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 an informal discussion led by the audience. « For more on this format, including tips for preparing a talk.»

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 (MTH 1310)

• Speaker: Paolo Piazza

• Title: The signature operator on Witt spaces

• Abstract: A Witt space is a particular kind of pseudomanifold. A singular complex projective variety is an example of Witt space; the space obtained as the quotient of a non-free actions of a compact Lie group on a smooth closed compact manifold is also a Witt space under certain additional assumptions on the action. Witt spaces enjoy extremely interesting properties, both from the point of view of topology and analysis. In this talk I will explain mainly the analytic side, concentrating on a very significant example: the signature operator. talk is based on joint work (scattered in several papers) with Pierre Albin, Markus Banagl, Eric Leichtnam, Rafe Mazzeo and Boris Vertman.

2025 Spring

• February 18

• Speaker: Dylan Galt (Stony Brook)

• Title: An Approach to Dimension Reduction for Generalized Anti-Self-Dual Instantons

• Abstract: In this talk, I will describe joint work with Langte Ma studying dimension reduction phenomena for absolute minimizers of the Yang-Mills functional. This work is motivated by special holonomy geometry and I will emphasize applications to gauge theory on special holonomy manifolds. I will explain the general approach to such phenomena that we develop, characterizing the moduli space of generalized anti-self-dual instantons on certain bundles over product Riemannian manifolds equipped with a parallel codimension-4 differential form. One outcome of this is an explicit description of instanton moduli spaces over certain product special holonomy manifolds.

• March 4

• Speaker: Duong Dinh (UPenn)

• Title: Probing moduli spaces with sub-line bundles

• Abstract: The moduli spaces of Higgs bundles and flat connections on a Riemann surface play important roles in several parts of mathematics and mathematical physics. I will explain how considering Higgs bundles/flat connections together with sub-line bundles of the underlying bundles is useful in yielding explicit descriptions of these moduli spaces. Curiously, this method is inspired by the Separation of Variables method for integrable systems on one hand and related to the extended Bogomolny equations on the other hand. As a result, for the GL_n and SL_n cases, we can construct certain Lagrangians in the moduli of Higgs bundles that are, in retrospect, natural from this point of view. Furthermore, for the rank-2 case, we can also explicitly describe components in the wobbly locus in the moduli of stable bundles.

• March 13 (at 2:00 PM room 1310)

• Speaker: Vlassis Mastrantonis and Zbigniew Blocki

• Title: From Bergman kernels to polarity: Perspectives on the Mahler, Bourgain, and Blocki conjectures

• Abstract: TBA

• March 14

• Speaker: Ritvik Vintipalli

• Title: Strong and weak positivity on super-forms

• Abstract: TBA

• March 31 (at 3:30pm room 1310)

• Speaker: Abdullah Ahmed and Vlassis Mastrantonis

• Title: Mixed volume and surface measures for convex bodies via superforms

• Abstract: TBA

• March 25, joint with JHU-UMD Complex Geometry Seminar (special time and place: 4.30pm--5.30pm, Maryland 110 at JHU)

• Speaker: Jeff Viaclovsky (UCI)

• Title: Fibrations on the 6-sphere and Clemens threefolds

• Abstract: Let Z be a compact, connected 3-dimensional complex manifold with vanishing first and second Betti numbers and non-vanishing Euler characteristic. We prove that there is no holomorphic mapping from Z onto any 2-dimensional complex space. In other words, Z can only possibly fiber over a curve. This result applies in particular to a class of threefolds, known as Clemens threefolds, which are diffeomorphic to a connected sum of k copies of S³ × S³ for k > 1. This result also gives a new restriction on any hypothetical complex structure on the 6-sphere S⁶. This is joint work with Nobuhiro Honda.

• April 15

• Speaker: Yuan Yao (Nantes)

• Title: Products on Fixed point Floer cohomology and closed string mirror symmetry for nodal curves

• Abstract: The fixed point Floer cohomology is a cochain complex generated by the fixed points of a symplectomorphism. It has been computed for all symplectomorphisms on (2 real-dimensional) surfaces, but it enjoys additional algebraic structure coming from a pair of pants "product". We explain the computation of this "product" in the case of iterations of a Dehn twists on a surface. We find surprisingly, the resulting structure forms a polynomial ring. Via the lens of mirror symmetry, we relate this ring to sections of tensor powers of a degree 1 line bundle on a nodal elliptic curve. Finally, using the product structure, we explain how to define a version of genus 0 Gromov Witten invariants for nodal Riemann surfaces and compute it. All of this joint work with Maxim Jeffs and Ziwen Zhao.

• May 2 (at 11:00am room MTH3206)

• Speaker: Vasanth Pidaparthy

• Title: Valuations on convex bodies and superforms

• Abstract: TBA

• May 5 (at 11:00am room MTH1310)

• Speaker: Ritvik Vantipalli

• Title: Characterization of strong and strongly homogeneous superforms I

• Abstract: TBA

• May 6 (Special time:10am, zoom link: https://umd.zoom.us/j/97463413907?pwd=vRRe4b3WPoKZ2QInm3q5bZKqgdef6x.1 zoom number:974 6341 3907; password: 404liu .)

• Speaker: Aron Wennman (KU Leuven)

• Title: On the hole event for random complex zeros

• Abstract: In this talk, based on joint work with Alon Nishry (Tel Aviv University) I will describe some curious phenomena related to rare events for random analytic functions, more precisely the Gaussian Entire Function (GEF). The GEF is a random Taylor series with i.i.d. centered Gaussian coefficients, whose variances are chosen so that the zero set of the GEF has translation invariant law. This zero set satisfies a number of interesting properties. For instance, the fluctuations of the number of zeros in a large ball behaves like that of a perturbed lattice (the zeros form a hyperuniform point process).

  We consider the following questions: Fix a region G in the plane of unit area, and let R be a large parameter. Rescaling the GEF to have zero intensity R, we would expect around R^2 zeros in G. Asymptotically as R tends to infinity, what is the probability of the rare event that there are no zeros in G (the “hole probability”)? And conditioned on this “hole event”, what is the limiting distribution of the remaining zeros, and how does it depend on the shape of G?

  Ghosh and Nishry (CPAM ’19) discovered that there emerges a forbidden region outside the hole G, where the zero density vanishes in the limit. In joint work with Alon Nishry, we studied the relationship between the geometry of the hole and that of the forbidden region. I plan to describe our results, including some striking rigidity properties of the forbidden region, and a curious connection to quadrature domains from potential theory.