Generalized Geometry Workshop

Swarthmore College, Swarthmore PA

Investigators

Abstract

This award provides support for the Generalized Geometry Workshop to be held at Swarthmore College on September 17-18, 2016. The goal is to bring together experts from mathematics and physics to consolidate the current development of the theory and extend its potential for applications to questions of importance in applied fields, as well as to train undergraduate and graduate students to work in this relatively young area with a wealth of accessible open problems. In the 20th century, the evolution of notions of geometric structures, such as Riemannian structures, complex structures, symplectic structures, and contact structures, has had an important impact both in the development of mathematics and in applications to many other areas of study. Generalized geometry is both an extension of these structures and a framework for analyzing the relationships between them. Although the concept is only about fifteen years old, it has made significant contributions both to understanding how various geometric structures on a space interact and has seen applications in string theory and supergravity. Generalized geometry is a contemporary approach to the study of differentiable manifolds in which the group of diffeomorphisms is extended to include additional symmetries known as B-field transforms. This enlarged symmetry group, which arises naturally from the point of view of loop spaces, has been studied extensively by geometers and string theorists and continues to provide an invaluable bridge connecting the two fields. In addition, since each classical geometry can be realized as a generalized geometry, it provides a framework for studying relationships between various types of classical geometric structures that can occur on a manifold and has been a source of surprising results in this regard. So this topic has intrinsic interest from a differential geometric point of view as well as with respect to possible physical applications. There has been a wealth of results produced in the first decade of activity in generalized geometry that has attracted much interest from mathematicians and theoretical physicists. So far, it has been primarily focused on even-dimensional manifolds, partly because of the special role played by generalized complex structures in string theory. However, recent work suggests that generalized geometry of odd-dimensional manifolds might be just as mathematically rich and physically useful as its even-dimensional counterpart. While significant steps toward a better understanding of odd-dimensional analogues of generalized complex structures (also known as generalized contact structures) have been made, the subject presents specific challenges and it is still in its infancy. To address these issues, this workshop has the following scientific goals: 1) Consolidate the progress made in odd-dimensional generalized geometry in the last two to three years by providing an accessible survey of the most recent developments, with particular emphasis on generalized contact geometry. 2) Bring together experts working in generalized complex geometry and generalized contact geometry with the aim of developing a unified understanding of generalized geometry. 3) Bring together physicists and mathematicians working in generalized geometry to better understand the role of odd-dimensional generalized geometry in string theory and identify open problems of interest for both communities. 4) Advertise generalized geometry, with particular emphasis on odd-dimensional aspects, to graduate students and advanced undergraduate students. Besides the obvious benefits of attracting new generations to the field, it offers excellent opportunities for involvement of students in research at an earlier stage. The very nature of this subject often requires deconstruction and critical rethinking of classical notions, and thereby makes it an invaluable training ground for any student interested in differential geometry. The website for the workshop is: https://sites.google.com/a/vcu.edu/ggw/