Informal Geometric Analysis Seminar
Academic Year 201819
Organizers: G. Liu, V. Tosatti, B. Weinkove, S. Zelditch
The seminar met on Thursdays from 3.00pm to 4.00pm in
Lunt 107.
Schedule

October 4
Bogdan Georgiev (MPIM)  Doubling and frequency for harmonic functions  Abstract
We present some classical as well as recent results describing doubling properties of harmonic functions and solutions to more general second order elliptic PDE. We also discuss some applications of such estimates to nodal geometry.

October 11
Jianchun Chu  Geometric estimates for complex MongeAmpère equations  Reference

October 18
Steve Zelditch  A relativistic DuistermaatGuillemin Trace formula  Abstract
Two cornerstones of spectral theory of Riemannian manifolds are Weyl's law and the DuistermaatGuillemin trace formula. They are manifestly nonrelativistic. In this talk, I give generalizations to stationary globally hyperbolic spacetimes. Joint work with Alex Strohmaier.

October 25
Sisi Shen  KählerRicci flow on manifolds with negative holomorphic sectional curvature  Reference

November 1
Siyi Zhang (Princeton)  A conformally invariant gap theorem characterizing complex projective space via the Ricci flow  Abstract
In this talk, we extend a sphere theorem proved by A. Chang, M. Gursky, and P. Yang to give a conformally invariant characterization of complex projective space.
In particular, we introduce a conformal invariant defined on closed fourmanifolds. We shall show the manifold is diffeomorphic to the complex projective space if the conformal invariant is pinched sufficiently closed to that of the FubiniStudy metric. This is a joint work with Alice Chang and Matthew Gursky.

November 8
Emmett Wyman  Lowfrequency generalized period integrals of eigenfunctions  Reference

November 15
Nick McCleerey  A Liouville theorem for the complex MongeAmpère equation  Reference

January 10
Oran Gannot  Black Holes

January 17
Perry Kleinhenz  Energy decay for the damped wave equation with Hölder continuous damping

January 24
Abe Rabinowitz  HermanKluk Semiclassical Approximation

January 31
Sisi Shen  KählerRicci flow on blowups along submanifolds  Reference

February 7
Mitchell Faulk (Columbia)  CalabiYau metrics on asymptotically conical manifolds with prescribed decay rates  Reference  Abstract
Yau's solution to Calabi's conjecture involves solving a complex Monge Ampere equation for a scalarvalued function on the manifold. A paper by Conlon and Hein states that in the case that the manifold is asymptotically conical, there still exist solutions to this equation, but the existence (and uniqueness) depends on the decay rate of the prescribed Ricci form appearing in the equation. In this talk, we discuss these existence results and focus on a small improvement with respect to the decay rate of solutions in the case that the Fredholm index of the Laplacian is the first negative value.

February 21
Eric Zaslow  Dimer models, quantum integrable systems, mirror symmetry, limit shapes and Mahler entropy  Abstract
In this sprawling and probably overambitious talk, I will state one theorem (because I'm supposed to) and then cite difficult work by other people to try demonstrate the interconnectivity of the topics of the title. The theorem is joint with David Treumann and Harold Williams.

February 28
Steve Zelditch  Nodal intersections and geometric control  Abstract
Suppose that a Riemannian surface has an isometric involution across an axis H (fixed point set), e.g. the equator of a sphere. Half of the eigenfunctions are even, half odd under the involution. The odd ones vanish on the axis, which is a union of geodesics. BourgainRudnick asked: is this the only kind of curve on which an infinite sequence of eigenfunctions can vanish? Toth and I posed a related problem: what if the L^{2} norms of the restrictions tend to zero. The main result: the only curves (or hypersurfaces, in higher dimensions) on which a positive proportion of eigenfunctions can vanish is 'very like' an axis of symmetry. The precise condition involves the dynamics of the geodesic flow. I will also explain how the proof combines with a recent result of DyatlovJin on the fractal uncertainty principle and control theory to replace 'positive proportion' by 'infinite sequence'.

March 7
Nick McCleerey  Lower bounds for the Calabi functional  References 1  2

April 4
Jianchun Chu  Stability of solutions of complex MongeAmpère equations  Reference

April 18
Sisi Shen  The continuity method for cscK metrics

May 2
Gang Liu  GromovHausdorff limits of Kähler manifolds with Ricci curvature lower bound, II  Reference  Abstract
We study noncollapsed GromovHausdorff limits of Kähler manifolds (not necessarily polarized) with Ricci curvature lower bound. Our main result states that any tangent cone is homeomorphic to a normal affine variety. This generalizes a result of DonaldsonSun. We also give application to CalabiYau manifolds. This is joint work with Gabor Székelyhidi.

May 9
Greg Edwards (Notre Dame)  The ChernRicci flow on primitive Hopf surfaces  Abstract
The ChernRicci flow is a flow of Hermitian metrics which generalizes the KählerRicci flow to nonKähler metrics. While solutions of the flow have been classified on some families nonKähler surfaces, the Hopf surfaces provide a family of nonKähler surfaces on which little is known about the ChernRicci flow. We use a construction of locally conformally Kähler metrics of GauduchonOrnea to study solutions of the ChernRicci flow on primitive Hopf surfaces of class 1. These solutions reach a volume collapsing singularity in finite time, and we show that the metric tensor satisfies a uniform upper bound, supporting the conjecture that the GromovHausdorff limit is isometric to a round S^{1}. Uniform C^{1+β} estimates are also established for the potential. Previous results had only been known for the simplest examples of Hopf surfaces.

May 16
ManChun Lee (UBC)  Existence of Kähler Ricci flow via nonKähler approximation  Abstract
In this talk, we will discuss the existence theory of Kähler Ricci flow when the initial metric is complete noncompact with possibly unbounded curvature. We will discuss some new extension of classical Shi's Kähler Ricci flow existence theory using the ChernRicci flow which was introduced by Gill and TosattiWeinkove. These are joint works with L.F. Tam and A. Chau.

May 23
Nick McCleerey  Analytic test configurations and geodesic rays
Schedule for the past years:
201213 
201314 
201415 
201516 
201617 
201718