Informal Geometric Analysis Seminar
University of Maryland

• September 20
  Prakhar Gupta (UMD)
  Title: Pluripotential solutions to the Complex Monge-Ampere flows
  Abstract: In a series of papers, Guedj-Lu-Zeriahi and Dang have laid the foundation for the pluripotential weak solutions to the complex Monge-Ampere flows. They used it to describe weak solutions to Kahler-Ricci flow where the classical smooth solutions do not exist. This work, thus, is the parabolic analog to the very fruitful elliptic theory of the pluripotential weak solution to the complex Monge-Ampere equations described by Bedford-Taylor and many others. In this talk, I'll explain how to define the pluripotential weak solutions to the Kahler-Ricci flow and show that weak solutions exist even when smooth solutions do not.


• September 22, 12:30 PM, MTH0302
  Jared Marx-Kuo (Stanford)
  Title:Variations of Renormalized Volume for Minimal Submanifolds of Poincare-Einstein Spaces
  Abstract: In this talk, we discuss the regularity and renormalized volume of minimal submanifolds, Y, of arbitrary codimension in Poincare-Einstein manifolds, M. Using geometric and microlocal techniques, we derive polyhomogeneous expansions for the minimal submanifold and variations along it. We then present formulae for the first and second variations of renormalized volume. We end with an orthogonality relationship between coefficients in the expansion of Y, for the specific case of M = H^{n+1}. In all theorems, we emphasize the Dirichlet-to-Neumann type map and its presence in formulae


• September 27
  Tamas Darvas (UMD)
  Title:Existence of KE metrics in big classes
  Abstract: We prove existence of twisted Kahler-Einstein metrics in big cohomology classes, using a divisorial stability condition. In particular, when -K_X is big, we obtain a uniform Yau-Tian-Donaldson existence theorem for Kahler-Einstein metrics. To achieve this, we build up from scratch the theory of Fujita-Odaka type delta invariants in the transcendental big setting, using pluripotential theory. This is joint work with Kewei Zhang.


• November 8
  Ronan Conlon (UT Dallas)
  Title: Shrinking Kahler-Ricci solitons
  Abstract: Shrinking Kahler-Ricci solitons model finite-time singularities of the Kahler-Ricci flow, hence the need for their classification. I will talk about the classification of such solitons in 4 real dimensions. This is joint work with Bamler-Cifarelli-Deruelle, Cifarelli-Deruelle, and Deruelle-Sun.


• November 22
  Chung-Ming Pan (Toulouse)
  Title: TBD
  Abstract: TBD


• December 8 (10 AM, over zoom)
  Yaxiong Liu (Tsinghua University)
  Title: Uniformly valuative stability of polarized varieties and applications
  Abstract: In the study of K-stability, Fujita and Li proposed the valuative criterion of K-stability on Fano varieties, which has played an essential role of the algebraic theory of K-stability. Recently, Dervan-Legendre considered the valuative criterion of polarized varieties, which is a generalization of Fujita-Li criterion on Fano varieties. We will show that valuative stability is an open condition. We would like to study the valuative criterion for the Donaldson's J-equation. Motivated by the beta-invariant of Dervan-Legendre, we introduce a notion, the so-called valuative J-stability and prove that J-stability implies valuative J-stability. If time permits, we show the upper bound of the volume of K-semistable polarized toric varieties as an application of valuative stability.


• February 7
  Rotem Assouline (Weizmann Institute)
  Title: Brunn-Minkowski for path spaces on Riemannian surfaces
  Abstract: We study the Brunn-Minkowski inequality on a Riemannian surface, where in the definition of the Minkowski sum, geodesics are replaced by an arbitrary family of constant-speed curves. We give a necessary and sufficient condition for such a family to satisfy the same Brunn-Minkowski inequality as the family of lines in the plane. Horocycles in the hyperbolic plane provide an example. Joint work with Bo'az Klartag.


• February 14, CANCELLED
  Chenzi Jin (UMD)
  Title: Okounkov bodies and Chebyshev transform
  Abstract: The Okounkov body is a convex body associated to a big line bundle on a projective variety; the Chebyshev transform is a convex function on the Okounkov body associated to a metric on the line bundle. In this talk, I will present their definitions and simple examples, as well as some basic properties - how they encode information about the line bundle and the metric, etc.


• February 21
  Liran Rotem (Technion)
  Title: The role of symmetry in Brunn-Minkowski type inequalities
  Abstract: The Brunn-Minkowski inequality, about the volume of the Minkowski sum of sets, is one of the cornerstones of convex geometry. Since the works of Borell in the 1970s, we know an exact characterization of all measures that satisfy a Brunn-Minkowski type inequality. It recently became clear that when the sets involved are convex and origin-symmetric, one can expect better inequalities than the ones guaranteed by Borell's theorem. Examples of this phenomenon are the proof of the B-conjecture for the Gaussian measure by Cordero-Erausquin, Fradelizi and Maurey, and the much more recent proof of the so-called Dimensional Gaussian Brunn-Minkowski conjecture by Eskenazis and Moschidis. In the non-Gaussian case much less is known, and we do not even have a good conjecture for a characterization theorem similar to Borell's. In this talk I will survey results in this direction, and in particular my contributions which are joint with D. Cordero-Erausquin. We will focus on the role of symmetry in such theorems and on open problems in the field.


• February 28,
  Chenzi Jin (UMD)
  Title: Okounkov bodies and Chebyshev transform
  Abstract: The Okounkov body is a convex body associated to a big line bundle on a projective variety; the Chebyshev transform is a convex function on the Okounkov body associated to a metric on the line bundle. In this talk, I will present their definitions and simple examples, as well as some basic properties - how they encode information about the line bundle and the metric, etc.


• March 9-10 (SPECIAL TIME, DLGA days)
  Sergiu Kleinerman (Princeton)
  

  LINK



• March 14 MTH3206 (joint with Numerical Analysis Seminar),
  Guillaume Bonnet (University of Maryland College Park)
  Title: Monotone discretization of the Monge-Ampere equation of optimal transport


• March 21, Spring Break
  


• March 28
  Ruobing Zhang (Princeton)
  Title: Metric geometry of Einstein 4-manifolds with special holonomy
  Abstract: This talk focuses on the recent resolutions of several well-known conjectures in studying the Einstein 4-manifolds with special holonomy. The main results include the following. (1) Any volume collapsed limit of unit-diameter Einstein metrics on the K3 manifold is isometric to one of the following: the quotient of a flat 3D torus by an involution, a singular special Kaehler metric on the topological 2-sphere, or the unit interval. (2) Any complete non-compact hyperkaehler 4-manifold with quadratically integrable curvature, namely gravitational instanton, must have an ALX model geometry with optimal asymptotic rate. (3) Any gravitational instanton is biholomorphic to a dense open subset of some compact algebraic surface.


• April 4, Tamas is away
  


• April 10, Joint with G&T seminar, Special place and time: MATH 3206, 3 PM
  Panos Dimakis (Stanford)
  Title: BAA branes on the Hitchin moduli space
  Abstract: BAA branes are complex Lagrangian submanifolds of the Hitchin space. Recently, there has been interest in these objects due to their appearance in mirror symmetry conjectures and due to their intimate connection with the geometry of the Hitchin space. In this talk I will introduce the above notions. Then I will introduce the extended Bogomolny equations and explain how their solutions lead to holomorphic data associated with a Riemann surface. I will show that the moduli of these holomorphic data is a BAA brane. Some of the BAA branes obtained this way are known but some are new.


• April 11, 4:30 PM (SPECIAL TIME)
  Mehdi Lejmi (CUNY)
  Title: Special metrics in almost-Hermitian geometry
  Abstract: Thanks to the work of Gauduchon and Ivanov, it is known that the only 4-dimensional compact Hermitian non-Kahler second-Chern-Einstein manifold is the Hopf surface. In this talk, we investigate the existence of such metrics in the almost-Hermitian setting. We also discuss the problem of existence of almost-Hermitian metrics with constant Hermitian scalar curvature


• May 9
  John Lott (UC Berkeley)
  Title: Solution of a Ricci pinching conjecture in three dimensions
  Abstract: The conjecture said that a complete Riemannian 3-manifold with pointwise pinched nonnegative Ricci curvature is compact or flat. It has been proved through the efforts of myself, Deruelle-Schulze-Simon and Lee-Topping. I'll describe the background to the conjecture and the main ideas of the proof, which uses Ricci flow.



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