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Frobenius Algebras and 2-D Topological Quantum Field Theories
This 2003 book describes a striking connection between topology and algebra, namely that 2D topological quantum field theories are equivalent to commutative Frobenius algebras. The precise formulation of the theorem and its proof is given in terms of monoidal categories, and the main purpose of the book is to develop these concepts from an elementary level, and more generally serve as an introduction to categorical viewpoints in mathematics. Rather than just proving the theorem, it is shown how the result fits into a more general pattern concerning universal monoidal categories for algebraic structures. Throughout, the emphasis is on the interplay between algebra and topology, with graphical interpretation of algebraic operations, and topological structures described algebraically in terms of generators and relations. The book will prove valuable to students or researchers entering this field who will learn a host of modern techniques that will prove useful for future work.
Higher Structures in Topology, Geometry, and Physics
This volume contains the proceedings of the AMS Special Session on Higher Structures in Topology, Geometry, and Physics, held virtually on March 26–27, 2022. The articles give a snapshot survey of the current topics surrounding the mathematical formulation of field theories. There is an intricate interplay between geometry, topology, and algebra which captures these theories. The hallmark are higher structures, which one can consider as the secondary algebraic or geometric background on which the theories are formulated. The higher structures considered in the volume are generalizations of operads, models for conformal field theories, string topology, open/closed field theories, BF/BV formalism, actions on Hochschild complexes and related complexes, and their geometric and topological aspects.
An Invitation to Quantum Cohomology
Elementary introduction to stable maps and quantum cohomology presents the problem of counting rational plane curves Viewpoint is mostly that of enumerative geometry Emphasis is on examples, heuristic discussions, and simple applications to best convey the intuition behind the subject Ideal for self-study, for a mini-course in quantum cohomology, or as a special topics text in a standard course in intersection theory
Clifford Algebras: An Introduction
A straightforward introduction to Clifford algebras, providing the necessary background material and many applications in mathematics and physics.
Mathematical Foundations of Information Flow
This volume is based on the 2008 Clifford Lectures on Information Flow in Physics, Geometry and Logic and Computation, held March 12-15, 2008, at Tulane University in New Orleans, Louisiana. The varying perspectives of the researchers are evident in the topics represented in the volume, including mathematics, computer science, quantum physics and classical and quantum information. A number of the articles address fundamental questions in quantum information and related topics in quantum physics, using abstract categorical and domain-theoretic models for quantum physics to reason about such systems and to model spacetime. Readers can expect to gain added insight into the notion of information flow and how it can be understood in many settings. They also can learn about new approaches to modeling quantum mechanics that provide simpler and more accessible explanations of quantum phenomena, which don't require the arcane aspects of Hilbert spaces and the cumbersome notation of bras and kets.
Dispersive Partial Differential Equations
Introduces nonlinear dispersive partial differential equations in a detailed yet elementary way without compromising the depth and richness of the subject.
The Riemann Hypothesis for Function Fields
This book provides a lucid exposition of the connections between non-commutative geometry and the famous Riemann Hypothesis, focusing on the theory of one-dimensional varieties over a finite field. The reader will encounter many important aspects of the theory, such as Bombieri's proof of the Riemann Hypothesis for function fields, along with an explanation of the connections with Nevanlinna theory and non-commutative geometry. The connection with non-commutative geometry is given special attention, with a complete determination of the Weil terms in the explicit formula for the point counting function as a trace of a shift operator on the additive space, and a discussion of how to obtain the explicit formula from the action of the idele class group on the space of adele classes. The exposition is accessible at the graduate level and above, and provides a wealth of motivation for further research in this area.
Fast Track to Forcing
For those who wonder if the forcing theory is beyond their means: no. Directions to research in forcing are given.
Topology and Field Theories
This book is a collection of expository articles based on four lecture series presented during the 2012 Notre Dame Summer School in Topology and Field Theories. The four topics covered in this volume are: Construction of a local conformal field theory associated to a compact Lie group, a level and a Frobenius object in the corresponding fusion category; Field theory interpretation of certain polynomial invariants associated to knots and links; Homotopy theoretic construction of far-reaching generalizations of the topological field theories that Dijkgraf and Witten associated to finite groups; and a discussion of the action of the orthogonal group on the full subcategory of an -category consisting of the fully dualizable objects. The expository style of the articles enables non-experts to understand the basic ideas of this wide range of important topics.
