The 26th SCGAS

Southern California Geometric Analysis Seminar

University of California, San Diego
January 26-27, 2019



Program

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All lectures to be held in Center 105. Click here for a pdf map.


Saturday morning, 01/26/19  
9:00am-9:45am Meet at Lobby of the Hotel (Best Western Del Mar, otherwise you need to let us know ahead of time so that proper arrangement can be made. Many Hotels also provide shuttle service to campus. Ask about it the night before) for possible ride to campus. Walking is also possible, but takes longer.
10:00am-10:50am Registration. Coffee and snacks served
10:50am-11:00am Welcome and a brief introduction
11:00am-11:50am X. Chen: On the constant scalar curvature Kaehler metric
   
Saturday afternoon, 01/26/19  
1:30pm-2:20pm G. Sacca: Hyper-K\"ahler manifolds deformation to O'Grady's 10-dimensional example
2:20pm-2:50pm Break, refreshments served
2:50pm-3:40pm F. Zheng: Hermitian manifolds with flat connections
3:40pm-4:10pm Break, refreshments served
4:10pm-5:00pm T. Collins: Stability and Nonlinear PDE in mirror symmetry
5:30pm Meet for ride to restaurant at APM building 1st floor
6:15pm Conference dinner at FungFung Yuen. Limited seats. (Cost is $40 per person, pay at the registration, closes when it hits the limit).
   
Sunday morning, 01/27/19  
8:15am-8:30am Meet at lobby of hotels for ride to campus. Walking is also possible, but takes longer.
8:40am-9:10am Coffee and snacks
9:10am-10:00am B. Wilking: The topology of fixed point components of Torus actions in positively curved manifolds
10:00am-10:25am Break, refreshments served
10:25am-11:15am J. Wang: Geometry of shrinking Ricci solitons
11:15am-11:40am Break, refreshments served
11:40am-12:30pm R. Haslhofer: Ancient low entropy flows and the mean convex neighborhood conjecture
   

 

Abstracts:

Xiuxiong Chen (Stony Brook).

Abstract: Inspired by the celebrated $C^0, C^2 $ and $C^3$ a priori estimate of Calabi, Yau and others on Kaehler Einstein metrics, we present a report on a priori estimates on constant scalar curvature Kaehler metrics. With this estimate, we prove the Donaldson conjecture on geodesic stability and the well known properness conjecture on the Mabuchi energy functional. This is a joint work with Cheng JingRui.


Tristan Collins (MIT).

Abstract: A longstanding problem in mirror symmetry has been to understand the relationship between the existence of solutions to certain geometric nonlinear PDES (the special Lagrangian equation, and the deformed Hermitian-Yang-Mills equation) and algebraic notions of stability, mainly in the sense of Bridgeland. I will discuss progress in this direction through ideas originating in infinite dimensional GIT. This is joint work with S.-T. Yau.


Robert Haslhofer (Toronto).

Abstract: In this talk, I will explain our recent proof of the mean convex neighborhood conjecture for the mean curvature flow of surfaces in R^3. Namely, if the flow has a spherical or cylindrical singularity at a space-time point X=(x,t), then there exists a positive ε=ε(X)>0 such that the flow is mean convex in a space-time neighborhood of size ε around X. The major difficulty is to promote the infinitesimal information about the singularity to a conclusion of macroscopic size. In fact, we prove a more general classification result for all ancient low entropy flows that arise as potential limit flows near X. As an application, we prove the uniqueness conjecture for mean curvature flow through spherical or cylindrical singularities. In particular, assuming Ilmanen's multiplicity one conjecture, we conclude that for embedded two-spheres the mean curvature flow through singularities is well-posed. This is joint work with Kyeongsu Choi and Or Hershkovits.


Giulia Sacca (Columbia).

Abstract: Compact hyper-Kähler manifolds are having an increasingly central role in algebraic geometry, both as higher dimensional analogues of K3 surfaces and in terms of their relationship to cubic hypersurfaces of dimension 4. In this talk I will discuss some recent and very recent results on these topics, focusing on O'Grady's 10-dimensional example.


Jiaping Wang (Minnesota).

Abstract: Self similar solutions to Ricci flows or Ricci solitons play a prominent role in the study of Ricci flow singularities. The talk concerns the geometry of shrinking Ricci solitons with an emphasis on the four dimension case. This is joint work with Ovidiu Munteanu.


Burkhard Wilking (Muenster).

Abstract:We show that each fixed point component of an isometric torus action of a 5 torus has the rational cohomology of a rank one symmetric space. We give various applications.


Fangyang Zheng (Ohio State).

Abstract:The classic Bieberbach Theorem states that any compact flat Riemannian manifold is finitely covered by a flat torus. If we consider a compact complex manifold, we may ask the question of when will it be flat? There are several ways to impose the question, here we restrict ourselves only to the canonical metric connections. For a given Hermitian manifold, there are several unqiuely determinded connections associated to the metric: the Riemannian (Levi-Civita) connection, the Chern connection, the Bismut connection, and the family of Gauduchon connections which is the line joining the Chern and Bismut connections. One can the question of what kind of compact Hermitian manifolds will have one of those connections being flat? Here we will report on some recent progress on this topic and also discuss some open questions.