JHU-UMD Complex Geometry Seminar

Departments of Mathematics

Johns Hopkins University & University of Maryland

Fall 2015 - Spring 2016


Date: Tuesdays at 4:30pm.
Room: Shaffer 304(JHU), Mathematics Building 3206 (UMD).

Organized by: T. Darvas, V.P. Pingali, H.J. Hein, Y.A. Rubinstein, B. Shiffman, R. Wentworth, S. Wolpert.
Coordinating organizers: Y.A. Rubinstein, B. Shiffman

The JHU-UMD Complex Geometry Seminar was founded in 2012 by Y. Rubinstein (UMD) and B. Shiffman (JHU). It serves to strengthen the ties between the historically vibrant complex geometry communities in Maryland. Starting with the academic year 2015/6, the seminar will meet (only) once a month, alternating locations, with the purpose of discussing recent developments in both complex and convex geometry and analysis.

The seminar is a combination of a learning and a research seminar. The first 15 minutes or so of each talk are a "trivial notions" talk defining all basic notions, giving examples and intuition to the subject, and should be accessible to a beginning graduate student. The next 50 minutes are a regular seminar talk.

Previous years: 2012-2013, 2013-2014, 2014-2015.

ˇ         September 15(UMD)
Gábor Székelyhidi, University of Notre Dame
Title: The J-flow on toric manifolds
Abstract: The J-flow is a parabolic equation for Kahler metrics, with relations to constant scalar curvature Kahler metrics. Work of Chen and Song-Weinkove give sufficient and necessary conditions for convergence of the flow, but they are hard to check in practice. With Mehdi Lejmi I conjectured that a simpler numerical criterion is an equivalent condition. I will discuss joint work with Tristan Collins, which proves this conjecture on toric manifolds.

ˇ         October  13, 4:30-6:00, JHU, Krieger 300
Ruadhaí Dervan (University of Cambridge)
Title: K-stability of finite covers
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.

ˇ         October 24
Gang Tian (Princeton) within MADGUYS at Howard University
Title: K-stability implies CM-stability
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 K-stability implies CM-stability. I will also discuss their relation to Geometric Invariant Theory and the problem on existence of constant scalar curvature Kahler metrics.

ˇ         November 10 3:30 PM, note special time (UMD),
Blaine Lawson (Stony Brook)
Title: Differential inequalities and generalized pluripotential theories
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.

ˇ         December 1 (JHU)
Zhiqin Lu (UC Irvine)
Title: On the L2 estimates on moduli space of Calabi-Yau manifolds
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.

ˇ         March 1 (JHU)
Tamas Darvas (UMD)
Title: Infinite dimensional geometry on the space of Kahler metrics and applications to canonical Kahler metrics
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)

ˇ         March 29 (JHU)
Joaquim Ortega-Cerdŕ (Barcelona)
Title: Sampling polynomials in algebraic varieties.

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.

ˇ         April 26 (JHU)
Dror Varolin (Stony Brook)
Title: Berndtsson's Convexity Theorem and the L^2 Extension Theorem
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.


Driving directions to JHU. Park in South Garage (see map) on any level (except the reserved spaces). Take a ticket when entering. The Department will provide a visitor parking pass to use when exiting.
Driving and parking directions to UMD. Park in Paint Branch Drive Visitor Lot (highlighted in yellow in the lower right corner of the second map in the previous link), or in Regents Drive Garage (highlighted in the upper right corner). If you arrive after 4pm you do not need to pay: see the instructions in the previous link.