Date: Tuesdays at 4:30pm.
Y.A. Rubinstein , B.
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.
Room: Krieger Hall 308 (JHU), Mathematics Building 3206 (UMD).
September 18, UMD
A generalised Monge-Ampere equation arising out a natural question from Chern-Weil 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 Chern-Weil theory, a computation of
Chern forms for certain trivial bundles with non-diagonal metrics shall be shown along with an
application or two.
Vamsi Pingali (Stony Brook),
Some analytic and computational aspects of Chern-Weil forms
October 2, JHU
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
Ioana Suivaina (Vanderbilt)
ALE Ricci-flat Kahler surfaces and weighted projective spaces
October 23, UMD, 3:30pm, room 2300 (note special time AND room)
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.
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 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.
Tamas Darvas (Purdue) at 4:40pm, room 3206 (usual room)
Morse theory and geodesics in the space of Kahler metrics
November 6 (rescheduled due to weather), JHU
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.
Gang Liu (Minnesota)
Some comparison theorems for Kahler manifolds with Ricci curvature bounded from below
November 13, UMD
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,
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.
Boris Hanin (Northwestern)
Correlations and Pairing of Zeros and Critical Points of Random
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 Kahler-Ricci
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
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.
Norm Levenberg (Indiana)
Characterization of meromorphic functions and projective hulls
February 12, UMD
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.
Slawomir Dinew (Rutgers)
Interior (ir)regularity for the complex Monge-Ampere equation
February 26, JHU
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.
Xiaochun Rong (Rutgers)
Continuity of extremal transitions and flops for Calabi-Yau
March 12, JHU
Dan Coman (Syracuse)
Convergence of the Fubini-Study currents for singular metrics on line bundles and
April 2, UMD
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.
Guangbo Xu (Princeton)
Gauged linear sigma model and adiabatic limits
April 16, UMD
Simon Donaldson (Imperial)
Informal talk on Kahler-Einstein geometry
April 30, JHU
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.
Ben Weinkove (Northwestern)
Geometric flows on complex surfaces
May 14, UMD
David Witt-Nystrom (Chalmers)
Local circle actions on Kahler manifolds and the Hele-Shaw flow