Informal Geometric Analysis Seminar

Academic Year 2018-19

Organizers: G. Liu, V. Tosatti, B. Weinkove, S. Zelditch

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The seminar met on Thursdays from 3.00pm to 4.00pm in Lunt 107.

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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 Monge-Ampère equations - Reference

• October 18
  Steve Zelditch - A relativistic Duistermaat-Guillemin Trace formula - Abstract

  Two cornerstones of spectral theory of Riemannian manifolds are Weyl's law and the Duistermaat-Guillemin trace formula. They are manifestly non-relativistic. In this talk, I give generalizations to stationary globally hyperbolic spacetimes. Joint work with Alex Strohmaier.

• October 25
  Sisi Shen - Kähler-Ricci 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 four-manifolds. We shall show the manifold is diffeomorphic to the complex projective space if the conformal invariant is pinched sufficiently closed to that of the Fubini-Study metric. This is a joint work with Alice Chang and Matthew Gursky.

• November 8
  Emmett Wyman - Low-frequency generalized period integrals of eigenfunctions - Reference

• November 15
  Nick McCleerey - A Liouville theorem for the complex Monge-Ampère 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 - Herman-Kluk Semiclassical Approximation

• January 31
  Sisi Shen - Kähler-Ricci flow on blowups along submanifolds - Reference

• February 7
  Mitchell Faulk (Columbia) - Calabi-Yau 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 scalar-valued 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 over-ambitious 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. Bourgain-Rudnick 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 L² 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 Dyatlov-Jin 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 Monge-Ampère equations - Reference

• April 18
  Sisi Shen - The continuity method for cscK metrics

• May 2
  Gang Liu - Gromov-Hausdorff limits of Kähler manifolds with Ricci curvature lower bound, II - Reference - Abstract

  We study noncollapsed Gromov-Hausdorff 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 Donaldson-Sun. We also give application to Calabi-Yau manifolds. This is joint work with Gabor Székelyhidi.

• May 9
  Greg Edwards (Notre Dame) - The Chern-Ricci flow on primitive Hopf surfaces - Abstract

  The Chern-Ricci flow is a flow of Hermitian metrics which generalizes the Kähler-Ricci flow to non-Kähler metrics. While solutions of the flow have been classified on some families non-Kähler surfaces, the Hopf surfaces provide a family of non-Kähler surfaces on which little is known about the Chern-Ricci flow. We use a construction of locally conformally Kähler metrics of Gauduchon-Ornea to study solutions of the Chern-Ricci 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 Gromov-Hausdorff limit is isometric to a round S¹. 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
  Man-Chun Lee (UBC) - Existence of Kähler Ricci flow via non-Kähler 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 Chern-Ricci flow which was introduced by Gill and Tosatti-Weinkove. These are joint works with L.F. Tam and A. Chau.

• May 23
  Nick McCleerey - Analytic test configurations and geodesic rays

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Schedule for the past years:

2012-13 - 2013-14 - 2014-15 - 2015-16 - 2016-17 - 2017-18