2019; 136 pp; Softcover
Print ISBN: 978-1-4704-4163-0
This new volume of What's Happening in the Mathematical Sciences features a rich selection of articles about recent topics in pure and applied mathematics.
“Expanding Horizons” and “Needles in an Infinite Haystack” explain new developments in the theory of expander graphs and in number theory (asymptotic Fermat's last theorem), respectively. “The SetR Game Has Met Its Match” presents a solution of the so-called Cap Set Conjecture, a statement about arithmetic progressions in finite vector spaces, which resulted from the mathematical analysis of the popular game “Set”.
“The Shape of Data” and “Quantum Computers and Golden Gates” present recent advances in theoretical computer science and related areas of data science. The mathematical aspects of one of the most fascinating recent developments in general relativity, the discovery of gravitational waves, is discussed in “When Black Holes Collide”.
Three articles talk about applications of mathematical methods in various aspects of everyday life: bike-sharing systems and ride-sharing services (like Lyft and Uber) in “The Mathematics of Commuting”, weight control in “The Calculus of Calories”, and an analysis of various partisan election practices in “Gerrymandering: Mathematics on Trial”.
We anticipate that many readers will find an interesting topic to read about and, hopefully, more than one.
General college and university audience; anyone interested in expository accounts of recent developments in mathematics.
Pure and Applied Undergraduate Texts, Volume: 41
2019; 607 pp; Hardcover
Print ISBN: 978-1-4704-5135-6
A Passage to Modern Analysis is an extremely well-written and reader-friendly invitation to real analysis. An introductory text for students of mathematics and its applications at the advanced undergraduate and beginning graduate level, it strikes an especially good balance between depth of coverage and accessible exposition. The examples, problems, and exposition open up a student's intuition but still provide coverage of deep areas of real analysis. A yearlong course from this text provides a solid foundation for further study or application of real analysis at the graduate level.
A Passage to Modern Analysis is grounded solidly in the analysis of RR and RnRn, but at appropriate points it introduces and discusses the more general settings of inner product spaces, normed spaces, and metric spaces. The last five chapters offer a bridge to fundamental topics in advanced areas such as ordinary differential equations, Fourier series and partial differential equations, Lebesgue measure and the Lebesgue integral, and Hilbert space. Thus, the book introduces interesting and useful developments beyond Euclidean space where the concepts of analysis play important roles, and it prepares readers for further study of those developments.
Undergraduate and graduate students interested in real analysis.
Student Mathematical Library, Volume: 90
2019; 273 pp; Softcover
Print ISBN: 978-1-4704-5298-8
Discrete Morse theory is a powerful tool combining ideas in both topology and combinatorics. Invented by Robin Forman in the mid 1990s, discrete Morse theory is a combinatorial analogue of Marston Morse's classical Morse theory. Its applications are vast, including applications to topological data analysis, combinatorics, and computer science.
This book, the first one devoted solely to discrete Morse theory, serves as an introduction to the subject. Since the book restricts the study of discrete Morse theory to abstract simplicial complexes, a course in mathematical proof writing is the only prerequisite needed. Topics covered include simplicial complexes, simple homotopy, collapsibility, gradient vector fields, Hasse diagrams, simplicial homology, persistent homology, discrete Morse inequalities, the Morse complex, discrete Morse homology, and strong discrete Morse functions. Students of computer science will also find the book beneficial as it includes topics such as Boolean functions, evasiveness, and has a chapter devoted to some computational aspects of discrete Morse theory. The book is appropriate for a course in discrete Morse theory, a supplemental text to a course in algebraic topology or topological combinatorics, or an independent study.
An instructor's manual for this title is available electronically to those instructors who have already adopted the textbook for classroom use. Please send email to textbooks@ams.org for more information.
Undergraduate and graduate students interested in discrete Morse theory.
Contemporary Mathematics, Volume: 738
2019; 147 pp; Softcover
Print ISBN: 978-1-4704-4735-9
This volume contains the proceedings of the International Conference on Algebra, Discrete Mathematics and Applications, held from December 9?11, 2017, at Dr. Babasaheb Ambedkar Marathwada University, Aurangabad (Maharashtra), India.
Contemporary topics of research in algebra and its applications to algebraic geometry, Lie groups, algebraic combinatorics, and representation theory are covered.
The articles are devoted to Leavitt path algebras, roots of elements in Lie groups, Hilbert's Nullstellensatz, mixed multiplicities of ideals, singular matrices, rings of integers, injective hulls of modules, representations of linear, symmetric groups and Lie algebras, the algebra of generic matrices and almost injective modules.
Graduate students and research mathematicians interested in algebra and its applications to algebraic geometry.
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Pure and Applied Undergraduate Texts, Volume: 42
2019; 389 pp; Hardcover
Print ISBN: 978-1-4704-5084-7
Linear Algebra for the Young Mathematician is a careful, thorough, and rigorous introduction to linear algebra. It adopts a conceptual point of view, focusing on the notions of vector spaces and linear transformations, and it takes pains to provide proofs that bring out the essential ideas of the subject. It begins at the beginning, assuming no prior knowledge of the subject, but goes quite far, and it includes many topics not usually treated in introductory linear algebra texts, such as Jordan canonical form and the spectral theorem. While it concentrates on the finite-dimensional case, it treats the infinite-dimensional case as well. The book illustrates the centrality of linear algebra by providing numerous examples of its application within mathematics. It contains a wide variety of both conceptual and computational exercises at all levels, from the relatively straightforward to the quite challenging.
Readers of this book will not only come away with the knowledge that the results of linear algebra are true, but also with a deep understanding of why they are true.
Undergraduate students interested in learning and teaching linear algebra.