Date: Tuesdays at 4:30pm.
Room: Krieger Hall 308 (JHU), Mathematics Building 3206 (UMD).
Organized by:
Y.A. Rubinstein , B.
Shiffman ,
R. Wentworth,
S. Wolpert,
Y. Yuan
.
The seminar is a combination of a learning and a research seminar.
The first 30 minutes 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.

September 18, UMD
Vamsi Pingali (Stony Brook),
Some analytic and computational aspects of ChernWeil forms
A generalised MongeAmpere equation arising out a natural question from ChernWeil theory shall
be presented. The difficulties involved with the analysis shall be discussed along with a
connection to Kahler geometry. On the computational side of ChernWeil theory, a computation of
Chern forms for certain trivial bundles with nondiagonal metrics shall be shown along with an
application or two.

October 2, JHU
Ioana Suivaina (Vanderbilt)
ALE Ricciflat Kahler surfaces and weighted projective spaces
I will give an explicit classification of the ALE Ricci flat Kahler surfaces, generalizing
previous classification results of Kronheimer. These manifolds are related to a special class of
deformations of quotient singularities of type C^2/G, with G a finite subgroup of U(2). I finish
the talk by explaining the relations with the TianYau construction of complete Ricci flat Kahler
manifolds.

October 23, UMD, 3:30pm, room 2300 (note special time AND room)
Robert Penner (Aarhaus and Caltech)
Counting chord diagrams
A linear chord diagram on some number b of backbones is a collection of n chords with distinct
endpoints attached to the interiors of b intervals.
Taking the intervals to lie in the real axis and the chords to lie in the upper halfplane
associates a fat graph to a chord diagram, which thus has its associated genus g. The numbers
of connected genus g chord diagrams on b backbones with n chords are of significance in
mathematics, physics and biology as we shall explain. Recent work using the topological recursion
of EynardOrantin has computed them perturbatively via a closed form expression for the free
energies of an Hermitian matrix model with potential V(x)=x^2/2stx/(1tx). Very recent work has
moreover shown that the partition function satisfies a second order nonlinear PDE which gives a
generalization of the HarerZagier equation that arises for one backbone.
Tamas Darvas (Purdue) at 4:40pm, room 3206 (usual room)
Morse theory and geodesics in the space of Kahler metrics
Given a compact Kahler manifold let H be the set of Kahler forms in a fixed cohomology class. As
observed by Mabuchi, this space has the structure of an infinite dimensional Riemannian manifold,
if one identifies it with a totally geodesic subspace of H, the set of Kahler potentials.
Following Donaldson's program, existence and regularity of geodesics in this space is of
fundamental interest. Supposing enough regularity of a geodesic
u : [0; 1]> H, we establish a Morse theoretic result relating the endpoints with the initial
tangent vector. As an application, we prove that on all
Kahler manifolds, connecting Kahler potentials with smooth geodesics
is not possible in general.

November 6 (rescheduled due to weather), JHU
Gang Liu (Minnesota)
Some comparison theorems for Kahler manifolds with Ricci curvature bounded from below
Comparison theorems are a fundamental tool in Riemannian geometry. When the Ricci curvature is
bounded from below, one has BishopGromov volume comparison, BonnetMyers theorem on the
diameter, comparison theorems on the spectrum of the Laplacian, and more. In the Kahler setting,
Li and Wang established analogous comparisons when the bisectional curvature has a lower bound.
In this talk, I will discuss some comparison theorems on Kahler manifolds when the Ricci
curvature has a lower bound.

November 13, UMD
Boris Hanin (Northwestern)
Correlations and Pairing of Zeros and Critical Points of Random
Polynomials
The goal of this talk is to explain how the zeros and
holomorphic critical points of random polynomials are correlated. The
motivation for studying this question comes from the GaussLucas theorem,
which states
that the critical points of a polynomial in one complex variable lie
inside the convex hull of its zeros. I will explain that, in fact, zeros
and critical points appear in rigid pairs. I will present some results
about the geometry of these pairs, and I will try to give some physical
intuition for why they should appear in the first place.

November 27, JHU
Jian Song (Rutgers)
Analytic minimal model program with Ricci flow
I will introduce the analytic minimal model program proposed
by Tian and me to study formation of singularities of the KahlerRicci
flow. We also construct geometric and analytic surgeries of
codimension one and higher codimensions equivalent to birational
transformations in algebraic geometry by Ricci flow.

December 11, UMD
Norm Levenberg (Indiana)
Characterization of meromorphic functions and projective hulls
Reese Harvey and Blaine Lawson introduced the notion of the projective hull of a closed subset in
a complex projective space with the hope of generalizing a result of John Wermer on the
polynomial hull of a realanalytic curve in a complex affine space. Both notions of "hull" can be
understood in terms of an extremal (quasi)plurisubharmonic function associated to the underlying
set. We begin by giving background motivation, definitions and examples of these hulls in the
setting of pluripotential theory; and we include a complex geometric interpretation of the
projective hull. Then we utilize these ideas to give conditions characterizing holomorphic and
meromorphic functions in the unit disk in the complex plane in terms of certain weak forms of the
maximum modulus principle. These characterizations are joint work with John Anderson, Joe Cima
and Tom Ransford.

February 12, UMD
Slawomir Dinew (Rutgers)
Interior (ir)regularity for the complex MongeAmpere equation
The complex MongeAmpere operator arises in many geometric problems.
When studying its local properties it is natural to ask for its interior
regularity theory. This is crucial if analysis is performed in coordinate
charts. Quite contrary to linear differential operators there is however
no general purely interior result. In the talk we shall present several
additional conditions under which such results can be obtained. We shall
give several examples suggesting what is the expected behavior under
different regularity assumptions.

February 26, JHU
Xiaochun Rong (Rutgers)
Continuity of extremal transitions and flops for CalabiYau
manifolds
We will discuss metric behavior of Ricciflat Kahler metrics on
CalabiYau manifolds under algebraic geometric surgeries: extremal
transitions or flops. We will prove a version of Candelas and de la Ossa's
conjecture: Ricciflat CalabiYau manifolds related via extremal
transitions and flops can be connected by a path consisting of continuous
families of Ricciflat CalabiYau manifolds and a compact metric space in
the GromovHausdorff topology. This is joint work with Yuguang Zhang.

March 12, JHU
Dan Coman (Syracuse)
Convergence of the FubiniStudy currents for singular metrics on line bundles and
applications
Abstract

April 2, UMD
Guangbo Xu (Princeton)
Gauged linear sigma model and adiabatic limits
The physics theory of gauged linear sigma model combines the theory of maps (the sigma model) and
gauge theory. In dimension 2, it is naturally related to holomorphic vector bundles over Riemann
surfaces and GromovWitten invariants of projective spaces (or more general varieties). In this
talk, I will discuss, from a mathematical perspective, of some simple examples in gauged linear
sigma model. I will also discuss about how to use the adiabatic limits of such theory to solve a
natural equation (the vortex equation) in gauged linear sigma model over the complex plane.

April 16, UMD
Simon Donaldson (Imperial)
Informal talk on KahlerEinstein geometry

April 30, JHU
Ben Weinkove (Northwestern)
Geometric flows on complex surfaces
I will discuss the behavior of the KahlerRicci flow and a new flow generalizing it, called the
ChernRicci flow, recently introduced by M. Gill. The ChernRicci flow can be defined on any
complex manifold. I will describe what is known about these flows in the case of complex
surfaces, with an emphasis on examples.

May 14, UMD
David WittNystrom (Chalmers)
Local circle actions on Kahler manifolds and the HeleShaw flow
