Introduction to Exceptional Geometry

Record: Yes

Level: Graduate

Language: English

Prerequisite

Linear algebra, basics of Riemmanian geometry

Abstract

The classification of Riemannian manifolds with special holonomy contains two “exceptional” cases: G2 and Spin(7). Manifolds with holonomy contained in G2 or Spin(7) are called G2-manifolds or Spin(7)-manifolds, respectively. In this course, I will introduce various topics of G2 and Spin(7) geometry, mainly focusing on the G2 case. We start from the linear algebra in G2 geometry. Then we study topics such as the structure of a G2-manifold and calibrated geometry/ gauge theory/mirror symmetry on a G2-manifold.

Reference

1. D. D. Joyce, Compact manifolds with special holonomy. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2000. xii+436 pp. ISBN: 0-19-850601-5

2. D. D. Joyce, Riemannian holonomy groups and calibrated geometry. Oxford Graduate Texts in Mathematics, 12. Oxford University Press, Oxford, 2007. x+303 pp. ISBN: 978-0-19-921559-1

3. S. Karigiannis, Deformations of G2 and Spin(7) structures. Canad. J. Math. 57 (2005), no. 5, 1012--1055.

Syllabus

Lecturer Intro

Kotaro Kawai got a bachelor's degree and a master's degree from the university of Tokyo, and received his Ph.D from Tohoku university in 2013. He was an assistant professor at Gakushuin university in Japan, then he moved to BIMSA this year. Gakushuin University was established as an educational institution for the imperial family and peers, and even today,

some members of the imperial family attend this university. There are many eminent professors and Kunihiko Kodaira worked at this university.

He majors in differential geometry, focusing on manifolds with exceptional holonomy. These manifolds are considered to be analogues of Calabi-Yau manifolds, and higher dimensional analogues of gauge theory are expected on these manifolds. This topic is also related to physics, and he think that this is an exciting research field.

Lecturer Email: kkawai@bimsa.cn

TA: Dr. Dongyu Wu, dow16@pitt.edu