JHU-UMD Complex Geometry Seminar Archives for Academic Year 2015


When: Tue, September 15, 2015 - 4:30pm
Where: Math 3206
Speaker: Gábor Székelyhidi (Notre Dame) - http://www3.nd.edu/~gszekely
Abstract: TBA

K-stability of finite covers

When: Tue, October 13, 2015 - 4:30pm
Where: Krieger 300 (JHU)
Speaker: Ruadhaí Dervan (University of Cambridge) -
Abstract: An important result of Chen-Donaldson-Sun and Tian relates the existence of Kaehler-Einstein metrics on Fano varieties to an algebro-geometric notion called K-stability. K-stability is however understood in very few cases. We show that certain finite covers of K-stable Fano varieties are K-stable.

K-stability implies CM-stability

When: Sat, October 24, 2015 - 11:00am
Where: Howard University
Speaker: Gang Tian (Princeton) - http://math.jhu.edu/~bernstein/MDGS/Schedule.html
Abstract: Both K-stability and CM-stability were first introduced on Fano manifolds in 90's and generalized to any polarized projective manifolds. In this talk, I will show how the K-stable implies CM-stable. I will also discuss their relation to Geometric Invariant Theory and the problem on existence of constant scalar curvature Kahler metrics.

Differential inequalities and generalized pluripotential theories

When: Tue, November 10, 2015 - 3:30pm
Where: 1308.0
Speaker: Blaine Lawson (Stony Brook) -
Abstract: I will focus on differential inequalities of the form D^2 u\in F where F is a closed subset of the symmetric matrices satisfying a weak ellipticity condition. Interesting cases arise in many areas of geometry -- for example, in studying Monge-Ampere equations, in Lagrangian geometry, calibrated geometry, Hessian equations, p-convexity, etc. To each such F there is an associated pluripotential theory based on the upper semi-continuous functions u which satisfy D^2 u\in F in a generalized sense. For such u, I will discuss the existence and uniqueness of tangents, monotonicity theorems, density functions Theta(u,x), and the structure of the sets {x : Theta(u,x)>= c} for c>0. I will also discuss the Dirichlet Problem with Prescribed Asymptotic Singularities in the interior of the domain, for the differential equation associated to F. In particular, this gives the existence of Green's Functions and multi-pole Green's functions for these nonlinear equations. Much of the analysis is based on the notion of the Riesz characteristic of F, which will be introduced near the outset of the talk.

On the L2 estimates on moduli space of Calabi-Yau manifolds

When: Tue, December 1, 2015 - 4:30pm
Where: JHU Krieger 300
Speaker: Zhiqin Lu (UC Irvine) -
Abstract: I will talk about my recent result on the L2 estimates joint with Hang Xu. In the first part of the talk, I will introduce the L2 estimates on noncomplete Kahler manifolds, after we prove that the self-adjoint extensions of holomorphic bundle-valued Laplacians on moduli space of polarized Kahler manifolds are unique. Then we shall use the result to prove that the holomorphic sections of the Hodge bundles over moduli spaces are of polynomial growth.

Infinite dimensional geometry on the space of Kahler metrics and applications to canonical Kahler metrics

When: Tue, February 9, 2016 - 4:30pm
Where: JHU Shaffer 304
Speaker: Tamas Darvas (UMD) -
Abstract: First we present the L^p Mabuchi structure of the space of Kahler metrics and give a comparison of the arising geometries. In the second part of the talk we will give applications to the long time behaviour of the Calabi flow, (and time permitting) properness of the K-energy and K-stability. (joint work R. Berman and L.H. Chinh)

Infinite dimensional geometry on the space of Kahler metrics and applications to canonical Kahler metrics

When: Tue, March 1, 2016 - 4:30pm
Where: Shaffer 304(JHU)
Speaker: Tamas Darvas (UMD) -

Sampling polynomials in algebraic varieties

When: Tue, March 29, 2016 - 4:30pm
Where: Shaffer 304 (at Hopkins)
Speaker: Joaquim Ortega-Cerdà (Barcelona) -
Abstract: I will present a joint work with Robert Berman where we consider the problem of sampling multivariate real polynomials of large degree in a general framework where the polynomials are defined on an affine real algebraic variety equipped with a weighted measure. It is shown that a necessary condition for sampling, in this general setting, is that the asymptotic density of sampling points is greater than the density of the corresponding weighted equilibrium measure, as defined in pluripotential theory. This result thus generalizes the well-known Landau-type results for sampling on the torus, where the corresponding critical density corresponds to the Nyqvist rate, as well as the classical result saying that zeroes of orthogonal polynomials become equidistributed with respect to the logarithmic equilibrium measure as the degree tends to infinity. NOTE: THIS TALK WILL BE AT HOPKINS. PLEASE CONTACT H. HEIN IF YOU WOULD LIKE TO ATTEND.

Berndtsson's Convexity Theorem and the L^2 Extension Theorem

When: Tue, April 26, 2016 - 4:30pm
Where: Shaffer 304 (at Hopkins)
Speaker: Dror Varolin (Stony Brook) - http://www.math.stonybrook.edu/~dror
Abstract: I will discuss a relatively new proof, due to Berndtsson and Lempert, of the L^2 Extension Theorem due to Ohsawa and Takegoshi. Loosely speaking, the theorem states that any holomorphic function with weighted L^2 estimates on a hypersurface in a pseudoconvex domain has an L^2 extension. The idea of the new proof is to degenerate the domain onto the hypersurface in a pseudoconvex way, and to show that the extension of minimal norm has its worst bounds when the domain is an infinitesimal neighborhood of the hypersurface. The latter fact uses a theorem of Berndtsson on the positivity of the curvature of Hilbert bundles of holomorphic L^2 spaces with psh weights over pseudoconvex domains. If time permits, I will explain this theorem.