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.

Spring 2024

date	speaker	title
Feb. 2	Alex Waldron	Łojasiewicz inequalities for maps of the 2-sphere
Feb. 9	Nianzi Li	Metric asymptotics on the irregular Hitchin moduli space
Feb. 16	Bing Wang (USTC)	On Kähler Ricci shrinker surfaces
Mar. 1	Hao Shen	Stochastic Yang-Mills flow
Mar. 8	Tristan Ozuch (MIT)	Instabilities of Einstein 4-metrics and selfduality along Ricci flow
Mar. 22	Max Stolarski (Warwick)	Singularities of Mean Curvature Flows with Mean Curvature Bounds
Apr. 1-5	Siarhei Finski (École Polytechnique)	Mini-course: Local version of the Riemann-Roch-Grothendieck Theorem (Time & Location below)
Apr. 12	Daniel Platt (King's college)	New examples of Spin(7)-instantons

Spring abstracts

Alex Waldron

Infinite-time convergence of geometric flows, as even for finite-dimensional gradient flows, is a notoriously subtle problem. The best (or only) bet is to get a ``Łojasiewicz(-Simon) inequality'' stating that a power of the gradient dominates the distance to the critical energy value. I'll discuss the recent proof of a Łojasiewicz inequality between the tension field and the Dirichlet energy of a map from the 2-sphere to itself, removing virtually all assumptions from an estimate of Topping (Annals '04). This gives us convergence of weak solutions of harmonic map flow from S^2 to S^2 assuming only that the body map is nonconstant.

Nianzi Li

For gauge-theoretic moduli spaces, the compactification and analysis of natural metrics are intriguing and challenging problems. In this talk, we consider the moduli space of rank-two Higgs bundles with irregular singularities over the projective line. Along a generic curve, we prove that Hitchin's hyperkähler metric is asymptotic to a simpler semi-flat metric at an arbitrary polynomial rate, based on the foundational works of Fredrickson, Mazzeo, Swoboda, Weiss, and Witt.  In our gluing construction of the harmonic metric, we introduce a new building block around a weakly parabolic singularity. In dimension four, we explicitly compute the asymptotic limit of the semi-flat metric, which is of type ALG or ALG*. Joint work with Gao Chen.

Bing Wang

We prove that any Kähler Ricci shrinker surface has bounded sectional curvature. Combining this estimate with earlier work by many authors, we provide a complete classification of all Kähler Ricci shrinker surfaces. This is joint work with Yu Li.

Hao Shen

We will discuss the stochastic Yang-Mills flow, which is the deterministic Yang-Mills flow driven by a (very singular) space-time white noise. It turns out that due to singularity, even construction of local solutions is challenging. We will discuss our construction for a trivial bundle over 2 and 3 dimensional tori, but starting with a gentle introduction to Stochastic PDE. In the end, I will also discuss the meaning of "gauge equivalence” and “orbit space" in the singular setting, and show that the flow has the gauge covariance property (in the sense of probability law), yielding a Markov process on the orbit space. Based on joint work with Ajay Chandra, Ilya Chevyrev and Martin Hairer.

Tristan Ozuch

Einstein metrics and Ricci solitons are the fixed points of Ricci flow and model the singularities forming. They are also critical points of natural functionals in physics. Their stability in both contexts is a crucial question, since one should be able to perturb away from unstable models.

I will present new results and upcoming directions about the stability of these metrics in dimension four in joint work with Olivier Biquard. The proofs rely on selfduality, a specificity of dimension four.

Max Stolarski

A hypersurface evolving by mean curvature flow generally encounters singularities in finite time. At such singularities, the second fundamental form of the hypersurface always blows up, but its trace, the mean curvature, can remain bounded. After reviewing examples of this pathological singularity formation, we demonstrate how to incorporate the theory of varifolds with bounded mean curvature to study the general structure of singularities for mean curvature flows with uniform mean curvature bounds. In particular, we show tangent flows are necessarily static flows of minimal cones, and the tangent flow is unique if the cone has smooth link.

Siarhei Finski

Monday-Thursday, 3-4:30pm, Birge 348

Friday, 1:10-2:10pm, Van Vleck 901 (as usual)

The main goal of this series of lectures is to present a curvature theorem of Bismut-Gillet-Soulé, which can be seen as a local version of the Riemann-Roch-Grothendieck theorem. Recall that for a proper holomorphic map between smooth quasi-projective manifolds, the Riemann-Roch-Grothendieck theorem gives a formula in the cohomology of the target manifold for the Chern characters of direct image sheaves in terms of the Chern and Todd classes of the fibration.  Bismut-Gillet-Soulé established that under some additional assumptions on the family, this statement holds on the level of differential forms.

More precisely, recall that Chern-Weil theory associates for any Hermitian vector bundle and a characteristic class a natural closed differential form, the de Rham cohomology of which coincides with the characteristic class of the vector bundle. The curvature formula states that one can construct a natural norm on the determinant of the direct images, called the Quillen norm, so that the Riemann-Roch-Grothendieck theorem holds pointwise for the differential forms constructed as the Chern-Weil representatives of both sides of the Riemann-Roch-Grothendieck theorem.

We will cover the basic elements of the proof of this result as well as the needed preliminaries including Hodge theory, Chern-Weil and Bott-Chern theories, Spectral theory of elliptic operators, Heat kernel asymptotics, local index theory and theory of superconnections.

Daniel Platt

Spin(7)-instantons are certain interesting principal bundle connections on 8-dimensional manifolds. Conjecturally, they can be used to define numerical invariants of 8-dimensional manifolds. However, not many examples of such instantons are known, which holds back the development of these invariants. In the talk I will explain a new construction method for Spin(7)-instantons generating more than 20,000 examples. The construction takes place on Joyce's first examples of compact Spin(7)-manifolds. No prior knowledge of Spin(7) or special holonomy is needed, this will be introduced in the talk. This is joint work with Mateo Galdeano, Yuuji Tanaka, and Luya Wang. (arXiv:2310.03451)