Where: MTH 3206, Colloquium

Speaker: Jan Swoboda (Bonn & Stanford) -

Abstract: I will review some known results and several open questions concerning the L^2 cohomology of various non-compact moduli spaces carrying a hyperkÃ¤hler metric. Particular attention will be given to the moduli space M of Higgs bundles over a Riemann surface, which arises as the space of solutions of a certain system of nonlinear PDEs, called self-duality equations. Although many aspects of M as a hyperkÃ¤hler manifold are by now well-established, it is still an open problem to describe the geometry of this non-compact manifold near its ends. In this talk, I shall present some preliminary results in that direction. (Joint work with Rafe Mazzeo, Hartmut WeiÃ, and Frederik Witt).

Where: Krieger Hall 300

Speaker: Tristan Collins (Columbia University) - http://www.math.columbia.edu/~tcollins/homepage.html

Abstract: We will discuss a geometric characterization of classes of positive volume on the

boundary of the Kahler cone of a compact Kahler manifold. As an application,

we will show that finite time singularities of the Kahler-Ricci flow always form along analytic subvarieties. This work is joint with Valentino Tosatti.

Where: Krieger Hall 300, JHU

Speaker: Chi Li (Stony Brook)

http://www.math.sunysb.edu/~chili/

Abstract: I will talk about how to determine the critical exponents of

some complex Monge-Ampere equations arising from Kahler-Einstein problem,

both in the global and local settings. Blow-up behaviors will also be

discussed.

Where: MATH 3206

Speaker: Luca F. Di Cerbo (Duke)

http://fds.duke.edu/db/aas/math/luca

Abstract: In this talk, I present a new approach to the study of cusped complex

hyperbolic manifolds through their compactifications. Among other things, I

give effective bounds on the number of complex hyperbolic manifolds with given

upper bound on the volume. Moreover, I estimate the number of cuspidal ends of

such manifolds in terms of their volume. Finally, I address the classification

problem for cusped complex hyperbolic surfaces with minimal volume. This is the

noncompact or logarithmic analogue of the well known classification problem for

fake projective planes.

Where: Math 3206

Speaker: Ronan J. Conlon

Abstract: Asymptotically Conical (AC) Calabi-Yau manifolds are Ricci-flat

Kahler manifolds that resemble a Ricci-flat Kahler cone at infinity. The

first result I will present is a refinement of an existence theorem of

Tian and Yau from the early '90's for these manifolds.

In more recent years, new examples of "irregular" Calabi-Yau cones have

been discovered by mathematicians and physicists alike. I will also

present the first example of an AC Calabi-Yau metric on a smoothing of

such a Calabi-Yau cone.

This is joint work with Hans-Joachim Hein (Nantes).

Where: JHU Krieger Hall 300

speaker: Jacob Sturm

Abstract: We shall discuss the proof of partial C^0 estimates for Kahler-Ricci solitons with bounded Futaki invariant, generalizing the recent work of Donaldson-Sun in the setting of Fano Kahler-Einstein manifolds. In particular, any sequence of bounded solitons has a convergent subsequence in the Gromov-Hausdorff topology to a Kahler-Ricci soliton on a Q-Fano variety with log terminal singularities. This is joint work with D.H. Phong and Jian Song.

Where: B0425 (basement)

Speaker: Zbigniew Blocki (Jagiellonian)

http://gamma.im.uj.edu.pl/~blocki/

Abstract: We discuss some applications of pluripotential

theory for the Bergman kernel and metric. In particular,

the pluricomplex Green function can be used to prove

Bergman completeness of a large class of domains

(much bigger that had been known before). It is also

applied to estimate the Bergman kernel from below,

and recently very accurate estimates of this kind

have been obtained. The main tools are the dbar-operator

and the Monge-Ampere equation. Most of the results

are new and nontrivial already in dimension one.

For convex domains such a lower bound can be used to

simplify Nazarov's complex analytic approach to the

Bourgain-Milman inequality in convex analysis.

Where: Math 3206

Speaker: Jesus Martinez-Garcia (JHU) -

Abstract: The existence of a Kahler-Einstein metric on a Fano variety is equivalent to the algebro-geometric concept of K-stability. However K-stability is very difficult to test. For those Fano varieties which are not K-stable, we can define a singular Kahler-Einstein metric known as Kahler-Einstein metric with edge singularities, depending on a parameter \beta\in (0,1]. These metrics also have a reformulation in terms of log K-stability. It is well known that a smooth del Pezzo surface admits a Kahler-Einstein metric if and only if it is not the blow-up of P^2 in one or two points. However they always admit a Kahler-Einstein edge metric. In this talk, after introducing all these topics, I explain how we can use birational geometry and log canonical thresholds to find Kahler-Einstein edge metrics on all del Pezzo surfaces. This is joint work with Ivan Cheltsov.

Where: JHU Krieger Hall 300

Speaker: Dror Varolin ((Stony Brook) ) -

Where: UMD Math 3206

Speaker: Darvas Tamas (Purdue)

Abstract: Given a compact Kahler

manifold, according to Mabuchi, the set of Kahler forms in a fixed

cohomology class has the natural structure of an in finite dimensional

Riemannian manifold. This space is not geodesically convex, as Kahler

forms along a geodesic segment connecting two arbitrary points are

typically not smooth. We address this issue by extending the Mabuchi

metric structure to Kahler forms with less regularity. In the resulting

metric space every two points can be joined by an honest geodesic, what is

more, this space will be non-positively curved in the sense of Alexandrov.

Where: JHU Ames 218

Speaker: Zhiqin Lu

http://www.math.uci.edu/~zlu/

Abstract: Let (M,L) be a polarized Kaehler manifold. We prove the

folklore theorem that the Bergman kernel of M is equal to the Poincare

series of the L^2 Bergman kernel on its universal cover. We apply this

result to give a simple proof of a theorem of Napier and the surjectivity

of the Poincare series. On a more general setting, I shall also discuss

the Agmon-type estimate when the injectivity radius of the manifold is

sufficiently small. This is the joint work with Steven Zelditch.

Where: Ames 218

Speaker: Ludmil Katzarkov, Univ. of Miami

Abstract:

Homological Mirror symmetry is a categorical correspondence originating

from string theory. In mathematics it is known as a way of counting holomorphic

curves. We will explain some less known connections of Homological

Mirror Symmetry with birational geometry and theory of algebraic cycles.

Where: Ames 218

Speaker: Steve Zelditch, Northwestern University,

http://mathnt.mat.jhu.edu/zelditch/

Abstract: Nodal (zero) sets of eigenfunctions are important in physics

and are also analogues of real algebraic varieties on Riemannian

manifolds. This analogy is close if the metric is real analytic, and I

will present results on the complex geometry of nodal sets. When the

metric is only smooth, much less is known. I will present recent results

with J. Jung proving that the number of nodal domains tends to infinity

with the eigenvalue if the geodesic flow is ergodic.