Informal Geometric Analysis Archives for Fall 2024 to Spring 2025
A gluing construction of constant scalar curvature Kähler metrics of Poincaré type
When: Tue, September 17, 2024 - 3:30pmWhere: Kirwan Hall 1313
Speaker: Yueqing Feng ( University of California, Berkeley) - https://fyqing.github.io/yueqing.com/
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.
Complete Calabi-Yau metrics and singular K\”ahler-Einstein metrics asymptotic to cones
When: Tue, October 22, 2024 - 3:30pmWhere: Kirwan Hall 1313
Speaker: Junsheng Zhang (Simons Laufer Mathematical Sciences Institute) - https://zhang-junsheng.github.io/
Abstract: We relate the geometry of complete Calabi-Yau metrics and singular K\”ahler-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.
Maximal submanifolds of pseudohyperbolic space
When: Tue, November 19, 2024 - 10:00amWhere: Kirwan Hall 1313
Speaker: Graham Andrew Smith () -
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..
The Diederich-Fornaess Exponent and the Worm Domain
When: Tue, November 19, 2024 - 4:00pmWhere: Kirwan Hall 1310
Speaker: Zbigniew Blocki () -
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.
The signature operator on Witt spaces
When: Tue, December 10, 2024 - 3:30pmWhere: Kirwan Hall 1313
Speaker: Paolo Piazza (Rome "La Sapienza") - https://www1.mat.uniroma1.it/people/piazza/
An Approach to Dimension Reduction for Generalized Anti-Self-Dual Instantons
When: Tue, February 18, 2025 - 3:30pmWhere: MTH 1310
Speaker: Dylan Galt (Stony Brook) - https://www.math.stonybrook.edu/~dgalt/
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.
Probing moduli spaces with sub-line bundles
When: Tue, March 4, 2025 - 3:30pmWhere: Kirwan Hall 1310
Speaker: Duong Dihn (UPenn) -
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.
Products on Fixed point Floer cohomology and closed string mirror symmetry for nodal curves
When: Tue, April 15, 2025 - 3:30pmWhere: MTH 1310
Speaker: Yuan Yao (Nantes) -
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.
On the hole event for random complex zeros
When: Tue, May 6, 2025 - 10:00amWhere: https://umd.zoom.us/j/97463413907?pwd=vRRe4b3WPoKZ2QInm3q5bZKqgdef6x.1 Zoom: 974 6341 3907 password:404liu
Speaker: Aron Wennman (KU Leuven) -
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.