##### Department of Mathematics,

University of California San Diego

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### Math 258

## Liao Yuan

#### UCSD

## Homogenous structures and the Ricci flow

##### Abstract:

This talk will focus on Böhm and Lafuente 2017's work on immortal homogeneous Ricci flow, where they prove that any sequence of blow-downs of such flow will subconverges to an expanding homogeneous Ricci soliton. For a \mathfrac{g}-homogeneous space M, Ricci flow of G-invariant metrics can be shown to be equivalent to a flow on Ad(H) invariant "bracket." We will show the existence of stratification of the space of brackets that induces curvature estimate on each strata, motivated by geometric invariant theory. The sharp case of the inequality corresponds to the limit case of an expanding G-invariant Ricci soliton, and we will show that as immortal homogeneous Ricci flow is of Type III, the blowdown will subconverges to such limit case. Finally if time allows, we will discuss Böhm and Lafuente recent proof of Alekseevskii's conjecture and its implication to homogeneous Ricci soliton.

### June 1, 2023

### 1:00 PM

APM 5829

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