Mathematical Sciences Research Institute - Introductory Workshop on Geometric Flows and Function Theory in Real and Complex Geometry

Various geometric evolution equations and function theory have the common goal of understanding the relations between the geometry, analysis, and topology of manifolds, submanifolds, vector bundles, maps, and other geometric structures. The fields of geometry, analysis, and topology are synthesized through the study of geometric and topological invariants via a priori estimates.

The goal of the Introductory Workshop is to survey current and recentdevelopments in geometric evolution equations and function theory in real and complex geometry.

Invited speakers include:
Albert Chau (University of Waterloo), Jaigyoung Choe (Seoul National University), Andre Neves (Princeton University), Lei Ni (University of California, San Diego), Natasa Sesum (New York University, Courant Institute), Jian Song (Johns Hopkins University), Jiaping Wang (University of Minnesota), Ben Weinkove (Imperial College, London), and Zhou Zhang (Massachusetts Institute of Technology).

The talks will focus on the following topics:

Geometric flows in complex geometry
Function theory on manifolds
Ricci flow and canonical metrics
Mean curvature flow and applications

Schedule of the Lectures

Monday September 11
9:30-10:30am Ben Weinkove: Geometric flows and moment maps, part I
Tea Break
11am-12pm Jiaping Wang: Stability of harmonic functions
Abstract: We aim to explain a result of P. Li and L. Tam relating the harmonic functions to the number of ends of a complete manifold. We start with their constructive proof of the existence of Green's function, and use the result to show the stability of the space of harmonic functions under the perturbation within a compact domain. We then reach the conclusion by carrying out a straightforward construction of the so-called barrier function on each end.
Lunch
2-3pm Andre Neves: Lagrangian mean curvature flow, part I
Tea Break
3:30-4:30pm Natasa Sesum: Convergence and stability results for the Ricci flow, part I

Tuesday September 12
9:30-10:30am Ben Weinkove: Geometric flows and moment maps, part II
Tea Break
11am-12pm Jiaping Wang: Sharp estimate of Green's function and applications
Abstract: We will establish a sharp integral decay estimate of the minimal positive Green's function on a complete manifold with positive spectrum. Some applications will also be discussed.
Lunch
2-3pm Natasa Sesum: Convergence and stability results for the Ricci flow, part II
Tea Break
3:30-4:30pm Lei Ni: Ricci flow and invariant cones

Wednesday September 13
9:30-10:30am Jaigyoung Choe: Rigidity of Capillary Surfaces and First Eigenvalue of Minimal Surfaces
Tea Break
11am-12pm Andre Neves: Lagrangian mean curvature flow, part II
Lunch
No talks in the afternoon

Thursday September 14
9:30-10:30am TBA
Tea Break
11am-12pm Jian Song: The Kaehler-Ricci flow on surfaces, part I
Abstract: We study the Kaehler-Ricci flow on minimal surfaces of Kodaira dimension one and show that the flow collapses and converges to a unique canonical metric on its canonical model. Such a canonical metric is a generalized Kaehler-Einstein metric. Combining the results of Cao, Tsuji, Tian and Zhang, we give a metric classification for Kaeher surfaces with a numerically effective canonical line bundle by the Kaehler-Ricci flow. In general, we propose to find canonical metrics on the canonical models of projective varieties of positive Kodaira dimension. This is a joint work with G. Tian.
Lunch
2-3pm Albert Chau: Uniformization of complete non-compact Kaehler manifolds and the Kaehler Ricci flow, part I

Friday September 15
9:30-10:30am Jian Song: The Kaehler-Ricci flow on surfaces, part II
Tea Break
11am-12pm Albert Chau: Uniformization of complete non-compact Kaehler manifolds and the Kaehler Ricci flow, part II
Lunch
2-3pm Zhou Zhang: Kaehler-Ricci Flows over Manifolds of General Type

Currently Available Videos

Ben Weinkove , Geometric flows and moment maps, part I September 11,2006, 09:30 AM to 10:30 AM
Jiaping Wang , Stability of harmonic functions September 11,2006, 11:00 AM to 12:00 PM
Andre Neves , Lagrangian mean curvature flow, part I September 11,2006, 02:00 PM to 03:00 PM
Natasa Sesum , Convergence and stability results for the Ricci flow, part I September 11,2006, 03:30 PM to 04:30 PM
Ben Weinkove , Geometric flows and moment maps, part II September 12,2006, 09:30 AM to 10:30 AM
Jiaping Wang , Sharp estimate of Green's function and applications September 12,2006, 11:00 AM to 12:00 PM
Natasa Sesum , Convergence and stability results for the Ricci flow, part II September 12,2006, 02:00 PM to 03:00 PM
Lei Ni , Ricci flow and invariant cones September 12,2006, 03:30 PM to 04:30 PM
Tom Ilmanen , TBA, part I September 13,2006, 09:30 AM to 10:30 AM
Andre Neves , Lagrangian mean curvature flow, part II September 13,2006, 11:00 AM to 12:00 PM
Tom Ilmanen , TBA, part II September 14,2006, 09:30 AM to 10:30 AM
Jian Song , The Kaehler-Ricci flow on surfaces, part I September 14,2006, 11:00 AM to 12:00 PM
Albert Chau , Uniformization of complete non-compact Kaehler manifolds and the Kaehler Ricci flow, part I September 14,2006, 02:00 PM to 03:00 PM
Jian Song , The Kaehler-Ricci flow on surfaces, part II September 15,2006, 09:30 AM to 10:30 AM
Albert Chau , Uniformization of complete non-compact Kaehler manifolds and the Kaehler Ricci flow, part II September 15,2006, 11:00 AM to 12:00 PM

You can find videos of other workshops and events on our VMath - Streaming Video page.

For more information:
Questions about this workshop should be sent either by email to

380@msri.org

or by regular mail to:

Introductory Workshop on Geometric Flows and Function Theory in Real and Complex Geometry Mathematical Sciences Research Institute 17 Gauss Way, Berkeley, CA 94720-5070. USA

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