Great Lakes Geometry Conference 2013

Northwestern University
April 20-21, 2013


All lectures to be held in Swift Hall 107. Click here for a campus map.

Saturday morning, 4/20/13  
8:45am-9:30am Registration. Breakfast and snacks served
9:30am-10:30 am Shing-Tung Yau On the pseudonorm project towards a birational classification of algebraic varieties
10:30am-11:00am Break, refreshments served
11:00am-12:00pm Christopher Sogge Harmonic analysis and nodal sets in negatively curved manifolds.
Saturday afternoon, 4/20/13  
1:30pm-2:30pm Rafe Mazzeo Spectral geometry of the Riemann moduli space
2:30pm-3:00pm Break, refreshments served
3:00pm-4:00pm Gábor Székelyhidi Filtrations and test-configurations
4:00pm-4:30pm Break, refreshments served
4:30pm-5:30pm Lydia Bieri Spacetime Geometry and Gravitational Waves
6:30pm Conference dinner at the Hilton Orrington Hotel
Sunday morning, 4/21/13  
8:45am-9:10am Breakfast and snacks served
9:10am-10:10am Bo Berndtsson Flatness of direct images and some uniqueness theorems in Kähler geometry
10:10am-10:35am Break, refreshments served
10:35am-11:35am Yanir Rubinstein Kähler-Einstein (edge) theory
11:35am-12:00pm Break, refreshments served
12:00pm-1:00pm Valentino Tosatti The boundary of the Kähler cone



Bo Berndtsson   (Chalmers). Title: Flatness of direct images and some uniqueness theorems in Kähler geometry.

Abstract: A volume form on a complex manifold X can be interpreted as a metric on the anticanonical line bundle -KX of X. If X is a compact Fano manifold the total volume has certain convexity properties when the metric varies in the space of all positively curved metrics on -KX. These convexity properties are formally similar to Brunn-Minkowski (type) inequalities for volumes of convex bodies in Rn and, as in Brunn-Minkowski theory, convexity is strict unless the variation is 'trivial'. We use this to prove some uniqueness theorems in Kähler geometry like the Bando-Mabuchi uniqueness theorem for Kähler-Einstein metrics and the Tian-Zhu theorem on uniqueness of Kähler-Ricci solitons. Generalizations of these theorems in several directions are also discussed.

Lydia Bieri   (Michigan). Title: Spacetime Geometry and Gravitational Waves

Abstract: Spacetime geometries arising from the Einstein equations coupled to various fields in General Relativity will be investigated. Foliations of the spacetimes into null hypersurfaces play a crucial role in the discussion. Results on their structure and asymptotic behavior yields insight into gravitational waves. In this talk, we focus on the geometric structure of radiative spacetimes, explain the geometric picture of gravitational radiation and discuss new results on their nonlinear phenomena.

Rafe Mazzeo   (Stanford). Title: Spectral geometry of the Riemann moduli space

Abstract: I will discuss a set of recent projects concerning the analysis of the Laplacian for the Weil-Peterson metric on the Riemann moduli space (joint with Ji, Mueller and Vasy), some related work with Swoboda about the fine asymptotics of the Weil Peterson metric, and consequences of this for the asymptotic structure of the heat kernel (due to Gell-Redman).

Yanir Rubinstein   (Maryland). Title: Kähler-Einstein (edge) theory

Abstract: We discuss some recent advances in the theory of Kähler-Einstein metrics on compact manifolds.

Christopher Sogge   (Johns Hopkins). Title: Harmonic analysis and nodal sets in negatively curved manifolds

Abstract: We summarize recent work linking Lp(M) estimates and restriction estimates of eigenfunctions. In particular, these estimates imply that, under the assumption of nonpositive curvature, there are no eigenfunctions maximally concentrating about geodesics (unlike the case of highest weight spherical harmonics on the round sphere). Based on this, under this curvature assumption, we can give slightly improved lower bounds for the size of nodal sets of eigenfunctions compared to the general ones of Colding and Minicozzi.

In the case of n=2 the main estimates are due to Bourgain, the speaker and Zelditch and the speaker. The results in higher dimensions are due to Blair and the speaker, as well as X. Chen and the speaker.

Gábor Székelyhidi   (Notre Dame). Title: Filtrations and test-configurations

Abstract: Test-configurations are certain degenerations of projective manifolds, used in the definition of K-stability, which in turn is related to the existence of special Kähler metrics. I will explain how filtrations of the homogeneous coordinate ring of a projective manifold can be thought of as sequences of test-configurations, and that they encode the limiting behavior of these sequences. Such filtrations arise naturally when studying the Calabi flow, or when trying to minimize the Calabi functional. I will also discuss how filtrations can be used to give a strengthening of the notion of K-stability, and why this is desirable.

Valentino Tosatti   (Northwestern). Title: The boundary of the Kähler cone

Abstract: We will discuss a geometric characterization of classes of positive volume on the boundary of the Kähler cone of a compact Kähler manifold. As an application, we will show that finite time singularities of the Kähler-Ricci flow always form along analytic subvarieties. Joint work with T. Collins.

Shing-Tung Yau   (Harvard). Title: On the pseudonorm project towards a birational classification of algebraic varieties