Proceedings of Symposia in Pure Mathematics, Volume: 96
2017; 297 pp; Hardcover
Print ISBN: 978-1-4704-2951-5
This volume contains the proceedings of the conference String-Math 2015, which was held from December 31, 2015?January 4, 2016, at Tsinghua Sanya International Mathematics Forum in Sanya, China. Two of the main themes of this volume are frontier research on Calabi-Yau manifolds and mirror symmetry and the development of non-perturbative methods in supersymmetric gauge theories. The articles present state-of-the-art developments in these topics.
String theory is a broad subject, which has profound connections with broad branches of modern mathematics. In the last decades, the prosperous interaction built upon the joint efforts from both mathematicians and physicists has given rise to marvelous deep results in supersymmetric gauge theory, topological string, M-theory and duality on the physics side, as well as in algebraic geometry, differential geometry, algebraic topology, representation theory and number theory on the mathematics side.
Advanced graduate students, post-docs, and post Ph.D. mathematicians and mathematical physicists interested in string theory.
Graduate Studies in Mathematics, Volume: 185
2017; 304 pp; Hardcover
Print ISBN: 978-1-4704-3660-5
This book is a self-contained introduction to braid foliation techniques, which is a theory developed to study knots, links and surfaces in general 3-manifolds and more specifically in contact 3-manifolds. With style and content accessible to beginning students interested in geometric topology, each chapter centers around a key theorem or theorems. The particular braid foliation techniques needed to prove these theorems are introduced in parallel, so that the reader has an immediate gtake-homeh for the techniques involved.
The reader will learn that braid foliations provide a flexible toolbox capable of proving classical results such as Markov's theorem for closed braids and the transverse Markov theorem for transverse links, as well as recent results such as the generalized Jones conjecture for closed braids and the Legendrian grid number conjecture for Legendrian links. Connections are also made between the Dehornoy ordering of the braid groups and braid foliations on surfaces.
All of this is accomplished with techniques for which only mild prerequisites are required, such as an introductory knowledge of knot theory and differential geometry. The visual flavor of the arguments contained in the book is supported by over 200 figures.
Graduate students and researchers interested in geometry and topology.
Mathematical Surveys and Monographs, Volume: 226
2017; 611 pp; Hardcover
Print ISBN: 978-1-4704-3711-4
This monograph studies the interplay between various algebraic, geometric and combinatorial aspects of real hyperplane arrangements. It provides a careful, organized and unified treatment of several recent developments in the field, and brings forth many new ideas and results. It has two parts, each divided into eight chapters, and five appendices with background material.
Part I gives a detailed discussion on faces, flats, chambers, cones, gallery intervals, lunes and other geometric notions associated with arrangements. The Tits monoid plays a central role. Another important object is the category of lunes which generalizes the classical associative operad. Also discussed are the descent and lune identities, distance functions on chambers, and the combinatorics of the braid arrangement and related examples.
Part II studies the structure and representation theory of the Tits algebra of an arrangement. It gives a detailed analysis of idempotents and Peirce decompositions, and connects them to the classical theory of Eulerian idempotents. It introduces the space of Lie elements of an arrangement which generalizes the classical Lie operad. This space is the last nonzero power of the radical of the Tits algebra. It is also the socle of the left ideal of chambers and of the right ideal of Zie elements. Zie elements generalize the classical Lie idempotents. They include Dynkin elements associated to generic half-spaces which generalize the classical Dynkin idempotent. Another important object is the lune-incidence algebra which marks the beginning of noncommutative Mobius theory. These ideas are also brought upon the study of the Solomon descent algebra.
The monograph is written with clarity and in sufficient detail to make it accessible to graduate students. It can also serve as a useful reference to experts.
Graduate students and researchers interested in hyperplane arrangements (of interest in several areas of mathematics).
Graduate Studies in Mathematics, Volume: 184
2017; 344 pp; Hardcover
Print ISBN: 978-1-4704-2556-2
This book is an introduction to the representation theory of quivers and finite dimensional algebras. It gives a thorough and modern treatment of the algebraic approach based on Auslander-Reiten theory as well as the approach based on geometric invariant theory. The material in the opening chapters is developed starting slowly with topics such as homological algebra, Morita equivalence, and Gabriel's theorem. Next, the book presents Auslander-Reiten theory, including almost split sequences and the Auslander-Reiten transform, and gives a proof of Kac's generalization of Gabriel's theorem. Once this basic material is established, the book goes on with developing the geometric invariant theory of quiver representations. The book features the exposition of the saturation theorem for semi-invariants of quiver representations and its application to Littlewood-Richardson coefficients. In the final chapters, the book exposes tilting modules, exceptional sequences and a connection to cluster categories.
The book is suitable for a graduate course in quiver representations and has numerous exercises and examples throughout the text. The book will also be of use to experts in such areas as representation theory, invariant theory and algebraic geometry, who want to learn about applications of quiver representations to their fields.
Graduate students and researchers interested in representation theory, quivers, and applications to categories.
Mathematical Surveys and Monographs, Volume: 225
2017; 478 pp; Hardcover
Print ISBN: 978-0-8218-7554-4
This monograph is devoted to the study of the dynamics of expanding Thurston maps under iteration. A Thurston map is a branched covering map on a two-dimensional topological sphere such that each critical point of the map has a finite orbit under iteration. It is called expanding if, roughly speaking, preimages of a fine open cover of the underlying sphere under iterates of the map become finer and finer as the order of the iterate increases.
Every expanding Thurston map gives rise to a fractal space, called its visual sphere. Many dynamical properties of the map are encoded in the geometry of this visual sphere. For example, an expanding Thurston map is topologically conjugate to a rational map if and only if its visual sphere is quasisymmetrically equivalent to the Riemann sphere. This relation between dynamics and fractal geometry is the main focus for the investigations in this work.
The book is an introduction to the subject. The prerequisites for the reader are modest and include some basic knowledge of complex analysis and topology. The book has an extensive appendix, where background material is reviewed such as orbifolds and branched covering maps.
Graduate students and researchers interested in dynamical systems and topological dynamics.