University Lecture Series, Volume: 73
2019; 118 pp; Softcover
MSC: Primary 22; Secondary 11; 20
Print ISBN: 978-1-4704-5231-5
Fifty years after it made the transition from mimeographed lecture notes to a published book, Armand Borel's Introduction aux groupes arithmetiques continues to be very important for the theory of arithmetic groups. In particular, Chapter III of the book remains the standard reference for fundamental results on reduction theory, which is crucial in the study of discrete subgroups of Lie groups and the corresponding homogeneous spaces.
The review of the original French version in Mathematical Reviews observes that “the style is concise and the proofs (in later sections) are often demanding of the reader.” To make the translation more approachable, numerous footnotes provide helpful comments.
Graduate students and researchers interested in the theory of algebraic groups.
This book is a reader-friendly introduction to the theory of symmetric functions, and it includes fundamental topics such as the monomial, elementary, homogeneous, and Schur function bases; the skew Schur functions; the Jacobi?Trudi identities; the involution ωω; the Hall inner product; Cauchy's formula; the RSK correspondence and how to implement it with both insertion and growth diagrams; the Pieri rules; the Murnaghan?Nakayama rule; Knuth equivalence; jeu de taquin; and the Littlewood?Richardson rule. The book also includes glimpses of recent developments and active areas of research, including Grothendieck polynomials, dual stable Grothendieck polynomials, Stanley's chromatic symmetric function, and Stanley's chromatic tree conjecture. Written in a conversational style, the book contains many motivating and illustrative examples. Whenever possible it takes a combinatorial approach, using bijections, involutions, and combinatorial ideas to prove algebraic results.
The prerequisites for this book are minimal?familiarity with linear algebra, partitions, and generating functions is all one needs to get started. This makes the book accessible to a wide array of undergraduates interested in combinatorics.
Undergraduate and graduate students interested in algebra and combinatorics.
Contemporary Mathematics, Volume: 740
2019; 248 pp; Softcover
MSC: Primary 11; 20; 05; 19; 14;
Print ISBN: 978-1-4704-3784-8
This volume contains the proceedings of the Arizona Winter School 2016, which was held from March 12?16, 2016, at The University of Arizona, Tucson, AZ.
In the last decade or so, analytic methods have had great success in answering questions in arithmetic geometry and number theory. The School provided a unique opportunity to introduce graduate students to analytic methods in arithmetic geometry.
The book contains four articles. Alina C. Cojocaru's article introduces sieving techniques to study the group structure of points of the reduction of an elliptic curve modulo a rational prime via its division fields. Harald A. Helfgott's article provides an introduction to the study of growth in groups of Lie type, with SL2(Fq)SL2(Fq) and some of its subgroups as the key examples. The article by Etienne Fouvry, Emmanuel Kowalski, Philippe Michel, and Will Sawin describes how a systematic use of the deep methods from ??-adic cohomology pioneered by Grothendieck and Deligne and further developed by Katz and Laumon help make progress on various classical questions from analytic number theory. The last article, by Andrew V. Sutherland, introduces Sato-Tate groups and explores their relationship with Galois representations, motivic LL-functions, and Mumford-Tate groups.
Graduate students and research mathematicians interested in arithmetic and algebraic geometry and analytic number theory.
Mathematical Surveys and Monographs, Volume: 245
2019; 250 pp; Hardcover
MSC: Primary 47; 34;
Print ISBN: 978-1-4704-5366-4
In 1910 Herman Weyl published one of the most widely quoted papers of the 20th century in Analysis, which initiated the study of singular Sturm-Liouville problems. The work on the foundations of Quantum Mechanics in the 1920s and 1930s, including the proof of the spectral theorem for unbounded self-adjoint operators in Hilbert space by von Neumann and Stone, provided some of the motivation for the study of differential operators in Hilbert space with particular emphasis on self-adjoint operators and their spectrum. Since then the topic developed in several directions and many results and applications have been obtained.
In this monograph the authors summarize some of these directions discussing self-adjoint, symmetric, and dissipative operators in Hilbert and Symplectic Geometry spaces. Part I of the book covers the theory of differential and quasi-differential expressions and equations, existence and uniqueness of solutions, continuous and differentiable dependence on initial data, adjoint expressions, the Lagrange Identity, minimal and maximal operators, etc. In Part II characterizations of the symmetric, self-adjoint, and dissipative boundary conditions are established. In particular, the authors prove the long standing Deficiency Index Conjecture. In Part III the symmetric and self-adjoint characterizations are extended to two-interval problems. These problems have solutions which have jump discontinuities in the interior of the underlying interval. These jumps may be infinite at singular interior points. Part IV is devoted to the construction of the regular Green's function. The construction presented differs from the usual one as found, for example, in the classical book by Coddington and Levinson.
Graduate students and researchers interested in differential equations, in particular, the Sturm-Liouville problem.
Contemporary Mathematics, Volume: 739
2019; 197 pp; Softcover
MSC: Primary 05; 11; 33; 35; 58; 60; 81;
Print ISBN: 978-1-4704-4145-6
This volume contains the proceedings of the CRM Workshops on Probabilistic Methods in Spectral Geometry and PDE, held from August 22?26, 2016 and Probabilistic Methods in Topology, held from November 14?18, 2016 at the Centre de Recherches Mathematiques, Universite de Montreal, Montreal, Quebec, Canada.
Probabilistic methods have played an increasingly important role in many areas of mathematics, from the study of random groups and random simplicial complexes in topology, to the theory of random Schrodinger operators in mathematical physics.
The workshop on Probabilistic Methods in Spectral Geometry and PDE brought together some of the leading researchers in quantum chaos, semi-classical theory, ergodic theory and dynamical systems, partial differential equations, probability, random matrix theory, mathematical physics, conformal field theory, and random graph theory. Its emphasis was on the use of ideas and methods from probability in different areas, such as quantum chaos (study of spectra and eigenstates of chaotic systems at high energy); geometry of random metrics and related problems in quantum gravity; solutions of partial differential equations with random initial conditions.
The workshop Probabilistic Methods in Topology brought together researchers working on random simplicial complexes and geometry of spaces of triangulations (with connections to manifold learning); topological statistics, and geometric probability; theory of random groups and their properties; random knots; and other problems.
This volume covers recent developments in several active research areas at the interface of Probability, Semiclassical Analysis, Mathematical Physics, Theory of Automorphic Forms and Graph Theory.
Graduate students and research mathematicians interested in probability theory and its applications to various areas of mathematics.
Graduate Studies in Mathematics, Volume: 203
2019; 356 pp; Hardcover
MSC: Primary 11;
Print ISBN: 978-1-4704-4754-0
Prime numbers have fascinated mathematicians since the time of Euclid. This book presents some of our best tools to capture the properties of these fundamental objects, beginning with the most basic notions of asymptotic estimates and arriving at the forefront of mathematical research. Detailed proofs of the recent spectacular advances on small and large gaps between primes are made accessible for the first time in textbook form. Some other highlights include an introduction to probabilistic methods, a detailed study of sieves, and elements of the theory of pretentious multiplicative functions leading to a proof of Linnik's theorem.
Throughout, the emphasis has been placed on explaining the main ideas rather than the most general results available. As a result, several methods are presented in terms of concrete examples that simplify technical details, and theorems are stated in a form that facilitates the understanding of their proof at the cost of sacrificing some generality. Each chapter concludes with numerous exercises of various levels of difficulty aimed to exemplify the material, as well as to expose the readers to more advanced topics and point them to further reading sources.
Undergraduate and graduate students and researchers interested in distribution of prime numbers.