvement of chapter editors, workshop participants, problem proposers, scribes, and referee—altogether drawing contributions from several hundred researchers, from early career mathematicians to senior figures in the field.
As in the influential problem lists compiled by R. Kirby in the 1970s and 1990s, the book is organized roughly by dimension: knot theory, surfaces,
-manifolds,
-manifolds, and a brief miscellanea chapter. It features close to 400 problems across a wide range of topics, with nearly a thousand subproblems, questions, and conjectures interspersed throughout.
Graduate students and research mathematicians interested in low-dimensional topology.
Table of Contents
Knot theory
Surfaces
-manifolds
-manifolds
Miscellany
Bibliography
Problems carried over from K2
Index
Softcover ISBN: 978-1-4704-8045-5
Product Code: CONM/836
Expected availability date: June 10, 2026
Contemporary Mathematics Volume: 836;
2026; 298 pp
MSC: Primary 13; 20; 05; 11
This volume is based on talks given at the Special Session at the second International Joint Meeting of the Unione Matematica Italiana and the American Mathematical Society, held at the Universitá degli Studi di Palermo on July 23–26, 2024.
During the last twenty years, the theory involving the structure of the arithmetic and ideal theory of various algebraic structures has been a popular topic and taken several important steps forward. Many applications of this theory, with particular attention to the multiplicative monoids of integral domains and their combinatorial or numerical applications to ring theory, have appeared throughout the mathematical literature.
The aim of this volume is to review recent developments in this area by bringing together researchers from different areas of algebra under the umbrella of commutative monoids, semigroups, and rings. Topics include multiplicative ideal theory and general ideal systems, arithmetic in Krull and Prüfer monoids, commutative monoid rings, integer-valued polynomials, numerical and congruence monoids, direct sum decompositions of modules, and various aspects of non-unique factorization.
Graduate students and research mathematicians interested in the arithmetic and ideal theory of commutative structures.
Alfred Geroldinger, Hwankoo Kim, and K. Alan Loper — On long-term problems in multiplicative ideal theory and factorization theory
Kai Steve Fan and Paul Pollack — Extremal elasticity of quadratic orders
Nathan Kaplan, Kaylee Kim, Cole McGeorge, Fabian Ramirez, and Deepesh Singhal — On the smallest partition associated to a numerical semigroup
S. Bonzio and P. A. García-Sánchez — When the divisibility poset of the ideal class monoid of a numerical semigroup is a lattice
Sogol Cyrusian, Alex Domat, Christopher O’Neill, Vadim Ponomarenko, Eric Ren, and Mayla Ward — On numerical semigroup elements and the
and norms of their factorizations
Scott T. Chapman, Felix Gotti, Marly Gotti, and Harold Polo — On three families of dense Puiseux monoids
Kamil Merito, Oscar Ordaz, and Wolfgang A. Schmid — The set of minimal distances of the monoid of plus-minus weighted zero-sum sequences and applications to the characterization problem
Andreas Reinhart — On counterexamples to Mordell’s Pellian Equation Conjecture and the AAC Conjecture: A non-computer-based approach
Jared Kettinger — A generalized Davenport constant of the second kind
Jesse Elliott and Neil Epstein — Additive subgroups of a module that are saturated with respect to a subset of the ring
Djamila AitElhadi and Ayman Badawi — The
-total graph of a commutative ring
Joseph Swanson — Radii of convergence of algebraic power series
Davide Castelnovo, Dikran Dikranjan, Anna Giordano Bruno, Dario Spirito, and Simone Virili — A length function of
-modules and Mahler measures
Damiano Saccone — Weakly Arf property for quadratic quotients of the Rees algebra
Softcover ISBN: 978-1-4704-7727-1
Product Code: CONM/837
Expected availability date: June 10, 2026
Contemporary Mathematics Volume: 837;
2026; 195 pp
MSC: Primary 16; 17; 14; 55; 13
This volume contains the proceedings of the AMS Special Session on Representation Theory and Flag Varieties, held at the AMS Sectional Meeting at SUNY Buffalo on September 9–10, 2023.
Representation theory, which originated with Frobenius's work on group representations in the late 19th century, explores symmetries in mathematics and natural sciences. It has evolved into a broad field since then. Its geometric approach, leverages flag and Nakajima quiver varieties, produces significant results such as the resolution of the Kazhdan-Lusztig conjecture, connects to disciplines such as enumerative geometry, algebraic geometry, and mathematical physics, and in return reveals hidden structures within these varieties. This volume compiles recent advancements in representation theory and flag/quiver variety geometry, offering original research and expository articles. Contributions include work on generalized Shubert calculus, quantum
-theory of semi-infinite flag variety, research on geometry of Springer fibers and their variants, combinatorial model for quiver varieties, and results on
-connections in the geometric Langlands program, providing fresh insights and methodologies.
Graduate students and research mathematicians interested in geometry related to representations of semisimple Lie groups.
Mee Seong Im, Shifra Reif, and Vera Serganova — The Grothendieck ring of the periplectic Lie supergroup and supersymmetric functions
Julianna S. Tymoczko — Divided difference operators for partial flag varieties
Daniele Rosso and Neil Saunders — Exotic Spaltenstein varieties
Rebecca Goldin and Martha Precup — Minimal semisimple Hessenberg schemes
Drew Meyer — On the pure dimensionality of Spaltenstein varieties: A family of counterexamples
Daniel S. Sage — Connections on the projective line whose differential Galois groups are as large as possible
Li Li — Nakajima’s quiver varieties and triangular bases of bipartitle cluster algebras
Changlong Zhong — Elliptic Schubert clases and the Poincaré duality
Yiqiang Li — Quasi-split symmetric pairs of type A and Steinberg varieties of classical type, II. Constructible functions
January 13, 2026
Paperback ISBN: 9780443439247
An Introduction to Writing Mathematical Proofs: Shifting Gears from Calculus to Advanced Mathematics addresses a critical gap in mathematics education, particularly for students transitioning from calculus to more advanced coursework. It provides a structured and supportive approach, guiding students through the intricacies of writing proofs while building a solid foundation in essential mathematical concepts. Sections introduce elementary proof methods, beginning with fundamental topics such as sets and mathematical logic, systematically develop the properties of real numbers and geometry from a proof-writing perspective, and delve into advanced proof methods, introducing quantifiers and techniques such as proof by induction, counterexamples, contraposition, and contradiction.
Finally, the book applies these techniques to a variety of mathematical topics, including functions, equivalence relations, countability, and a variety of algebraic activities, allowing students to synthesize their learning in meaningful ways. It not only equips students with essential proof-writing skills but also fosters a deeper understanding of mathematical reasoning. Each chapter features clearly defined objectives, fully worked examples, and a diverse array of exercises designed to encourage exploration and independent learning. Supplemented by an Instructors' Resources guide hosted online, this text is an invaluable companion for undergraduate students eager to master the art of writing mathematical proofs.
1. Introduction
Section I: Elementary Proof Methods: Our First Bicycle
2. Sets and Notation - Introduction to basic set theory
3. Mathematical Logic - Basic logic needed to be able to be able to write proofs.
4. Properties of Real Numbers - Systematically builds up the properties of real numbers
5. Geometry Revisited - Approaches topics from high school geometry from the point of view of proof writing.
Section II: Advanced Proof Methods: Bicycles with Multiple Gears
6. Quantifiers and Induction - Introduces quantifiers and the technique of proof by induction
7. The Three C's: Counterexamples, Contraposition, and Contradiction - Introduces these indirect proof methods
Section III: Using Our Techniques: A Mathematical Tour de France
8. Fun with Functions and Relations - An Exploration and Opportunities for writing proofs involving functions and relations
9. An Amalgam of Algebraic Activities - Opportunities to write proofs for algebraic topics
10. Appendix
11. Index
Published: February 2026
Format: Hardback
ISBN: 9781009579520
Discover the foundations of classical and quantum information theory in the digital age with this modern introductory textbook. Familiarise yourself with core topics such as uncertainty, correlation, and entanglement before exploring modern techniques and concepts including tensor networks, quantum circuits and quantum discord. Deepen your understanding and extend your skills with over 250 thought-provoking end-of-chapter problems, with solutions for instructors, and explore curated further reading. Understand how abstract concepts connect to real-world scenarios with over 400 examples, including numerical and conceptual illustrations, and emphasising practical applications. Build confidence as chapters progressively increase in complexity, alternating between classic and quantum systems. This is the ideal textbook for senior undergraduate and graduate students in electrical engineering, computer science, and applied mathematics, looking to master the essentials of contemporary information theory.
A self-contained introduction that can be tailored for course use or self-study
Introduces Information Theory with a stronger focus on practical quantification of uncertainty, information, correlation, and entanglement
Highlights definitions and key results for quick reference and includes chapter appendices containing proofs and supplementary material
Preface
References
Acknowledgements
Notations
Acronyms
Part I. Classical Information:
1. Uncertainty, information, and entropy
2. Information quantification by asking, compressing, or sampling: Shannon entropy
3. Information quantification by predicting or guessing: Tsallis entropy and Rényi entropy
4. Relative entropy
Part II. Quantum Information:
5. From classical to quantum information
6. Quantum uncertainty: measured entropy, coherence, and the uncertainty principle
7. Classical and quantum uncertainty: mixed states and quantum entropy
8. Quantum relative entropy
Part III. Dynamic Information:
9. Dynamic classical information
10. Quantum dynamic information in closed systems
11. Quantum dynamic information in open systems
Part IV. Bipartite Classical Information:
12. Bipartite classical information as correlation
13. Bipartite classical information via residual uncertainty
Part V. Bipartite Quantum Information:
14. Bipartite classical and quantum Information for pure states
15. Bipartite dynamic classical and quantum information for pure states
16. Bipartite quantum and classical information for mixed states
Part VI. Multipartite Classical and Quantum Information:
17. A brief introduction to tensor networks
18. Multipartite classical information: fragmentation, scale, and strength
19. Multipartite classical information: structure
20. Multipartite quantum information: fragmentation, scale, and strength
21. Multipartite quantum information: structure
Index.
Hardback
ISBN 9781041197140
448 Pages 58 B/W Illustrations
April 24, 2026 by Chapman & Hall
Few other books purport to be a second course in complex analysis. This book differs in that it covers more modern topics and is more geometric in focus. Most texts on complex variable theory contain the same material. However, complex analysis is a vast and diverse subject with a long history and many aspects. A second course will benefit students and introduce these new topics that they might not otherwise experience.
Lars Ahlfors alone invented many new parts of the subject; Lipman Bers made decisive contributions, and there are many others. It is easy to justify a “second course” in complex analysis. That is what this book purports to be.
Some of the topics presented here are:
• harmonic measure
• extremal length
• Riemann surfaces
• uniformization
• automorphism groups
• the Schwarz lemma and its generalizations
• analytic capacity
• the Bergman theory
• invariant metrics
• Picard’s theorem
• the boundary Schwarz lemma
The goal is to expose the reader to unfamiliar parts of the subject of complex variables and perhaps to pique interest in further reading. As with the authors’ other books, not only theorems and proofs are included, but also many examples and some exercises. Numerous graphics illustrate the key ideas.
1. Preliminaries
1.1 Holomorphic Functions
1.2 Meromorphic Functions
1.3 Algebraic Functions
1.4 The Mittag-Leffler Problem in an Open Planar Domain
1.5 The Compact Support Case
1.6 The General Case
1.7 Paths and Curves
1.8 Continuous Choice of Argument
1.9 Homotopic Curves
1.10 Index of a Loop
1.11 The Fundamental Group
1.12 Connectivity
1.13 The Brouwer Fixed Point Theorem
1.14 Gamma Function
2. Extremal Length
2.1 Some Definitions
2.2 The Conformal Invariance of Extremal Length
2.3 Further Properties of Extremal Lengths
2.4 Some Examples
3. Harmonic Measure
3.1 The Idea of Harmonic Measure
3.2 A Discussion of Interpolation of Linear Operators
3.3 The F. and M. Riesz Theorem
4. Riemann Surfaces
4.1 The Riemann Surface of a Function
4.2 Constructions of Riemann Surfaces
4.3 Analytic Continuation of Complex Functions
4.4 Riemann Surfaces of Functions
4.4.5 Global Topological Properties
4.5 Puiseux Series
4.6 Algebraic Curves
4.7 The Connectedness of Algebraic Curves
4.8 Riemann Surfaces Associated to a Polynomial
4.9 Smooth Projective Plane Curves
4.10 Compact Riemann Surfaces
4.11 Projective Algebraic Curves
4.12 Projective Closure and Affine Restriction
4.13 Example 4.4.12 (continuation)
4.14 Smooth and Singular Points on Affine and Projective Curves
4.15 Chow’s Theorem
5. Abstract Riemann Surfaces
5.1 Basic Definitions
5.2 Examples of Riemann Surfaces
5.3 Genus of a Compact Riemann Surfaces
5.4 Triangulations of Riemann Surfaces
5.5 A Short Proof that Compact 2-Manifolds Can Be Triangulated
5.6 Holomorphic and Meromorphic Functions
5.7 Elliptic Functions
5.8 Rational Functions on Riemann Surfaces
5.9 The Argument Principle on Riemann Surfaces
5.10 Construction Conceived by Riemann
5.11 The Riemann–Hurwitz Formula — Applications
5.12 Any Two Meromorphic Functions are Algebraically Related
5.13 Holomorphic and Meromorphic Differential Form
5.14 Harmonic Differential Forms
5.15 The Dimension of the Space of Holomorphic Differential Forms Ω1(X)
5.16 Every Riemann Surface Admits a Non-constant Meromorphic Function
6. The Riemann–Roch Theorem
6.1 Introduction
6.2 Baby Proof of the Riemann–Roch Theorem
6.3 The Residue Theorem on Compact Riemann Surfaces
6.4 Dependence Among the Linear Constraints
6.5 Next Steps Towards the Riemann–Roch Theorem
6.6 Higher Order Poles
6.7 Divisors
6.8 The Riemann–Roch Problem in Terms of Divisors
6.9 The Canonical Divisor
6.10 The Riemann–Roch Formula
6.11 The Geometric Riemann–Roch Theorem
6.12 Embedding into Projective Space
6.13 Compact Riemann Surfaces and Algebraic Curves
6.14 Hyperelliptic Integral
6.15 Abel’s Theorem
6.16 The Jacobi Inversion Problem
7. Covering Surfaces and Classical Plane Geometries
7.1 Normal Families and Automorphisms
7.2 The Basic Examples
7.3 Riemann Surfaces and Covering Spaces
7.4 Covering Spaces and Invariant Metrics, I: Quotients of C
7.5 Covering Spaces and Invariant Metrics, II: Quotients of D
7.6 Compact Quotients of D and Their Automorphisms
7.7 The Automorphism Group of a Riemann Surface of Genus at Least 2
7.8 Automorphisms of Multiply Connected Domains
8. The Uniformization Theorem
8.1 Preliminary Results
8.2 Green’s Functions
8.3 Bipolar Green’s Functions
9. Analytic Capacity
9.1 Calculating Analytic Capacity
9.2 Analytic Capacity and Removability
10. The Bergman Kernel
10.1 Smoothness to the Boundary of KΩ
10.2 Calculating the Bergman Kernel
10.3 The Poincaré-Bergman Metric on the Disc
10.4 Appendix to Section 10.2: The Biholomorphic Inequivalence of the Ball and the Polydisc
11. Appendix
11.1 Covering Spaces
11.2 The Sheaf of Germs of Holomorphic Functions
11.3 Sheaves
11.4 Polynomials
Closing Remarks
Hardback
ISBN 9781041253228
176 Pages 64 B/W Illustrations
April 20, 2026 by Chapman & Hall
Convexity is an ancient idea going back to Archimedes. Used sporadically in mathematical literature over the centuries, today it is a flourishing area of research. Convexity is used in optimization theory, functional analysis, complex analysis, and other parts of mathematics.
This text, popular in its first edition, introduces analytic tools for studying convexity and provides analytical applications of the concept. It includes a general background on classical geometric theory, revealing a glimpse of how modern mathematics is developed and how geometric ideas may be studied analytically.
The book contains copious examples, many applications, and plenty of figures. It also includes an appendix which offers the technical tools needed to understand certain arguments in the book, a table of notation, and a thorough glossary to help readers with unfamiliar terms.
The book presents an analytic way to think about convexity theory. Although this means of doing things is well known to the experts, it is not well documented in the literature. The reader with only a basic background in real analysis (and perhaps a little linear algebra) can get a lot out of this book. This book is a definitive introductory text to the concept of convexity in the context of mathematical analysis and a suitable resource for students and faculty alike.
1. Basic Ideas
1.0 Introduction
1.1 The Classical Theory
1.2 Separation Theorems
1.3 Approximation
2. Functions
2.1 Defining Function
2.2 Analytic Definition
2.3 Convex Functions
2.4 Exhaustion Functions
3. More on Functions
3.1 Other Characterizations
3.2 Convexity of Finite Order
3.3 Extreme Points
3.4 Support Functions
3.5 Approximation from Below
3.6 Bumping
4. Applications
4.1 Nowhere Differentiable Functions
4.2 The Krein-Milman Theorem
4.3 The Minkowski Sum
4.4 Brunn-Minkowski
5. Sophisticated Ideas
5.1 The Polar of a Set
5.2 Optimization
5.3 Generalizations
5.4 Integral Representation
5.5 The Gamma Function
5.6 Hard Analytic Facts
5.7 Sums and Projections
6. The MiniMax Theorem
6.1 von Neuman’s Theorem
7. Concluding Remarks
Appendix: Technical Tools
Table of Notation
Glossary