Differential Geometry Seminar

The Differential Geometry Seminar (formerly Geometry and Topology seminar) meets in room 901 of Van Vleck Hall on Thursdays from 1:15pm - 2:15pm (with some exceptions).

For more information, contact Sean Paul or Alex Waldron.

Spring 2025

date	speaker	title
Jan. 24	Jialong Deng (YMSC, Tsinghua)	The Nonexistence of Positive Scalar Curvature Metrics
Jan. 31	Dylan Galt (Stony Brook)	A Dimension Reduction Result for Generalized Anti-Self-Dual Instantons
Feb. 14	Yujie Wu (Stanford)	From \mu-bubble to \theta-bubble: Geometry of Mean Convex Manifolds with NNSC
Feb. 21	Yifan Chen (Berkeley)	More Complete Calabi-Yau Metrics of Calabi Type
Mar. 7	Yannick Sire (Hopkins)	Recent results on harmonic maps with free boundaries and applications
Mar. 14	Guangbo Xu (Rutgers)	Transversality on Orbifolds
Apr. 4	Juan Munoz-Echaniz (Stony Brook)	Boundary Dehn twists on 4-manifolds and Milnor fibrations of surface singularities
Apr. 11-13	Mini-conference	https://geometryworkshop.wiscweb.wisc.edu/speakers-and-schedule/
Apr. 25	Georgios Daskalopoulos (Brown)	Analytic questions arising in the theory of Best Lipschitz Maps
May 2	Yu Li (USTC)	Non-collapsing of Ricci shrinkers with bounded curvature
May 9	Wilderich Tuschmann (KIT, Germany)	Moduli Spaces of Riemannian Metrics

Spring 2025 Abstracts

Jialong Deng

Determining whether a closed smooth manifold admits a Riemannian metric with positive scalar curvature is a fundamental question in geometry and general relativity. In this talk, I will demonstrate that certain classes of manifolds, including some locally CAT(0) manifolds and almost nonpositively curved manifolds, do not admit metrics of positive scalar curvature.

Dylan Galt

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.

Yujie Wu

We introduce a method of constructing (generalized) capillary surfaces called "\theta-bubble", via Gromov's celebrated "\mu-bubble" method. Using this, we study manifolds with nonnegative scalar curvature (NNSC) and strictly mean convex boundary. We prove a fill-in question of Gromov, a band-width estimate, and a compactness conjecture of M. Li in the case of surfaces.

Yifan Chen

We construct more complete Calabi-Yau metrics asymptotic to Calabi ansatz. They are the higher-dimensional analogues of two dimensional ALH* gravitational instantons. Our work builds on and generalizes the results of Tian-Yau and Hein-Sun-Viaclovsky-Zhang, creating Calabi-Yau metrics that are only polynomially close to the model space. We will also show the uniqueness of such metrics in a given cohomology class with fixed asymptotic behavior.

Yannick Sire

I will report on an old topic in geometry, namely harmonic maps with free boundary,  insisting on latest developments and on a new viewpoint reformulating a class of those maps into a suitable framework of pseudo-differential equations. Using the latest developments on PDE techniques for integral equations, I will explain how to tackle some questions about existence, regularity and blow-up analysis for both elliptic and parabolic problems with these geometric free boundaries . I will also try to explain how those maps can be used to investigate  a variety of issues related to rigidity/flexbility for manifolds with boundary.

Guangbo Xu

In gauge theory and symplectic geometry, in order to define numerical invariants such as Donaldson invariants or Gromov-Witten invariants, one often needs to perturb the geometric data to make the relevant moduli spaces transverse. In symplectic geometry, the additional difficulty comes from the fact that the moduli spaces are stacks (orbifolds) and the failure of generic transversality on orbifolds. I will review these issues and describe a refined notion of transversality on orbifolds which allows us to define refined, integer-valued Gromov-Witten invariants, which leads to new applications in symplectic geometry. This is a joint work with Shaoyun Bai.

Juan Munoz-Echaniz

A 4-manifold with boundary on a Seifert-fibered space admits a `boundary Dehn twist’ diffeomorphism, obtained by a fibered version of the classical 2-dimensional Dehn twist. This diffeomorphism arises naturally as (a power of) the monodromy of Milnor fibrations of surface singularities. In this talk I will discuss non-triviality results for boundary Dehn twists on symplectic fillings of Seifert-fibered spaces, using tools from Seiberg—Witten theory. Some applications include:

- The ADE singularities are the only weighted-homogeneous isolated hypersurface singularities in complex dimension 2 whose monodromy has finite order in the smooth mapping class group.

- There are exotic R^4’s which admit exotic (compactly-supported) diffeomorphisms.

This is joint work with Hokuto Konno, Jianfeng Lin and Anubhav Mukherjee.

Georgios Daskalopoulos

Karen Uhlenbeck and I have been studying best Lipschitz maps from surfaces to surfaces. While this is motivated by Thurston’s distance function in Teichmuller space, it has connections with older ideas.  I will give a bit of the history about Lipschitz extensions, remind the listeners about infinity harmonic functions, and describe our construction for infinity harmonic mappings.  The goal is to motivate several interesting, new and I believe hard questions in analysis and their connection with Teichmuller theory.

Yu Li

Ricci shrinkers serve as singularity models for the Ricci flow. In this talk, we show that any Ricci shrinker with bounded curvature and certain topological constraints has a uniform entropy bound. This is joint work with Conghan Dong.

Wilderich Tuschmann

Focussing on lower curvature bounds and, in particular, the flat and Ricci flat case, I will present results and open questions about the global topological properties of moduli spaces of Riemannian metrics.

Fall 2024

date	speaker	title
Sep. 13	Pei-Ken Hung (UIUC)	Thom's gradient conjecture for geometric PDEs
Sep. 20	Hongyi Liu (Princeton)	A compactness theorem for hyperkähler 4-manifolds with boundary
Oct. 4	Lei Ni (UCSD)	The holonomy groups of Hermitian manifolds
Oct. 18	Keaton Naff (Lehigh)	Area estimates and intersection properties for minimal hypersurfaces in space forms
Oct. 25	Baozhi Chu (Rutgers)	Liouville theorems for second order conformally invariant equations and their applications
Nov. 1	Andoni Royo-Abrego (Tübingen)	Sobolev conformal structures on closed 3-manifolds
Nov. 15	Gioacchino Antonelli (Courant)	Concavity of isoperimetric profiles and applications to geometry
Dec. 6	Matthew Gursky (Notre Dame)	Some rigidity results for asymptotically hyperbolic Einstein metrics

Fall 2024 Abstracts

Pei-Ken Hung

Understanding solutions near their singularities is a fundamental topic in PDE.  The pioneering works of Leon Simon established the uniqueness of blow-ups for a broad class of geometric PDEs. Subsequently, the investigation of higher-order behavior becomes a crucial step for further analysis. In this talk, we will provide a complete description of the second-order asymptotic based on analytic gradient flows. Consequently, we prove Thom's gradient conjecture in the context of geometric PDEs. This talk is based on joint work with B. Choi.

Hongyi Liu

A hyperkähler triple on a compact 4-manifold with boundary is a triple of symplectic 2-forms that are pointwise orthonormal with respect to the wedge product. It defines a Riemannian metric of holonomy contained in SU(2) and its restriction to the boundary defines a framing. In this talk, I will show that a sequence of hyperkähler triples converges smoothly up to diffeomorphims if their restrictions to the boundary converge smoothly up to diffeomorphisms, under certain topological assumptions and the “positive mean curvature” condition of the boundary framings. I will also demonstrate a short proof of the surjectivity of period maps on the K3 manifold.

Lei Ni

The holonomy group of the Levi-Civita connection (denoted by $D$), which is called the Riemannian holonomy,  is an important object for the study of Riemannian manifolds. Roughly the size of the group measures how much the manifold locally  is deviated from being Euclidean. I shall discuss the progress made, some with F. Zheng,  in understanding the holonomy of Hermitian manifolds for Hermitian connections.

Keaton Naff

In this talk, I wish to discuss recent work (joint with Jonathan J. Zhu) on area estimates for minimal submanifolds and ``half-space" Frankel properties for minimal hypersurfaces in space forms. In the first setting, we will discuss sharp area estimates for minimal submanifolds in the curved space forms which pass through a prescribed point (building on work of Brendle-Hung). Our work settles the question in the hyperbolic setting, but leaves open an interesting outstanding question in the sphere. This leads naturally to the question of stability of minimal submanifolds in the hemisphere and in this direction we will demonstrate a Frankel property for the hemisphere (and other related settings).

Baozhi Chu

I will present optimal Liouville-type theorems for second order conformally invariant equations. A crucial new ingredient in proving these theorems is our enhanced understanding of solution behaviors near isolated singularities of such equations.  These Liouville-type theorems lead to optimal local gradient estimates for a wide class of fully nonlinear elliptic equations involving Schouten (Ricci) tensors. As an application of these Liouville-type theorems and gradient estimates, we establish new existence and compactness results for conformal metrics on a closed Riemannian manifold with prescribed symmetric functions of the Schouten (Ricci) tensor, allowing the scalar curvature of the conformal metrics to have varying signs.  This talk is based on a joint work with YanYan Li and Zongyuan Li.

Andoni Royo-Abrego

It is well-known in differential geometry that harmonic coordinates can be used to find the most regular expression for the components of a Riemannian metric. In this talk we will discuss a conformal analogue problem. More precisely, we will study the following question: given a Riemannian metric of limited regularity, does it exist a more regular (even smooth) representative in its conformal class? This problem is naturally linked to the Yamabe problem and finds applications in General Relativity.

Gioacchino Antonelli

In this talk, I shall discuss two results that show how the isoperimetric structure of a space is connected to its geometry.

First, I will present a sharp and rigid spectral generalization of the Bishop-Gromov volume comparison theorem. The proof of this result builds on a concavity property of an unequally weighted isoperimetric profile on the manifold. I will discuss how this volume estimate has been recently used by L. Mazet, following contributions by O. Chodosh, C. Li, P. Minter, and D. Stryker, to settle a well-known open problem in the theory of minimal surfaces: the stable Bernstein problem in R^n, with n<=6.

Second, I will show a sharp concavity property of the isoperimetric profile of noncompact manifolds with Ricci lower bounds. Although the statement is set in the smooth context, its proof relies on tools from non-smooth geometry that have been developed in recent years. I will explain how this concavity result interplays with the existence of isoperimetric regions in spaces with lower curvature bounds.

Matthew Gursky

In this talk I will describe some gap estimates for ’even’ and self-dual AHE metrics in dimension four. Even AHE metrics naturally arise from a non-local variational problem, and there is an interesting parallel between uniqueness questions for Einstein metrics on S^4 and even AHE metrics on the ball. I will also explain a connection to the study of self-dual AHE metrics. This is joint work with S. McKeown and A. Tyrrell.

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)

Archive of past Geometry seminars

2023-2024 Geometry_and_Topology_Seminar_2023_2024

2022-2023 Geometry_and_Topology_Seminar_2022_2023

2021-2022 Geometry_and_Topology_Seminar_2021_2022

2020-2021 Geometry_and_Topology_Seminar_2020-2021

2019-2020 Geometry_and_Topology_Seminar_2019-2020

2018-2019 Geometry_and_Topology_Seminar_2018-2019

2017-2018 Geometry_and_Topology_Seminar_2017-2018

2016-2017 Geometry_and_Topology_Seminar_2016-2017

2015-2016: Geometry_and_Topology_Seminar_2015-2016

2014-2015: Geometry_and_Topology_Seminar_2014-2015

2013-2014: Geometry_and_Topology_Seminar_2013-2014

2012-2013: Geometry_and_Topology_Seminar_2012-2013

2011-2012: Geometry_and_Topology_Seminar_2011-2012

Fall-2010-Geometry-Topology

Dynamics_Seminar_2020-2021