This volume contains the proceedings of an NSF-CBMS Conference held at Texas Christian University in Fort Worth, Texas, May 18-22, 2009. The papers, written especially for this volume by well-known mathematicians and mathematical physicists, are an outgrowth of the talks presented at the conference. Topics examined are highly interdisciplinary and include, among many other things, recent results on D-brane charges in $K$-homology and twisted $K$-homology, Yang-Mills gauge theory and connections with non-commutative geometry, Landau-Ginzburg models, $C^*$-algebraic non-commutative geometry and ties to quantum physics and topology, the rational homotopy type of the group of unitary elements in an Azumaya algebra, and functoriality properties in the theory of $C^*$-crossed products and fixed point algebras for proper actions. An introduction, written by Jonathan Rosenberg, provides an instructive overview describing common themes and how the various papers in the volume are interrelated and fit together. The rich diversity of papers appearing in the volume demonstrates the current interplay between superstring theory, geometry/topology, and non-commutative geometry. The book will be of interest to graduate students, mathematicians, mathematical physicists, and researchers working in these areas.

String theory is the leading candidate for a physical theory that combines all the fundamental forces of nature, as well as the principles of relativity and quantum mechanics, into a mathematically elegant whole. The mathematical tools used by string theorists are highly sophisticated, and cover many areas of mathematics. As with the birth of quantum theory in the early 20th century, the mathematics has benefited at least as much as the physics from the collaboration. In this book, based on CBMS lectures given at Texas Christian University, Rosenberg describes some of the most recent interplay between string dualities and topology and operator algebras. The book is an interdisciplinary approach to duality symmetries in string theory. It can be read by either mathematicians or theoretical physicists, and involves a more-or-less equal mixture of algebraic topology, operator algebras, and physics. There is also a bit of algebraic geometry, especially in the last chapter. The reader is assumed to be somewhat familiar with at least one of these four subjects, but not necessarily with all or even most of them. The main objective of the book is to show how several seemingly disparate subjects are closely linked with one another, and to give readers an overview of some areas of current research, even if this means that not everything is covered systematically.

This volume contains the proceedings of the 2016 AMS von Neumann Symposium on Topological Recursion and its Influence in Analysis, Geometry, and Topology, which was held from July 4–8, 2016, at the Hilton Charlotte University Place, Charlotte, North Carolina. The papers contained in the volume present a snapshot of rapid and rich developments in the emerging research field known as topological recursion. It has its origin around 2004 in random matrix theory and also in Mirzakhani's work on the volume of moduli spaces of hyperbolic surfaces. Topological recursion has played a fundamental role in connecting seemingly unrelated areas of mathematics such as matrix models, enumeration of Hurwitz numbers and Grothendieck's dessins d'enfants, Gromov-Witten invariants, the A-polynomials and colored polynomial invariants of knots, WKB analysis, and quantization of Hitchin moduli spaces. In addition to establishing these topics, the volume includes survey papers on the most recent key accomplishments: discovery of the unexpected relation to semi-simple cohomological field theories and a solution to the remodeling conjecture. It also provides a glimpse into the future research direction; for example, connections with the Airy structures, modular functors, Hurwitz-Frobenius manifolds, and ELSV-type formulas.

This book is a collection of articles written in memory of Boris Dubrovin (1950–2019). The authors express their admiration for his remarkable personality and for the contributions he made to mathematical physics. For many of the authors, Dubrovin was a friend, colleague, inspiring mentor, and teacher. The contributions to this collection of papers are split into two parts: “Integrable Systems” and “Quantum Theories and Algebraic Geometry”, reflecting the areas of main scientific interests of Dubrovin. Chronologically, these interests may be divided into several parts: integrable systems, integrable systems of hydrodynamic type, WDVV equations (Frobenius manifolds), isomonodromy equations (flat connections), and quantum cohomology. The articles included in the first part are more or less directly devoted to these areas (primarily with the first three listed above). The second part contains articles on quantum theories and algebraic geometry and is less directly connected with Dubrovin's early interests.

This book contains the proceedings of the 2009-2011 Southeastern Lie Theory Workshop Series, held October 9-11, 2009 at North Carolina State University, May 22-24, 2010, at the University of Georgia, and June 1-4, 2011 at the University of Virginia. Some of the articles, written by experts in the field, survey recent developments while others include new results in Lie algebras, quantum groups, finite groups, and algebraic groups.