A classic problem in mathematics is solving systems of polynomial equations in several unknowns. Polynomial models are ubiquitous and widely used across the sciences. They arise in robotics, coding theory, optimization, mathematical biology, computer vision, game theory, statistics, and numerous other areas. This work furnishes a bridge across mathematical disciplines and exposes many facets of systems of polynomial equations. It covers a wide spectrum of mathematical techniques and algorithms, both symbolic and numerical. The set of solutions to a system of polynomial equations is an...

The book surveys some recent developments in the arithmetic of modular elliptic curves. It places a special emphasis on the construction of rational points on elliptic curves, the Birch and Swinnerton-Dyer conjecture, and the crucial role played by modularity in shedding light on these two closely related issues. The main theme of the book is the theory of complex multiplication, Heegner points, and some conjectural variants. The first three chapters introduce the background and prerequisites: elliptic curves, modular forms and the Shimura-Taniyama-Weil conjecture, complex multiplication and...

Graph algebras are a family of operator algebras, associated to directed graphs. The first part of this book provides an introduction to the subject. The second part, surveys the literature on the structure theory of graph algebras, and highlights some app

This volume presents the expanded notes from ten lectures given by the author at the NSF/CBMS conference held at California State University (Bakersfield). The author describes what he calls single orbit dynamics, which is an approach to the analysis of dynamical systems via the study of single orbits, rather than the study of a system as a whole. He presents single orbit interpretations of several areas of topological dynamics and ergodic theory and some new applications of dynamics to graph theory.

Suitable for readers who want to learn about the Newtonian $N$-body problem, this book contains simple explanations of the apparent 'looping' orbit of Mars and the unexpected 'Sunrise, Sunset' behavior as viewed from Mercury. It also covers the weird dynam

Based on lectures given at the CBMS Workshop on the Combinatorics of Large Sparse Graphs, this work presents fresh perspectives in graph theory and helps to contribute to a sound scientific foundation for our understanding of discrete networks that permeat

This monograph deals with the theory of tight closure and its applications. The contents are based on ten talks given at a CBMS conference held at North Dakota State University in June 1995.

This CBMS lecture series, held in Albany, New York in June 1994 aimed to introduce the audience to the literature on complex dynamics in higher dimension. Some of the lectures are updated versions of earlier lectures given jointly with Nessim Sibony in Montreal 1993. the authro's intent in this book is to give an expansion of the Montreal lectures, basing complex dynamics in higher dimension systematically on pluripotential theory.

Modular forms appear in many ways in number theory. This book details various roles that modular forms and $q$-series play in number theory, such as applications and connections to basic hypergeometric functions, Gaussian hypergeometric functions, super-congruences, Weierstrass points on modular curves, singular moduli, and class numbers.

The last ten years have seen a number of significant advances in Hopf algebras. The best known is the introduction of quantum groups, which are Hopf algebras that arose in mathematical physics and now have connections to many areas of mathematics. In addition, several conjectures of Kaplansky have been solved, the most striking of which is a kind of Lagrange's theorem for Hopf algebras. Work on actions of Hopf algebras has unified earlier results on group actions, actions of Lie algebras, and graded algebras. This book brings together many of these recent developments from the viewpoint of...