##### Department of Mathematics,

University of California San Diego

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### Special Colloquium

## Richard Bamler

#### Stanford University

## Long-time behavior of Ricci flows and construction of Einstein metrics

##### Abstract:

In this talk I will survey current results on the long-time existence and behavior of Ricci flows in dimensions 2, 3 and higher. Moreover, I will point out analogies with construction techniques for Einstein metrics. In dimension 3, the Ricci flow together with a certain surgery process has been used by Perelman, amongst many others, to establish the PoincarÃƒÂ© and Geometrization Conjectures. Despite the depth of this result, a precise description of the long-time behavior of this flow has remained unknown. For example, it was only conjectured by Perelman that it suffices to carry out a finite number of surgeries and that the geometric decomposition of the manifold is exhibited by the flow as $t \to \infty$. Recently I was able to confirm Perelman's first conjecture and I partially answered his second one. I will first give a brief overview of Ricci flows with surgery and explain the finite surgery theorem. Next, I will present long-time existence results in dimensions 4 and higher and describe possible further directions in this field.

Hosts: Ben Chow and Lei Ni

### December 10, 2013

### 2:00 PM

### AP&M 6402

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