### JHU-UMD Complex Geometry Seminar Archives for Academic Year 2013

#### Some analytic and computational aspects of Chern-Weil forms

When: Tue, September 18, 2012 - 4:30pm
Where: Math 3206
Speaker: Vamsi Pingali (Stony Brook) -
Abstract: http://www2.math.umd.edu/~yanir/cgs.html

#### ALE Ricci-flat Kahler surfaces and weighted projective spaces

When: Tue, October 2, 2012 - 4:30pm
Where: JHU Shaffer Hall 303
Speaker: Ioana (Suvaina) -
Abstract: 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 Tian-Yau construction of complete Ricci flat Kahler manifolds.

#### Counting chord diagrams

When: Tue, October 23, 2012 - 3:30pm
Where: Math 2300
Speaker: Robert Penner (Aarhaus and Caltech) -
Abstract: 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 half-plane
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 Eynard-Orantin has computed them perturbatively via a closed form expression for the free
energies of an Hermitian matrix model with potential  V(x)=x^2/2-stx/(1-tx). Very recent work has
moreover shown that the partition function satisfies a second order non-linear PDE which gives a
generalization of the Harer-Zagier equation that arises for one backbone.

#### Morse theory and geodesics in the space of Kahler metrics

When: Tue, October 23, 2012 - 4:30pm
Where: Math 3206
Speaker: Tamas Darvas (Purdue) -
Abstract: 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.

#### Some comparison theorems for Kahler manifolds with Ricci curvature bounded from below

When: Tue, October 30, 2012 - 4:30pm
Where: JHU Shaffer Hall 303
Speaker: Gang Liu (Minnesota) -
Abstract: Comparison theorems are a fundamental tool in Riemannian geometry. When the Ricci curvature is bounded from below, one has Bishop-Gromov volume comparison, Bonnet-Myers 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.

#### Some comparison theorems for Kahler manifolds with Ricci curvature bounded from below

When: Tue, November 6, 2012 - 4:30pm
Where: JHU Shaffer Hall 303
Speaker: Gang Liu (Minnesotra) -
Abstract: Comparison theorems are a fundamental tool in Riemannian geometry. When the Ricci curvature is
bounded from below, one has Bishop-Gromov volume comparison, Bonnet-Myers 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.

#### Correlations and Pairing of Zeros and Critical Points of Random Polynomials

When: Tue, November 13, 2012 - 4:30pm
Where: Math 3206
Speaker: Boris Hanin (Northwestern) -
Abstract: 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 Gauss-Lucas 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.

#### Analytic minimal model program with Ricci flow

When: Tue, November 27, 2012 - 4:30pm
Where: JHU Shaffer Hall 303
Speaker: Jian Song (Rutgers) -
Abstract: I will introduce the analytic minimal model program proposed
by Tian and me to study formation of singularities of the Kahler-Ricci
flow. Â We also construct geometric and analytic surgeries of
codimension one and higher codimensions equivalent to birational
transformations in algebraic geometry by Ricci flow.

#### Characterization of meromorphic functions and projective hulls

When: Tue, December 11, 2012 - 4:30pm
Where: 3206.0
Speaker: Norm Levenberg (Indiana) -
Abstract: 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 real-analytic 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.

#### Interior (ir)regularity for the complex Monge-Ampere equation

When: Tue, February 12, 2013 - 4:30pm
Where: 3206.0
Speaker: Slawomir Dinew (Rutgers, Newark) -
Abstract: The complex Monge-Ampere 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.

#### Continuity of extremal transitions and flops for Calabi-Yau manifolds

When: Tue, February 26, 2013 - 4:30pm
Where: Shaffer Hall 303 (JHU)
Speaker: Xiaochun Rong (Rutgers) -
Abstract: We will discuss metric behavior of Ricci-flat Kahler metrics on
Calabi-Yau manifolds under algebraic geometric surgeries: extremal
transitions or flops. We will prove a version of Candelas and de la Ossa's
conjecture: Ricci-flat Calabi-Yau manifolds related via extremal
transitions and flops can be connected by a path consisting of continuous
families of Ricci-flat Calabi-Yau manifolds and a compact metric space in
the Gromov-Hausdorff topology. This is joint work with Yuguang Zhang.

#### Convergence of the Fubini-Study currents for singular metrics on line bundles and applications

When: Tue, March 12, 2013 - 4:30pm
Where: JHU Shaffer Hall 303
Speaker: Dan Coman (Syracuse) -
Abstract: http://www2.math.umd.edu/~yanir/cgs.html

#### Gauged linear sigma-model and adiabatic limits

When: Tue, April 2, 2013 - 4:30pm
Where: 3206.0
Speaker: Guangbo Xu (Princeton) -
Abstract: 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 Gromov-Witten 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.

#### Geometric flows on complex surfaces

When: Tue, April 30, 2013 - 4:30pm
Where: Krieger Hall 308 (JHU)
Speaker: Ben Weinkove (Northwestern) -
Abstract: I will discuss the behavior of the Kahler-Ricci flow and a new flow generalizing it, called the Chern-Ricci flow, recently introduced by M. Gill. The Chern-Ricci 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.

#### Local circle actions on Kahler manifolds and the Hele-Shaw flow

When: Tue, May 14, 2013 - 4:30pm
Where: 3206.0
Speaker: David Witt Nystrom (Chalmers) -