Joachim Kock - Böcker

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.

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.

This book is an elementary introduction to some ideas and techniques that have revolutionized enumerative geometry: stable maps and quantum cohomology. A striking demonstration of the potential of these techniques is provided by Kont- vich's famous formula, which solves a long-standing question: How many plane rational curves of degree d pass through 3d — 1 given points in general position? The formula expresses the number of curves for a given degree in terms of the numbers for lower degrees. A single initial datum is required for the recursion, namely, the case d = I, which simply amounts to the fact that through two points there is but one line. Assuming the existence of the Kontsevich spaces of stable maps and a few of their basic properties, we present a complete proof of the formula, and use the formula as a red thread in our Invitation to Quantum Cohomology. For more information about the mathematical content, see the Introduction. The canonical reference for this topic is the already classical Notes on Stable Maps and Quantum Cohomology by Fulton and Pandharipande [29], cited henceforth as FP-NOTES. We have traded greater generality for the sake of introducing some simplifications. We have also chosen not to include the technical details of the construction of the moduli space, favoring the exposition with many examples and heuristic discussions.

This volume contains the proceedings of the Workshop on Higher Segal Spaces and their Applications to Algebraic $K$-Theory, Hall Algebras, and Combinatorics, held from January 21 to 26, 2024, at the Banff International Research Station, Banff (AB), Canada. This book is a collection of ten expository papers providing an introduction to the theory and applications of higher Segal spaces, a rather new topic in the interface between homotopy theory and higher category theory, finding applications in homological algebra, representation theory, geometry, combinatorics, algebraic $K$-theory, and other areas. The papers are written by experts on the various topics, in an engaging style welcoming newcomers to the field, and are also designed to help specialists in one application area make the transition to another.