Informal Geometric Analysis Archives for Fall 2023 to Spring 2024
Pluripotential solutions to the Complex Monge-Ampère flows
When: Tue, September 20, 2022 - 3:30pmWhere: Kirwan Hall 1313
Speaker: Prakhar Gupta (UMD) -
Abstract: In a series of papers, Guedj-Lu-Zeriahi and Dang have laid the foundation for the pluripotential weak solutions to the complex Monge-Ampère flows. They used it to describe weak solutions to Kähler-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-Ampère equations described by Bedford-Taylor and many others. In this talk, I'll explain how to define the pluripotential weak solutions to the Kähler-Ricci flow and show that weak solutions exist even when smooth solutions do not.
Existence of Kahler Einstein metrics in big classes
When: Tue, September 27, 2022 - 3:30pmWhere: Kirwan Hall 1313
Speaker: Tamas Darvas (University of Maryland) - http://math.umd.edu/~tdarvas
Abstract: We prove existence of twisted Kähler-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 Kähler-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.
Shrinking Kahler-Ricci solitons
When: Tue, November 8, 2022 - 3:30pmWhere: Kirwan Hall 1313
Speaker: Ronan Conlon (UT Dallas) -
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.
Families of degenerate complex Monge-Ampère equations
When: Tue, November 22, 2022 - 3:30pmWhere: https://umd.zoom.us/j/94631653106?pwd=b3kya3ZYRUl4ZUJOcmhVbC9ENHhSUT09
Speaker: Chung-Ming Pan (Toulouse) -
Abstract: This talk aims to explain a uniform $L^\infty$ estimate of
complex Monge-Ampère equations on families of singular hermitian
varieties. I will first overview some generalized results of Yau’s
celebrated solution to Calabi’s conjecture. In a general framework, I
will introduce some pluripotential results to establish an $L^\infty$
estimate which is one of the most challenging parts of the proof. Then
we shall focus on the family setting and chase the important dependence
on varieties.
Uniformly valuative stability of polarized varieties and applications
When: Thu, December 8, 2022 - 10:00amWhere: https://umd.zoom.us/j/4131293393
Speaker: Yaxiong Liu (Tsinghua University) -
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.
Brunn Minkowski inequalities for path spaces on Riemannian surfaces
When: Tue, February 7, 2023 - 3:30pmWhere: https://umd.zoom.us/j/8467540632?pwd=VHFKbFJiVjBhSFRsQ3lRazNhbVUxZz09 passcode:590899
Speaker: Rotem Assouline (Weizmann Insititute (Israel)) -
Abstract: The Minkowski average of two sets on a Riemannian manifold can be defined by replacing straight lines with geodesics. The Brunn Minkowski inequality is then equivalent to nonnegative Ricci curvature. We propose a generalization of this operation in which geodesics are replaced by an arbitrary family of curves. We show that horocycles in the hyperbolic plane satisfy the Brunn Minkowski inequality, in stark contrast to geodesics. Our main tool is needle decomposition. Joint work with Bo'az Klartag.
Okounkov bodies and Chebyshev transform
When: Tue, February 14, 2023 - 3:30pmWhere: Kirwan Hall 1313
Speaker: Chenzi Jin (UMD) -
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.
The role of symmetry in Brunn-Minkowski type inequalities
When: Tue, February 21, 2023 - 3:30pmWhere: https://umd.zoom.us/j/8467540632?pwd=VHFKbFJiVjBhSFRsQ3lRazNhbVUxZz09
Speaker: Liran Rotem (Technion) -
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
The role of symmetry in Brunn-Minkowski type inequalities Transform
When: Tue, February 28, 2023 - 3:30pmWhere: https://umd.zoom.us/j/8467540632?pwd=VHFKbFJiVjBhSFRsQ3lRazNhbVUxZz09
Speaker: Liran Rotem (Technion) -
Abstract: The Brunn-Minkowski inequality, about the volume of the Minkowskisum 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.
Zoom link: https://umd.zoom.us/j/8467540632?pwd=VHFKbFJiVjBhSFRsQ3lRazNhbVUxZz09Meeting ID: 846 754 0632
Passcode: 590899
Metric geometry of Einstein 4-manifolds with special holonomy
When: Tue, March 28, 2023 - 3:30pmWhere: Kirwan Hall 1313
Speaker: Ruobing Zhang (Princeton) -
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.
BAA branes on the Hitchin moduli space from solutions to the extended Bogomolny equations
When: Mon, April 10, 2023 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Panagiotis Dimakis (Stanford) -
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.
Special metrics in almost-Hermitian geometry
When: Tue, April 11, 2023 - 4:30pmWhere: Kirwan Hall 1313
Speaker: Mehdi Lejmi (CUNY) -
Abstract: Thanks to the work of Gauduchon and Ivanov, it is known that the only 4-dimensional compact Hermitian non-Kähler 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
Solution of a Ricci pinching conjecture in three dimensions
When: Tue, May 9, 2023 - 3:30pmWhere: Kirwan Hall 1313
Speaker: John Lott (UC Berkeley) -
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.