Where: Math 3206

Speaker: Vamsi Pingali (Stony Brook) -

Abstract: http://www2.math.umd.edu/~yanir/cgs.html

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Where: JHU Shaffer Hall 303

Speaker: Dan Coman (Syracuse) -

Abstract: http://www2.math.umd.edu/~yanir/cgs.html

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.

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.

Where: 3206.0

Speaker: David Witt Nystrom (Chalmers) -