005 Recent Progress in Geometric Analysis

2023-04-04 08:40:01



1. 8:30-9:30


Title:Introduction to Milnor's Conjecture

Abstract:In this talk, we will review the history of Milnor's conjecture, such as the low-dimensional case, the almost nilpotent properties of fundamental groups and the finite generation of local fundamental groups, etc.

2. 9:50-10:50 

SpeakerAaron Naber,Northwestern University

TitleRicci Curvature, Fundamental Group and the Milnor Conjecture

AbstractIt was conjectured by Milnor in 1968 that the fundamental group of a complete manifold with nonnegative Ricci curvature is finitely generated. The main result of this paper is a counterexample, which provides an example M^7 with Ric>= 0 such that \pi_1(M)=Q/Z is infinitely generated.

There are several new points behind the result. The first is a new topological construction for building manifolds with infinitely generated fundamental groups, which can be interpreted as a smooth version of the fractal snowflake. The ability to build such a fractal structure will rely on a very twisted gluing mechanism. Thus the other new point is a careful analysis of the mapping class group \pi_0Diff(S^3\times S^3) and its relationship to Ricci curvature. In particular, a key point will be to show that the action of \pi_0Diff(S^3\times S^3) on the standard metric g_{S^3\times S^3} lives in a path connected component of the space of metrics with Ric>0.

Tencent Meeting: 448 331 498

PASSCODE: 230409

3. 11:20-12:10


TitleOn the Kahler-Ricci flow on spherical Fano manifolds

AbstractI will talk about the recent progress of the Kahler-Ricci flow on Fano manifolds. Especially, I will give some more precise results for spherical Fano manifolds.