Informal Geometric Analysis Seminar (2024.09-2025.06)

The seminar aims to be a relaxed forum for researchers and graduate students interested in geometric analysis, and students of all levels are encouraged to attend and ask questions. Talks will focus on the "big picture" and key ideas and encourage interaction. Talks will be in the format: 30-minute talks followed by a ~ 15-30 minute discussion.

2024 Fall

• September 10, joint with Hopkins-Maryland geometry seminar (special time and place: 4.40pm--5.40pm, MATH 3206)

• Speaker: Zbigniew Blocki (Jagiellonian University)

• Title: \bar\partial and ODEs

• Abstract: Hörmander's L^2-estimate for \bar\partial and its generalizations can be used to construct holomorphic functions and prove various quantitative results in several complex variables. We will present some known and new \bar\partial-estimates and their relations with ODEs.

• September 17

• Speaker: Yueqing Feng (Berkeley)

• Title: A gluing construction of constant scalar curvature Kähler metrics of Poincaré type

• Abstract: In this talk, we construct new examples of constant scalar curvature Kähler(cscK) metrics of Poincaré type from existing cscK ones. The construction is obtained via gluing a cscK metric on a compact Kähler manifold to a complete scalar-flat Kähler metric of Poincaré type on $\mathbb{C}^n$ removing the origin. Assuming the compact Kähler manifold has no non-trivial holomorphic vector field, we prove the existence of cscK metrics of Poincaré type on this compact manifold removing finitely many points.

• October 10

• Speaker: Rotem Assouline (Weizmann)

• Title: Magnetic Brunn-Minkowski and Optimal Transport

• Abstract: We study the Brunn-Minkowski and Borell-Brascamp-Lieb inequalities on a Riemannian manifold endowed with an exact magnetic field, replacing geodesics by minimizers of an action functional given by the length minus the integral of the magnetic potential. We also review earlier work, joint with Klartag, in the special case of the hyperbolic plane.

• October 22

• Speaker: Junsheng Zhang (SLMath)

• Title: Complete Calabi--Yau metrics and singular Kähler--Einstein metrics asymptotic to cones

• Abstract: We relate the geometry of complete Calabi--Yau metrics and singular Kähler--Einstein metrics to their tangent cones under certain assumption on the curvature. Building on Donaldson--Sun’s 2-step degeneration theory, we proved an asymmetric phenomenon between the global and local setting. Part of the talk is based on joint work with Song Sun.

• October 29

• Speaker: Rolf Andreasson (Chalmers)

• Title: Arithmetic Fano Varieties and K-stability

• Abstract: This talk will give an overview of joint work with R. Berman that gives sharp bounds on the height of K-semistable (arithmetic) Fano varieties

• November 7

• Speaker: Rolf Andreasson (Chalmers)

• Title: SYZ Conjecture and real Monge-Ampere equations

• Abstract: Given a reflexive polytope with a height function, we prove a necessary and sufficient condition for solvability of the associated Monge-Ampère equation. We also improve on existing results regarding the SYZ conjecture for the Fermat family by showing regularity of the limiting potential. Joint work with J. Hultgren.

• November 19 (10-11am, not usual time)

• Speaker: Graham Andrew Smith (PUC-Rio)

• Title: Maximal submanifolds of pseudohyperbolic space

• Abstract: In joint work with A. Seppi and J. Toulouse we solve the Plateau problem for complete, maximal spacelike submanifolds of pseudohyperbolic space $\Bbb{H}^{p,q}$, with interesting applications to the study of Anosov representations in $\text{O}(p,q+1)$. In this talk, I will discuss some of the analytic aspects of our construction..

• November 19, 4pm in room 1310

• Speaker: Zbigniew Blocki (Jagiellonian University)

• Title: The Diederich-Fornaess Exponent and the Worm Domain

• Abstract: By a classical result of Diedrich and Fornaess for a bounded pseudoconvex domain in C^n with smooth boundary one can find a smooth defining function \rho and b>0 such that -(-\rho)^b is plurisubharmonic. Such a b is called a Diederich-Fornaess exponent. We will give a quantitative version of this result and also present the situation for the worm domains, generalizing a result of B. Liu.

• November 21, 3.15pm, joint with Complex Geometry & Algebraic Geometry Seminar in room 1310.

• Speaker: Ziquan Zhuang (JHU)

• Title: Boundedness of singularities and discreteness of local volumes

• Abstract: The local volume of a Kawamata log terminal (klt) singularity is an invariant that plays a central role in the local theory of K-stability. By the stable degeneration theorem, every klt singularity has a volume preserving degeneration to a K-semistable Fano cone singularity. I will talk about a joint work with Chenyang Xu on the boundedness of Fano cone singularities when the volume is bounded away from zero. This implies that local volumes only accumulate around zero in any given dimension.

• November 26, Thanksgiving break, no talk.

• December 10

• Speaker: Paolo Piazza

• Title: TBA

• Abstract: TBA

2025 Spring