This volume is based on lectures delivered at the 2019 AMS Short Course “Sum of Squares: Theory and Applications”, held January 14–15, 2019, in Baltimore, Maryland.
This book provides a concise state-of-the-art overview of the theory and applications of polynomials that are sums of squares. This is an exciting and timely topic, with rich connections to many areas of mathematics, including polynomial and semidefinite optimization, real and convex algebraic geometry, and theoretical computer science.
The six chapters introduce and survey recent developments in this area; specific topics include the algebraic and geometric aspects of sums of squares and spectrahedra, lifted representations of convex sets, and the algorithmic and computational implications of viewing sums of squares as a meta algorithm. The book also showcases practical applications of the techniques across a variety of areas, including control theory, statistics, finance and machine learning
Graduate students and researchers interested in various aspects of the theory of non-negative polynomials and applications.
Proceedings of Symposia in Applied Mathematics, Volume: 77
2020; 142 pp; Softcover
MSC: Primary 05; 14; 52; 68; 90;
Print ISBN: 978-1-4704-5025-0
Product Code: PSAPM/77
This book introduces a new research direction in set theory: the study of models of set theory with respect to their extensional overlap or disagreement. In Part I, the method is applied to isolate new distinctions between Borel equivalence relations. Part II contains applications to independence results in Zermelo–Fraenkel set theory without Axiom of Choice.
The method makes it possible to classify in great detail various paradoxical objects obtained using the Axiom of Choice; the classifying criterion is a ZF-provable implication between the existence of such objects. The book considers a broad spectrum of objects from analysis, algebra, and combinatorics: ultrafilters, Hamel bases, transcendence bases, colorings of Borel graphs, discontinuous homomorphisms between Polish groups, and many more. The topic is nearly inexhaustible in its variety, and many directions invite further investigation.
Graduate students and researchers interested in current research in axiomatic set theory.
Mathematical Surveys and Monographs, Volume: 248
2020; 330 pp; Softcover
MSC: Primary 03; 05; 11; 37;
Print ISBN: 978-1-4704-5462-3
Product Code: SURV/248
The French expression “dessins d'enfants” means children's drawings. This term was coined by the great French mathematician Alexandre Grothendieck in order to denominate a method of pictorial representation of some highly interesting classes of polynomials and rational functions. The polynomials studied in this book take their origin in number theory. The authors show how, by drawing simple pictures, one can prove some long-standing conjectures and formulate new ones. The theory presented here touches upon many different fields of mathematics.
The major part of the book is quite elementary and is easily accessible to an undergraduate student. The less elementary parts, such as Galois theory or group representations and their characters, would need a more profound knowledge of mathematics. The reader may either take the basic facts of these theories for granted or use our book as a motivation and a first approach to these subjects.
Graduate students and researchers interested in learning about combinatorics of polynomials as part of the new theory of dessins d'enfants.
Mathematical Surveys and Monographs, Volume: 249
2020; 187 pp; Softcover
MSC: Primary 11; Secondary 05; 12; 20
Print ISBN: 978-1-4704-5634-4
Product Code: SURV/249
This book contains the proceedings of the AMS Special Session, in honor of S. K. Jain's 80th birthday, on Categorical, Homological and Combinatorial Methods in Algebra held from March 16–18, 2018, at Ohio State University, Columbus, Ohio.
The articles contained in this volume aim to showcase the current state of art in categorical, homological and combinatorial aspects of algebra.
Graduate students and research mathematicians interested in noncommutative algebra and homological algegra.
Contemporary Mathematics, Volume: 751
2020; 357 pp; Softcover
MSC: Primary 16; 18;
Print ISBN: 978-1-4704-4368-9
Product Code: CONM/751
This volume contains the proceedings of Simon Fest, held in honor of Simon Thomas's 60th birthday, from September 15–17, 2017, at Rutgers University, Piscataway, New Jersey.
The topics covered showcase recent advances from a variety of main areas of set theory, including descriptive set theory, forcing, and inner model theory, in addition to several applications of set theory, including ergodic theory, combinatorics, and model theory.
Graduate students and research mathematicians interested in set theory and its applications.
Contemporary Mathematics, Volume: 752
2020; 207 pp; Softcover
MSC: Primary 03; 28; 05;
Print ISBN: 978-1-4704-4332-0
Product Code: CONM/752
This volume contains the proceedings of the International Conference on Vertex Operator Algebras, Number Theory, and Related Topics, held from June 11–15, 2018, at California State University, Sacramento, California.
The mathematics of vertex operator algebras, vector-valued modular forms and finite group theory continues to provide a rich and vibrant landscape in mathematics and physics. The resurgence of moonshine related to the Mathieu group and other groups, the increasing role of algebraic geometry and the development of irrational vertex operator algebras are just a few of the exciting and active areas at present.
The proceedings center around active research on vertex operator algebras and vector-valued modular forms and offer original contributions to the areas of vertex algebras and number theory, surveys on some of the most important topics relevant to these fields, introductions to new fields related to these and open problems from some of the leaders in these areas.
Graduate students and research mathematicians interested in vertex algebras and representation theory of infinite dimensional Lie algebras.
Contemporary Mathematics, Volume: 753
2020; 250 pp; Softcover
MSC: Primary 17; 11;
Print ISBN: 978-1-4704-4938-4
Product Code: CONM/753
This text develops linear algebra with the view that it is an important gateway connecting elementary mathematics to more advanced subjects, such as advanced calculus, systems of differential equations, differential geometry, and group representations. The purpose of this book is to provide a treatment of this subject in sufficient depth to prepare the reader to tackle such further material.
The text starts with vector spaces, over the sets of real and complex numbers, and linear transformations between such vector spaces. Later on, this setting is extended to general fields. The reader will be in a position to appreciate the early material on this more general level with minimal effort.
Notable features of the text include a treatment of determinants, which is cleaner than one often sees, and a high degree of contact with geometry and analysis, particularly in the chapter on linear algebra on inner product spaces. In addition to studying linear algebra over general fields, the text has a chapter on linear algebra over rings. There is also a chapter on special structures, such as quaternions, Clifford algebras, and octonions.
Readership
Undergraduate and graduate students interested in linear algebra.
Pure and Applied Undergraduate Texts, Volume: 45
2020; 306 pp; Softcover
MSC: Primary 15;
Print ISBN: 978-1-4704-5670-2
Product Code: AMSTEXT/45
This text was produced for the second part of a two-part sequence on advanced calculus, whose aim is to provide a firm logical foundation for analysis. The first part treats analysis in one variable, and the text at hand treats analysis in several variables.
After a review of topics from one-variable analysis and linear algebra, the text treats in succession multivariable differential calculus, including systems of differential equations, and multivariable integral calculus. It builds on this to develop calculus on surfaces in Euclidean space and also on manifolds. It introduces differential forms and establishes a general Stokes formula. It describes various applications of Stokes formula, from harmonic functions to degree theory. The text then studies the differential geometry of surfaces, including geodesics and curvature, and makes contact with degree theory, via the Gauss–Bonnet theorem.
The text also takes up Fourier analysis, and bridges this with results on surfaces, via Fourier analysis on spheres and on compact matrix groups.
Undergraduates interested in analysis in several variables.
Pure and Applied Undergraduate Texts, Volume: 46
2020; 445 pp; Softcover
MSC: Primary 26;
Print ISBN: 978-1-4704-5669-6
Product Code: AMSTEXT/46
This is the second of three volumes that, together, give an exposition of the mathematics of grades 9–12 that is simultaneously mathematically correct and grade-level appropriate. The volumes are consistent with CCSSM (Common Core State Standards for Mathematics) and aim at presenting the mathematics of K–12 as a totally transparent subject.
The first part of this volume is devoted to the study of standard algebra topics: quadratic functions, graphs of equations of degree 2 in two variables, polynomials, exponentials and logarithms, complex numbers and the fundamental theorem of algebra, and the binomial theorem. Having translations and the concept of similarity at our disposal enables us to clarify the study of quadratic functions by concentrating on their graphs, the same way the study of linear functions is greatly clarified by knowing that their graphs are lines. We also introduce the concept of formal algebra in the study of polynomials with complex coefficients. The last three chapters in this volume complete the systematic exposition of high school geometry that is consistent with CCSSM. These chapters treat the geometry of the triangle and the circle, ruler and compass constructions, and a general discussion of axiomatic systems, including non-Euclidean geometry and the celebrated work of Hilbert on the foundations.
This book should be useful for current and future teachers of K–12 mathematics, as well as for some high school students and for education professionals.
Teachers of middle school mathematics; students and professionals interested in mathematical education.
2020; 504 pp; Softcover
MSC: Primary 97; 00;
Print ISBN: 978-1-4704-5676-4
Product Code: MBK/132