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.