Geometry and Topology Seminar 2023 2024
Fall 2023
| date | speaker | title |
|---|---|---|
| Sep. 29 | Sean Paul | The Mahler Measure of the X-discriminant |
| Oct. 6 | Junsheng Zhang (Berkeley) | On complete Calabi-Yau manifolds asymptotic to cones |
| Oct. 13 | Richard Wentworth (Maryland) | Compactifications of Hitchin's moduli space |
| Oct. 20 | Gorapada Bera (Stony Brook) | Conically singular associatives in counting associative submanifolds |
| Oct. 27 | Siarhei Finski (École Polytechnique) | Asymptotic study of filtrations on section rings and geodesic rays of metrics |
| Nov. 3 | Liuwei Gong (Rutgers) | Conformal metrics of constant scalar curvature with unbounded volumes |
| Nov. 10 | Gayana Jayasinghe (UIUC) | An extension of the Lefschetz fixed point theorem |
| Nov. 17 | Sean Paul (UW) | The Mahler Measure of the classical X-discriminant II |
| Dec. 1 | Gavin Ball (UW) | The Morse index of quartic minimal hypersurfaces |
| Dec. 8 | Ilyas Khan (Duke) | Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G_2-Laplacian Flow |
Fall abstracts
Sean Paul (09/29/2023)
Let P be a homogeneous polynomial in several complex variables. The (logarithmic) Mahler measure of P is the integral of log|P| over the unit sphere with respect to the standard unitary invariant measure of the sphere. The Mahler measure is extraordinary difficult to compute, even for simple polynomials. This is the first of perhaps three talks devoted to outlining a strategy to compute the asymptotic behavior of the Mahler measure of the X-discriminant of a projective manifold of large degree.
Despite the completely elementary definition of the measure, the mathematics required to compute it turns out to be of surprising depth and technical complexity.
The talk(s) are designed so as to require very little background to appreciate.
Junsheng Zhang
We proved a ``no semistability at infinity" result for complete Calabi-Yau metrics asymptotic to cones, by eliminating the possible appearance of an intermediate K-semistable cone in the 2-step degeneration theory developed by Donaldson-Sun. As a consequence, a classification result for complete Calabi-Yau manifolds with Euclidean volume growth and quadratic curvature decay is given. Moreover a byproduct of the proof is a polynomial convergence rate to the asymptotic cone for such manifolds. Joint work with Song Sun.
Richard Wentworth
The moduli space of rank 2 Higgs bundles has a much studied very rich structure related to integrable systems, hyperkaehler reduction, mirror symmetry, and supersymmetric gauge theory. The space has several compactifications arising from the nonabelian Hodge theorem. In this talk, I will present specific results on two of them: one from the algebraic geometry of the C-star action, and another from the analytic "limiting configurations" of solutions to the Hitchin equations. I will discuss how the nonabelian Hodge correspondence extends as a map between these compactifications. Somewhat surprisingly, the extension is not continuous.
Gorapada Bera
In the spirit of counting holomorphic curves (or special Lagrangians) in Calabi-Yau 3-folds, there are proposals to define enumerative invariants of G_2-manifolds by counting closed associative submanifolds. Here, G_2-manifolds can be thought of as 7-dimensional analogues of Calabi-Yau 3-folds, where associative submanifolds are 3-dimensional analogues of holomorphic curves (or special Lagrangians). The naive counting does not lead to an invariant due to degenerations of smooth associatives into singular associatives, and raises the natural question of finding all possible singular associatives and their desingularisations. In this talk, after a brief introduction to this field, we will restrict ourselves to the simplest singular associative submanifolds, which are conically singular only at a finite number of points, and address the above questions. The answers to these questions thus contribute to the above proposals.
Siarhei Finski
For a complex projective manifold polarised by an ample line bundle, we study the asymptotic properties of submultiplicative filtrations on the associated section ring and show that these are related to the geometry at infinity of the space of Kähler metrics on the manifold. This establishes a certain metric relation between test configurations, filtrations and geodesic rays in the space of Kähler metrics.
Luiwei Gong
When n>24, Brendle and Marques constructed a smooth metric on S^n such that there exists a sequence of conformal metrics with the same positive constant scalar curvature but with unbounded Ricci curvatures. We prove a “worse” blowup phenomenon when n>24: a smooth metric on S^n such that there exists a sequence of conformal metrics with the same positive constant scalar curvature but with unbounded volumes (and, in particular, unbounded Ricci curvatures). This is a joint work with Yanyan Li.
Gayana Jayasinghe
Atiyah and Bott generalized the Lefschetz fixed point theorem to elliptic complexes on smooth manifolds, and its various incarnations now appear in many areas of mathematics and physics. I will describe a generalization of this theorem for Hilbert complexes associated to Dirac type operators on stratified pseudomanifolds, comparing the local and global formulas for some complexes as the domains of operators change, as well as with related results including the Lefschetz-Riemann-Roch formulas of Baum-Fulton-Quart on singular algebraic varieties. I will show how one can compute indices of spin-Dirac operators, self-dual and anti-self dual complexes and other important invariants in mathematics and physics. This is based on the work in https://arxiv.org/abs/2309.15845.
Sean Paul (11/17/2023)
The Mahler Measure of the classical X-discriminant II
In this talk we will make the connection between the Mahler measure of the X-discriminant and the work of Mathai-Quillen on the Thom form and J.M. Bismut's work on Quillen's super connection currents.
The talk will be accessible to graduate students.
Gavin Ball
Given a minimal hypersurface N in a compact Riemannian manifold, its Morse index is the number of variations of N that are area-decreasing to second order. In practice, computing the Morse index of a given minimal hypersurface is difficult. Indeed, even for the simplest case in which the ambient space is the round sphere and N is homogeneous, the Morse index of N is not known in general. In this talk, I will describe recent work (joint with Jesse Madnick and Uwe Semmelmann) where we compute the Morse index of two such minimal hypersurfaces. In this setting the Morse index is determined by the Laplace spectrum, and for these examples we are able to give an algorithm to determine the spectrum. Moreover, we find that in both of our examples, the spectra contain eigenvalues not expressible in radicals, a phenomenon not present in other examples.
Ilyas Khan
Riemannian 7-manifolds with holonomy equal to the exceptional Lie group G_2 are objects of great interest in diverse domains of mathematics and physics. One approach to understanding such manifolds is through natural flows of 3-forms called G_2-structures, the most prominent of which is Bryant's Laplacian flow. In general, Laplacian flow is expected to encounter finite-time singularities and, as in the case of other flows, self-similar solutions should play a major role in the analysis of these singularities. In this talk, we will discuss recent joint work with M. Haskins and A. Payne in which we prove the uniqueness of asymptotically conical gradient shrinking solitons of the Laplacian flow of closed G_2 structures. We will particularly emphasize the unique difficulties that arise in the setting of Laplacian flow (in contrast to the Ricci flow, where an analogous result due to Kotschwar and Wang is well-known) and how to overcome these difficulties.