Softcover ISBN: 978-1-4704-8152-0
Product Code: MBK/154
2025; 292 pp
MSC: Primary 11
This is an introduction to analytic number theory developed through the study of the distribution of prime numbers, highlighting how analytic number theorists think. The central focus is on the Prime Number Theorem, presented through a proof selected to balance conceptual understanding with technical depth, alongside a sketch of Riemann's classical approach to highlight the subject's elegance. Providing a wide range of further directions (e.g., sieve methods, the anatomy of integers, primes in arithmetic progressions, prime gaps, smooth numbers, and extensive discussion of probabilistic heuristics which play an important role in guiding research goals), the emphasis throughout the book is on clarity of argument and the development of technique using a conversational style of writing. The book ends with 13 short introductions to hot topics.
The book assumes familiarity with elementary number theory and basic complex analysis, though it provides helpful review material. Boxed equations highlight the most memorable formulas; exercises, some embedded directly in the proofs, are designed to deepen understanding without becoming overwhelming. Its flexible structure makes the book suitable for various course designs, whether emphasizing core theory or incorporating optional sections on combinatorics, arithmetic progressions, or open research problems. By blending classical results with current perspectives, this book prepares advanced undergraduates and beginning graduate students to not just learn analytic number theory, but to acquire contemporary ways of thinking about the subject.
Undergraduate and graduate students interested in modern analytic number theory.
Background in analytic number theory
How many primes are there?
Unconditional estimates for sums over primes
Partial summation, and consequences of the Prime Number Theorem
What should be true about primes?
The modified Gauss-Cramér heuristic
Multiplicative functions and Dirichlet series
Anatomies of mathematical objects
Counting irreducibles
The average number of indecomposables
The typical number of indecomposables
Normal distributions
The multiplication table
With two or more parts
Poisson and beyond
Sieves and primes
The Chinese Remainder Theorem as a sieve
A first look at sieve methods
Background in analysis
Fourier series, Fourier analysis, and Poisson summation
Complex analysis
Analytic continuation of the Riemann zeta-function
Perron’s formula
The use of Perron’s formula
The proof of the Prime Number Theorem
Riemann’s plan for proving the Prime Number Theorem
Technical remarks
Zeros of
with
Proof of the Prime Number Theorem
The Riemann Hypothesis without zeros of
Primes in arithmetic progressions
Primes in arithmetic progressions
Dirichlet characters
Dirichlet
-functions
The Prime Number Theorem for arithmetic progressions
The Generalized Riemann Hypothesis
A dozen and one different directions
Exceptional zeros and primes in arithmetic progressions
Selberg’s small sieve
Equidistribution in arithmetic progressions?
Distribution of the error in the Prime Number Theorem
Chebyshëv’s bias
Primes in short intervals
Smooths, factoring, and large gaps between primes
Short gaps between primes
The circle method
Primes missing digits
Towards the prime
-tuplets conjecture
Prime values of higher-degree polynomials
Primes in sparse sequences
Probability primer
Couting prime factors with multiplicity
Analytic continuation for certain Dirichlet series
Different proofs of the Prime Number Theorem
Sketch of technical proofs
Circle method primer
Image credits
Bibliography
Index
Softcover ISBN: 978-1-4704-7437-9
Product Code: PSPUM/112.1
Expected availability date: January 28, 2026
Proceedings of Symposia in Pure Mathematics, Volume: 112;
2025; 552 pp
MSC: Primary 11; 14; 22
These three volumes comprise the proceedings of a summer school on the
Langlands Program held at the Institut des Hautes Études Scientifiques
during the three weeks of July 11–29, 2022. The twenty-five articles in
these proceedings capture the content of lectures given by thirty-one leading
experts at the summer school. They showcase the state of the art in some
(but not all) aspects of the Langlands program, such as the theory of endoscopy,
the trace formula, the local Langlands correspondence, Shimura varieties,
and shtukas. In addition, the volumes highlight several emerging unifying
themes and new connections that are expected to be influential in the future
development of the subject, such as the ideas of geometrization and categorification
and the evolving Relative Langlands Program. The broad spectrum of topics
reflects the continued growth of the Langlands Program, and the articles
are written to help overcome the language barrier that sometimes exists
between different subfields, while highlighting the connecting threads
that run through different parts of it. These volumes should serve as a
useful resource for beginning PhD students as well as more seasoned researchers
who are interested in learning about the new directions and developments
of this fascinating subject.
Graduate students and researchers Interested in various aspects of algebraic geometry.
Part 1. Representations, automorphic forms, and endoscopy
Olivier Taïbi — The local Langlands conjecture
Jessica Fintzen — Supercuspidal representations: Construction, classification, and characters
Lucas Mason-Brown — Arthur’s conjectures and the orbit method for real reductive groups
Tasho Kaletha — A brief introduction to the trace formula and its stabilization
Erez Lapid — Some perspectives on Eisenstein series
Part 2. Shimura varieties and shtukas
Sophie Morel — Shimura varieties
Zhiwei Yun — Introduction to shtukas and their moduli
Jared Weinstein — Shtukas and the Langlands program: A bird’s eye view
Ana Caraiani and Sug Woo Shin — Recent progress on Langlands reciprocity for general linear groups: Shimura varieties and beyond
Cong Xue — Cohomology sheaves of stacks of shtukas
Sam Raskin — An arithmetic application of geometric Langlands
Softcover ISBN: 978-1-4704-7438-6
Product Code: PSPUM/112.2
Expected availability date: January 28, 2026
Proceedings of Symposia in Pure Mathematics
Volume: 112; 2025; Estimated: 551 pp
MSC: Primary 11; 14; 22
These three volumes comprise the proceedings of a summer school on the Langlands Program held at the Institut des Hautes Études Scientifiques during the three weeks of July 11–29, 2022. The twenty-five articles in these proceedings capture the content of lectures given by thirty-one leading experts at the summer school. They showcase the state of the art in some (but not all) aspects of the Langlands program, such as the theory of endoscopy, the trace formula, the local Langlands correspondence, Shimura varieties, and shtukas. In addition, the volumes highlight several emerging unifying themes and new connections that are expected to be influential in the future development of the subject, such as the ideas of geometrization and categorification and the evolving Relative Langlands Program. The broad spectrum of topics reflects the continued growth of the Langlands Program, and the articles are written to help overcome the language barrier that sometimes exists between different subfields, while highlighting the connecting threads that run through different parts of it. These volumes should serve as a useful resource for beginning PhD students as well as more seasoned researchers who are interested in learning about the new directions and developments of this fascinating subject.
Graduate students and researchers Interested in various aspects of algebraic geometry.
Part 3. Geometrization of the Langlands correspondence
Jean-François Dat — Moduli spaces of local Langlands parameters
Xinwen Zhu — Coherent sheaves on the stack of Langlands parameters
Laurent Fargues and Peter Scholze — The Langlands program and the moduli of bundles on the curve
Matthew Emerton, Toby Gee and Eugen Hellmann — An introduction to the categorical
-adic Langlands program
David Ben-Zvi, Harrison Chen, David Helm and David Nadler — Between coherent and constructible local Langlands correspondences
Tony Feng and Michael Harris — Derived structures in the Langlands correspondence
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Softcover ISBN: 978-1-4704-7439-3
Product Code: PSPUM/112.3
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Book Details
Proceedings of Symposia in Pure Mathematics
Volume: 112; 2025; 441 pp
MSC: Primary 11; 14; 22
These three volumes comprise the proceedings of a summer school on the Langlands Program held at the Institut des Hautes Études Scientifiques during the three weeks of July 11–29, 2022. The twenty-five articles in these proceedings capture the content of lectures given by thirty-one leading experts at the summer school. They showcase the state of the art in some (but not all) aspects of the Langlands program, such as the theory of endoscopy, the trace formula, the local Langlands correspondence, Shimura varieties, and shtukas. In addition, the volumes highlight several emerging unifying themes and new connections that are expected to be influential in the future development of the subject, such as the ideas of geometrization and categorification and the evolving Relative Langlands Program. The broad spectrum of topics reflects the continued growth of the Langlands Program, and the articles are written to help overcome the language barrier that sometimes exists between different subfields, while highlighting the connecting threads that run through different parts of it. These volumes should serve as a useful resource for beginning PhD students as well as more seasoned researchers who are interested in learning about the new directions and developments of this fascinating subject.
Graduate students and researchers Interested in various aspects of algebraic geometry.
Part 4. Periods,
-functions, and the relative trace formula
Raphaël Beuzart-Plessis — Introduction to the Relative Langlands program
Wee Teck Gan — Explicit constructions of automorphic forms: theta correspondence and automorphic descent
Dipendra Prasad — Homological aspects of branching laws
Pierre-Henri Chaudouard — Introduction to the (relative) trace formula
Ngô Bảo Châu — On generalized Hitchin fibrations and orbital integrals
Yiannis Sakellaridis — Local and global questions “beyond endoscopy”
Wei Zhang — High dimensional Gross–Zagier formula: a survey
Chao Li — Geometric and arithmetic theta correspondences
Hardcover ISBN: 978-1-4704-6124-9
Product Code: PCMS/29
Expected availability date: January 18, 2026
IAS/Park City Mathematics Series Volume: 29;
2025; 322 pp
MSC: Primary 11
This book is a collection of lecture notes from the Graduate Summer School “Number Theory Informed by Computation” held at the IAS/Park City Mathematics Institute 2022. The lectures address topics at the interface between number theory and computation, both terms interpreted broadly. Included are lectures on polynomial-time algorithms for problems in algebraic number theory, algorithms for counting points on mod
reduction of a variety working with many values of
simultaneously, arithmetic statistics, the theory of lattices, Brauer–Manin obstruction, the theory of rigid cocycles, and the inverse Galois problem. The volume starts with the notes of Hendrik Lenstra’s public lecture, aimed at a general audience, where the speaker explains how modern abstract algebra can be used to explain concrete properties of integers.
This volume is aimed at students with a background in graduate level number theory. For some expositions, undergraduate abstract algebra may be sufficient, while others might require basic algebraic geometry.
Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute.
Undergraduate and graduate students and researchers interested in computational number theory.
Hendrik Lenstra — Algebra in the real world
Daniël M. H. van Gent — Polynomial-time algorithms in algebraic number theory
David Harvey — Counting points on hyperelliptic curves over finite fields
Melanie Matchett Wood — Number field counting, class group heuristics, and computation
Joseph H. Silverman — An introduction to lattices, lattice reduction, and lattaice-based cryptography
Bianca Viray — Rational points on varieties and the Brauer–Manin obstruction
J. B. Vonk — Rigid cocycles and singular moduli for real quadratic fields
Olivier Wittenberg — Park City lecture notes: Around the inverse Galois problem
Softcover ISBN: 978-1-4704-5079-3
Product Code: HMATH/49
Expected availability date: January 28, 2026
History of Mathematics Volume: 49;
2025; Estimated: 372 pp
MSC: Primary 81; 01
This book offers a historical analysis of Bartel Leendert van der Waerden’s early contributions to quantum mechanics, in particular, focusing on his role in the development and application of group-theoretic methods around 1930. While van der Waerden is widely known for his modern algebra, his engagement with quantum theory has received little attention in the historical literature.
Through careful study of published and archival sources, the author reconstructs the contexts in which van der Waerden worked, and examines his networks as well as his interactions with contemporaries such as Hermann Weyl, Eugene Wigner, Paul Ehrenfest, and John Slater. By comparing the different emerging approaches of adapting group-theoretic methods to problems in quantum mechanics, the author shows that van der Waerden stands out as a rather pragmatic and undogmatic thinker. Special attention is given to reconstructing a more nuanced picture of the arising controversy around the “group plague” in quantum mechanics.
Combining technical exposition with biographical and historical analysis, the book is intended for mathematicians, physicists, and historians of science. The reader is expected to have beginning, graduate-level knowledge of mathematics—mainly algebra and, in particular, representation theory—as well as some background in quantum physics. The study sheds light on a complex and fascinating period in the development of mathematical physics. It explores the dynamics of the interactions between mathematics and physics, and thus contributes to ongoing discussions about mathematization processes in science in the twentieth century.
Undergraduate and graduate students and researchers interested in van der Waerden and the history of quantum mechanics.
Background: The development until circa 1928
The history of representation theory
On the development of quantum mechanics
Group theory and quantum mechanics until 1928
van der Waerden’s scientific development until 1928
van der Waerden’s first steps in quantum mechanics in Groningen
van der Waerden as a Professor in Groningen (1928-1931)
The spinor calculus as a commissioned work for Ehrenfest
Spinor calculus and wave equation
van der Waerden and his work on quantum mechanics in Leipzig
van der Waerden as a Professor in Leipzig (1931-1945)
Survey of van der Waerden’s monograph on the group-theoretic method in quantum mechanics
Representation theory via groups with operators
Constructing representations
Applications of group theory in quantum mechanics
Dealing with Slater’s group-free method
Molecular spectra
Spinors in general relativity
Reprecussions on mathematics: The Casimir operator
Future prospects: van der Waerden and physics after 1945
Turn towards applied mathematics (1945-1951)
van der Waerden’s Professorship in Zurich (1951-1972)
List of archives consulted
Credits
Bibliography
Softcover ISBN: 978-1-4704-8201-5
Product Code: HMATH/50
Expected availability date: January 18, 2026
History of Mathematics Volume: 50;
2025; 145 pp
MSC: Primary 11
Hermann Minkowski (1864–1909) was a prominent member of the famous golden epoch of mathematics at Göttingen University in Germany, which started with Karl Friedrich Gauss in 1807 and continued into the twentieth century (until the beginning of World War II). Minkowski is well known for his foundational work introducing the “Minkowski spacetime” in special relativity. He made significant contributions to number theory, especially in geometry of numbers and other topics. Minkowski was also known as an excellent teacher.
The present book is an English translation of notes (never published) from the course on number theory taught by Minkowski in Winter 1907/08. The main topic of the course was the Fermat theorem and various methods used to prove it at that time. The translation is appended by the foreword written by John Little, who also edited the translation.
The book can be helpful and interesting to students in number theory and to anyone interested in mathematics and its history.
Undergraduate and graduate students and researchers interested in the development of important ideas and notions of algebraic number theory and in early attempts to prove the Fermat theorem.
Fermat’s theorem
The theory of algebraic numbers
Special theory of prime ideals
Fermat’s theorem revisited
Index
Softcover ISBN: 978-1-4704-7619-9
Product Code: CONM/829
Expected availability date: January 04, 2026
Contemporary Mathematics Volume: 829;
2025; 291 pp
MSC: Primary 11; 15; 16; 17; 18
From July 3 to 7, 2024, Taras Shevchenko National University of Kyiv, A.S. Makarenko Sumy State Pedagogical University, and the Institute of Mathematics of the National Academy of Sciences of Ukraine held the 14th Ukraine Algebra Conference as part of a series of biannual conferences originally initiated by V. Kirichenko, V. Sushchansky, and V. Usenko.
The 14th Ukraine Algebra Conference was held online due to Russian bombings. It brought together 350 participants from nearly 40 countries and featured 150 talks.
The conference proceedings consist of two volumes: Volume 1 contains papers on Representation Theory, and Volume 2 (Contemporary Mathematics, Volume 830) contains papers on Groups and Algebras.
Graduate students and research mathematicians interested in topics in algebra.
Tomoyuki Arakawa, Thomas Creutzig and Kazuya Kawasetsu — On Lisse non-admissible minimal and principal W-algebras
Juan Camilo Arias, Vyacheslav Futorny and André de Oliveira — The category of reduced imaginary Verma modules
R. Bezrukavnikov and K. Tolmachov — Exterior powers of a parabolic Springer sheaf on a Lie algebra
Igor Burban and Yuriy Drozd — Some aspects of the theory of nodal orders
Igor Burban and Semyon Klevtsov — Norms of wave functions for FQHE models on a torus
Rocco Chirivì, Xin Fang and Peter Littelmann — Schubert valuations on Grassmann varieties
Kevin Coulembier, Pavel Etingof, Victor Ostrik and Daniel Tubbenhauer — Fractal behavior of tensor powers of the two dimensional space in prime characteristic
Gabriella D’Este and H. Melis Tekin Akcin — Bijections between
-rigid and
-tilting modules
Joel Gibson, Daniel Tubbenhauer and Geordie Williamson — Equivariant neural networks and piecewise linear representation theory
Maria Gorelik and Victor G. Kac — On simplicity of universal minimal
-algebras
Alex Martsinkovsky — The defect, the Malgrange functor, and linear control systems
Volodymyr Mazorchuk — The tale of Kostant’s problem
Ian M. Musson — Weyl groupoids and superalgebraic sets
Oksana S. Yakimova — Some remarks on periodic gradings
Softcover ISBN: 978-1-4704-7620-5
Product Code: CONM/830
Expected availability date: January 04, 2026
Contemporary Mathematics Volume: 830;
2025; 285 pp
MSC: Primary 03; 14; 16; 15; 20
From July 3 to 7, 2024, Taras Shevchenko National University of Kyiv, A.S. Makarenko Sumy State Pedagogical University, and the Institute of Mathematics of the National Academy of Sciences of Ukraine held the 14th Ukraine Algebra Conference as part of a series of biannual conferences originally initiated by V. Kirichenko, V. Sushchansky, and V. Usenko.
The 14th Ukraine Algebra Conference was held online due to Russian bombings. It brought together 350 participants from nearly 40 countries and featured 150 talks.
The conference proceedings consist of two volumes: Volume 1 (Contemporary Mathematics, Volume 829) contains papers on Representation Theory, and Volume 2 contains papers on Groups and Algebras.
Graduate students and research mathematicians interested in topics in algebra.
Nicolás Andruskiewitsch and Ken A. Brown — On the bosonization of the enveloping algebra of a finite dimensional Lie superalgebra
Taras Banakh and Andriy Rega — Polyboundedness of zero-closed semigroups
Ayşe Berkman and Alexandre Borovik — Primitive permutation groups of finite Morley rank and affine type
Oksana Bezushchak — On Clifford algebras of infinite dimensional vector spaces
Mikhailo Dokuchaev and Nicola Sambonet — Partial actions in terms of category theory
Alexander Dranishnikov — Distributional topological complexity of finitely generated groups
Fatma Azmy Ebrahim, Sergio R. López-Permouth and Majed Zailaee — Symmetry questions about amenability and simplicity of bases
Rostislav Grigorchuk and Ville Salo — SFT covers for actions of the first Grigorchuk group
Misha Gromov — Algebraisation of combinatorial isoperimetry
Hanspeter Kraft and Mikhail Zaidenberg — Automorphism groups of affine varieties and their Lie algebras
L. A. Kurdachenko and I. Ya. Subbotin — On the structure of some pre-Lie algebras
Andriy Oliynyk and Veronika Prokhorchuk — Free products of groups defined by
-automata
Ivan Shestakov and Efim Zelmanov — Simple Jordan superalgebras with the even parts of Clifford type
Vasyl Ustimenko — On Schubert cells of Lie geometries and public keys of multivariate cryptography
Softcover ISBN: 978-1-4704-7838-4
Product Code: CONM/831
Expected availability date: January 29, 2026
Contemporary Mathematics Volume: 831;
2025; Estimated: 271 pp
MSC: Primary 13
This volume consists of edited lecture notes from two events at the International Centre for Theoretical Physics (ICTP) in Trieste, Italy, organized in honor of Melvin Hochster and Craig Huneke. The two events were the online International Graduate course on Tight Closure and Its Applications, spread over summer 2022 and the May 2023 School on Commutative Algebra and Algebraic Geometry in Prime Characteristics. The unifying basis of these events was the theory of tight closure, the brainchild of Melvin Hochster and Craig Huneke in the late 1980s, which has had a dramatic effect on the field of commutative algebra, giving unified proofs and strong generalizations of many major theorems in commutative algebra, and stimulating much research, including recent proofs of longstanding conjectures. The aim of the two events as well as of this volume is to provide training in the foundations of commutative algebra and algebraic geometry in prime characteristic and to present some of the exciting recent developments in and beyond tight closure. The lecture notes for the online school also come with exercises and solutions.
The topics in the volume include characteristic p methods, test ideals, direct summands, singularities, Hilbert-Kunz multiplicity, Briançon-Skoda theorems, big Cohen-Macaulay algebras, the localization problem, uniform Artin-Rees results, vector bundles and tight closure, singularity invariants. The intended audience is graduate students learning the material as well as researchers wanting the latest advances in one reference.
Graduate students and research mathematicians interested in various aspects of commutative algebra.
Online Inernational Graduate course on Tight Closure and Its Applications, summer 2022
Neil Epstein — An introduction to the tight closure operation
Florian Enescu — F-rational rings and rational singularities
Kevin Tucker — Three lectures on test ideals
Vijayalaxmi Trivedi — Hilbert-Kunz multiplicity
Ian M. Aberbach — Briançon-Skoda Theorems in positive characteristic
Thomas Polstra — On the applications of Big Cohen-Macaulay modules and algebras
Irena Swanson — Uniform Artin Rees results
Wenliang Zhang — Lectures on the localization problem in tight closure
School on Commutative Algebra and Algebraic Geometry in Prime Characteristics, May 2023
Ilya Smirnov — An invitation to equimultiplicity of F-invariants
Holger Brenner — Vector bundles and tight closure (Triest 2023)
Kei-ichi Watanabe — F-singuralities; characteristic
methods in commutative ring theory and algebraic geometry
Linquan Ma and Kevin Tucker — An introduction to singularities in commutative algebra via perfectoid big Cohen-Macaulay algebras