Geometry, Topology, and Physics Seminar, Fall 2016

Part of the NSF/UCSB ‘Research Training Group’ in Topology and Geometry

Organizers: Dave Morrison and Zhenghan Wang.
Meets 4:00 - 5:30 p.m. Fridays in South Hall 6635.

September 30

### Introduction to K3 surfaces

Abstract: K3 surfaces have played an important role in algebrac geometry, complex analytic geometry, and differential geometry for the past 60 years. They also have important applications to physics. I will give a series of lectures this fall on K3 surfaces, of which this is the first.

Audio [ mp3, m4a ]; Lecture notes.

October 7

October 14

October 21

### Introduction to K3 surfaces, II

Abstract: K3 surfaces have played an important role in algebrac geometry, complex analytic geometry, and differential geometry for the past 60 years. They also have important applications to physics. I will give a series of lectures this fall on K3 surfaces, of which this is the second.

Audio [ mp3, m4a ]; Lecture notes..

October 28

November 4

### K3 surfaces, III

Abstract: In this third and final introductory lecture about K3 surfaces, I will explain how the global Torelli theorem allows one to translate many interesting problems about K3 surfaces into problems about embeddings of lattices, and I will illustrate this with geometric examples, such as the Kummer construction.

Audio [ mp3, m4a ]; Lecture notes.

November 11

November 18

November 25

December 2

### Twisted connected sum constructions of $G_2$ manifolds

Abstract: At present, there are only two known constructions for $G_2$ manifolds: desingularizations of toroidal orbifolds (constructed by Joyce), and the twisted connected sum construction (originally introduced by Kovalev, and extended by Corti, Haskins, Nordström and Pacini). The latter construction begins from algebraic geometry -- a semi-Fano 3-fold and a K3 surface on it -- and also involves a nontrivial application of the "matching" of two non-isomorphic algebraic K3 surfaces whose underlying Ricci-flat metrics are the same. Although we will be applying results from the theory of K3 surfaces which were covered in earlier lectures this fall, this lecture is independent of the earlier ones.

Audio [ mp3, m4a ]; Lecture notes.