A publication of the Tata Institute of Fundamental Research
Tata Institute of Fundamental Research Publications Volume: 19
2018; 400 pp; Hardcover
This volume contains the proceedings of the international colloquium organized by the Tata Institute of Fundamental Research in January 2016, one of a series of colloquia going back to 1956.
The talks at the colloquium covered a wide spectrum of mathematics, ranging over algebraic geometry, topology, algebraic K
-theory and number theory. Algebraic theory, A1A-homotopy theory and topological K-theory formed important sub-streams in this colloquium.
Several branches of K-theory, like algebraic cycles, triangulated categories of motives, motivic cohomology, motivic homotopy theory, Chow groups of varieties, Euler class theory, equivariant K-theory as well as classical K-theory have developed considerably in recent years, giving rise to newer directions to the subject as well as proving results of “classical” interest. The colloquium brought together experts in these various branches and their talks covered this wide spectrum, highlighting the interconnections and giving a better perspective of the whole subject area.
This volume contains refereed articles by leading experts in these fields and includes original results as well as expository materials in these areas.
Graduate students and researchers interested in geometry, topology, and number theory.
Spectrum Volume: 93
2018; Softcover
MSC: Primary 01; 05; 57; 58; 37; 97;
Print ISBN: 978-1-4704-4828-8
Hassler Whitney was a giant of twentieth-century mathematics. This biography paints a picture of him and includes dozens of revealing anecdotes. Mathematically, he had a rare detector that went off whenever he spotted a piece of mathematical gold, and he would then draw countless pictures, gradually forging a path from hunch to proof. This geometric path is seldom reflected in the rigor of his formal papers, but thanks to a close friendship and many conversations over decades, author Kendig was able to see how he actually worked. This book shows this through accessible accounts of his major mathematical contributions, with figures copiously supplied.
Whitney is probably best known for introducing the grandfather of today's innumerable embedding theorems?his strong embedding theorem stating that any smooth manifold can be smoothly embedded in a Euclidean space of twice the manifold's dimension. This in turn led to several standard techniques used every day in algebraic topology. Whitney also established the fundamentals of graph theory, the four-color problem, matroids, extending smooth functions, and singularities of smooth functions. He almost never used complicated technical machinery, so most of his work is accessible to a general reader with a modest mathematical background.
His math-music connection was intense: He played piano, violin, and viola and won “best composition of the year” while earning a Bachelor's degree in music at Yale. He was an accomplished mountain climber, and as a tinkerer, at age sixteen he built the large-format camera used to take this book's cover photograph. Whitney's family generously provided dozens of photographs appearing here for the very first time. This biography is a revealing portrait of a fascinating personality and a titan of twentieth-century mathematics.
Undergraduate and graduate students and researchers interested in history, biography, and the history of topology.
Mathematical Surveys and Monographs Volume: 234
2018; 286 pp; Hardcover
MSC: Primary 60; 28; 46; 54;
Print ISBN: 978-1-4704-4738-0
This book provides a thorough exposition of the main concepts and results related to various types of convergence of measures arising in measure theory, probability theory, functional analysis, partial differential equations, mathematical physics, and other theoretical and applied fields. Particular attention is given to weak convergence of measures. The principal material is oriented toward a broad circle of readers dealing with convergence in distribution of random variables and weak convergence of measures.
The book contains the necessary background from measure theory and functional analysis. Large complementary sections aimed at researchers present the most important recent achievements. More than 100 exercises (ranging from easy introductory exercises to rather difficult problems for experienced readers) are given with hints, solutions, or references. Historic and bibliographic comments are included.
The target readership includes mathematicians and physicists whose research is related to probability theory, mathematical statistics, functional analysis, and mathematical physics.
Graduate students and researchers interested in probability theory and functional analysis.
This volume contains the proceedings of the QMATH13: Mathematical Results in Quantum Physics conference, held from October 8?11, 2016, at the Georgia Institute of Technology, Atlanta, Georgia.
In recent years, a number of new frontiers have opened in mathematical physics, such as many-body localization and Schrodinger operators on graphs. There has been progress in developing mathematical techniques as well, notably in renormalization group methods and the use of Lieb?Robinson bounds in various quantum models.
The aim of this volume is to provide an overview of some of these developments. Topics include random Schrodinger operators, many-body fermionic systems, atomic systems, effective equations, and applications to quantum field theory. A number of articles are devoted to the very active area of Schrodinger operators on graphs and general spectral theory of Schrodinger operators. Some of the articles are expository and can be read by an advanced graduate student.
Graduate students and research mathematicians interested in current developments in mathematical physics.
Contemporary Mathematics Volume: 718
2018; Softcover
MSC: Primary 18; 53; 55; 81;
Print ISBN: 978-1-4704-4243-9
This volume contains the proceedings of the NSF-CBMS Regional Conference on Topological and Geometric Methods in QFT, held from July 31-August 4, 2017, at Montana State University in Bozeman, Montana.
In recent decades, there has been a movement to axiomatize quantum field theory into a mathematical structure. In a different direction, one can ask to test these axiom systems against physics. Can they be used to rederive known facts about quantum theories or, better yet, be the framework in which to solve open problems? Recently, Freed and Hopkins have provided a solution to a classification problem in condensed matter theory, which is ultimately based on the field theory axioms of Graeme Segal.
Papers contained in this volume amplify various aspects of the Freed?Hopkins program, develop some category theory, which lies behind the cobordism hypothesis, the major structure theorem for topological field theories, and relate to Costello's approach to perturbative quantum field theory. Two papers on the latter use this framework to recover fundamental results about some physical theories: two-dimensional sigma-models and the bosonic string. Perhaps it is surprising that such sparse axiom systems encode enough structure to prove important results in physics. These successes can be taken as encouragement that the axiom systems are at least on the right track toward articulating what a quantum field theory is.
Graduate students and research mathematicians interested in topology, geometry, and mathematical physics.
AMS/MAA Textbooks Volume: 41
2018; 388 pp; Hardcover
MSC: Primary 97;
Print ISBN: 978-1-4704-4696-3
Journey into Discrete Mathematics is designed for use in a first course in mathematical abstraction for early-career undergraduate mathematics majors. The important ideas of discrete mathematics are included?logic, sets, proof writing, relations, counting, number theory, and graph theory?in a manner that promotes development of a mathematical mindset and prepares students for further study. While the treatment is designed to prepare the student reader for the mathematics major, the book remains attractive and appealing to students of computer science and other problem-solving disciplines.
The exposition is exquisite and engaging and features detailed descriptions of the thought processes that one might follow to attack the problems of mathematics. The problems are appealing and vary widely in depth and difficulty. Careful design of the book helps the student reader learn to think like a mathematician through the exposition and the problems provided. Several of the core topics, including counting, number theory, and graph theory, are visited twice: once in an introductory manner and then again in a later chapter with more advanced concepts and with a deeper perspective.
Owen D. Byer and Deirdre L. Smeltzer are both Professors of Mathematics at Eastern Mennonite University. Kenneth L. Wantz is Professor of Mathematics at Regent University. Collectively the authors have specialized expertise and research publications ranging widely over discrete mathematics and have over fifty semesters of combined experience in teaching this subject.
Undergraduate students interested in discrete mathematics and computer science.