# Complete noncompact metrics of special and exceptional holonomy: the first 40 years. Simons Center Weekly seminar. Simons Center for Geometry and Physics. September 11, 2018 - September 11, 2018

Keynote/Named Lecture

I will attempt to give a (biased) overview of recent progress on the construction of complete noncompact metrics of exceptional holonomy. Along the way I will describe some of the most important historical developments since the field began (in the late 1970s). Throughout its history, the field has seen a fruitful back-and-forth between physicists and mathematicians, some of which I will describe. I will try to explain some of the similarities and differences between the more familiar special holonomy metrics — hyperkaehler and Calabi—Yau metrics— and the exceptional cases G_2 and Spin_7 holonomy, and why the latter are much more difficult to construct. In the early 2000s M theorists predicted the existence of various new complete noncompact Riemannian metrics with holonomy group the compact exceptional Lie group G_2. Very recently mathematicians have constructed many, but by no means all, of these physically predicted G_2 metrics and also other G_2 metrics not necessarily anticipated by physics. It will turn out the construction of these complete noncompact metrics of exceptional holonomy relies on some of the most recent developments on constructing complete noncompact hyperkaehler and Calabi—Yau metrics with controlled asymptotic geometry. If (by some miracle) time permits, I will describe some of the future prospects for the field.

### Service Performed By

- Haskins, Mark Professor of Mathematics

### Date

- September 11, 2018

### Service or Event Name

- Simons Center Weekly seminar

### Host Organization

- Simons Center for Geometry and Physics

### Location or Venue

- Stony Brook University