Geometry and Topology Seminar: Difference between revisions

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The [[Geometry and Topology|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). [[Image:Hawk.jpg|thumb|300px]]For more information, contact [https://people.math.wisc.edu/~stpaul/ Sean Paul] or [https://people.math.wisc.edu/~awaldron3/ Alex Waldron].
Link to [[Differential Geometry Seminar]]
== Spring 2025 ==
 
{| cellpadding="8"
! align="left" | date
! align="left" | speaker
! align="left" | 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 ==
 
{| cellpadding="8"
! align="left" | date
! align="left" | speaker
! align="left" | 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 ==
 
{| cellpadding="8"
! align="left" | date
! align="left" | speaker
! align="left" | 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<nowiki>''</nowiki> 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)
 
== Fall 2023 ==
 
{| cellpadding="8"
! align="left" | date
! align="left" | speaker
! align="left" | 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 2023 ==
 
=== Ruobing Zhang Minicourse: Topics in Metric Riemannian Geometry ===
 
Mon May 01​: 2:25 pm - 3:50 pm in Van Vleck B235
 
Tue May 02​:  2:15 pm - 4:00 pm in Birge 348
 
Wed May 03:​10:00 am - 11:45 am in Van Vleck B123
 
Thu May 04:​  2:15 pm - 4:00 pm in Birge 348 
 
Fri May 05​:  10:00 am - 11:45 am in Van Vleck B123
 
Lecture notes are available here: [[File:Topics in Metric Riemannian geometry.pdf]]
 
 
== Fall 2022 ==
 
{| cellpadding="8"
!align="left" | date
!align="left" | speaker
!align="left" | title
|-
|Sept. 23
|Ruobing Zhang (Princeton)
|Metric geometry of hyperkähler four-manifolds
|-
|Oct. 14
|Min Ru (University of Houston) (joint w/ Analysis seminar)
|The K-stability and Nevanlinna/Diophantine theory
|-
|Nov. 4
|Jesse Madnick (University of Oregon)
|Cohomogeneity-One Lagrangian Mean Curvature Flow
|-
|Nov. 11
|Gavin Ball
|Associative submanifolds of some nearly parallel G2-manifolds
|}
 
== Fall abstracts ==
 
===Ruobing Zhang===
 
This talk focuses on the recent resolution of the following three well-known conjectures in the field.
 
(1) Any volume collapsed limit of unit-diameter K3 metrics is isometrically classified as: the quotient of a flat 3D torus by an involution, a singular special Kähler metric on the topological 2-sphere, or the unit interval.
 
(2) Any complete non-compact hyperkähler 4-manifold with quadratically integrable curvature, must have a classified model end.
 
(3) Any gravitational instanton can be compactified to an open dense subset of certain compact algebraic surface.
 
Therefore, in the hyperkähler setting, we obtain a rather complete picture of the metric geometry on all scales.
 
=== Min Ru ===
In the recent paper with P. Vojta, we introduced the so-called beta-constant, and used it to extend the Cartan's Second Main Theorem in Nevanlinna theory and Schmidt's subspace theorem in Diophantine approximation. It turns out the beta-constant is also used in the algebro-geometric stability criterion in the Fano's case. In this talk, I'll describe and explore the somewhat mysterious connection. The talk is based on the recent joint paper with Yan He entitled "The stability threshold and Diophantine approximation", Proc. AMS, 2022.
 
===Jesse Madnick===
In complex n-space, mean curvature flow preserves the class of Lagrangian submanifolds, a fact known as “Lagrangian mean curvature flow” (LMCF).  As LMCF typically forms finite-time singularities, it is of interest to understand the blowup models of such singularities, as well as the possible soliton solutions.
 
In this talk, we'll consider the mean curvature flow of Lagrangians that are cohomogeneity-one under the action of a compact Lie group.  Interestingly, each such Lagrangian lies in a level set \mu^{-1}(\xi) of the moment map, and mean curvature flow preserves this containment.  Using this fact, we'll classify all cohomogeneity-one shrinking, expanding, and translating solitons.  Further, in the zero level set \mu^{-1}(0), we'll classify the Type I and Type II blowup models of cohomogeneity-one LMCF singularities. Finally, given any cohomogeneity-one special Lagrangian in \mu^{-1}(0), we'll show that it arises as the Type II blowup model of an LMCF singularity, thereby yielding infinitely many new singularity models.  This is joint work with Albert Wood.
 
=== Gavin Ball ===
A nearly parallel G2-structure is the natural geometric structure induced on the seven-dimensional link of a conical manifold with holonomy Spin(7). The link of a conical Cayley submanifold gives an associative submanifold of the nearly parallel G2-manifold, and thus associative submanifolds of nearly parallel G2-manifolds provide models for conically singular Cayley submanifolds. In my talk I will give an introduction to nearly parallel G2-manifolds and associative manifolds and, if time permits, explain a construction of certain associative submanifolds in two settings: in the Berger space SO(5)/SO(3) with its homogenous nearly parallel G2-structure, and in squashed 3-Sasakian manifolds. In both cases the submanifolds are ruled by a special class of geodesics and arise from a construction based on holomorphic curves in the spaces of rulings. This is joint work with Jesse Madnick.
 
== Spring 2022 ==
 
{| cellpadding="8"
!align="left" | date
!align="left" | speaker
!align="left" | title
|-
|Jan. 28
|
|Organizational meeting (includes graduate reading seminar)
|-
|Feb. 4
|Daniel Stern (U Chicago)
|Steklov-maximizing metrics on surfaces with many boundary components
|-
|Feb. 11
|Autumn Kent (NOTE: starts at 1:00pm)
|Deformations of hyperbolic manifolds and a theorem of Tian
|-
|Feb. 18
|Alex Waldron
|Strict type-II blowup in harmonic map flow
|-
|Mar. 4
|Sean Paul
|Geometric Invariant Theory, Stable Pairs, Canonical Kähler metrics & Heights
|-
|Mar. 11
|Tian-Jun Li (U Minnesota, REMOTE)
|Enhancing gauge theory invariants via generalized cohomologies
|-
|Mar. 25
|Max Engelstein (U Minnesota)
|Winding for Wave Maps
|-
|Apr. 8
|Matthew Stover (Temple)
|How to use, and prove, a superrigidity theorem
|-
|Apr. 15
|Aleksander Doan (Columbia)
|Holomorphic Floer theory and the Fueter equation
|-
|Apr. 22
|McFeely Goodman (Berkeley)
|Moduli Spaces of Nonnegative Curvature on Exotic Spheres
|-
|Apr. 29
|Aaron Kennon (UCSB)
|On the Laplacian Flow and its Soliton Solutions
|}
 
== Spring abstracts ==
 
===Daniel Stern===
 
Just over a decade ago, Fraser and Schoen initiated the study of the maximization problem for the first Steklov eigenvalue among all metrics of fixed boundary length on a given compact surface. Drawing inspiration from the maximization problem for Laplace eigenvalues on closed surfaces–where extremal metrics are induced by minimal immersions into spheres–they showed that Steklov-maximizing metrics are induced by free boundary minimal immersions into Euclidean balls, and laid the groundwork for an existence theory (recently completed by Matthiesen-Petrides). In this talk, I’ll describe joint work with Mikhail Karpukhin, characterizing the limiting behavior of these metrics on surfaces of fixed genus g and k boundary components as k becomes large. In particular, I’ll explain why the associated free boundary minimal surfaces converge to the closed minimal surface of genus g in the sphere given by maximizing the first Laplace eigenvalue, with areas converging at a rate of (log k)/k.
 
===Autumn Kent===
 
(NOTE: talk will start at 1:00pm)
 
A closed 3-manifold with pinched negative curvature admits a bona fide hyperbolic metric thanks to Perelman's proof of geometrization. Unfortunately, the proof doesn't tell us anything about the global geometry of the metric. An unpublished theorem of Tian says that if the curvature is very close to 1, the injectivity radius is bounded below, and a certain weighted L^2-norm of the traceless Ricci curvature is also small, then the metric is actually close to the unique hyperbolic metric up to third derivatives. The remarkable thing about his theorem is that there is no hypothesis on the volume.
 
I'll talk about some applications of this theorem to hyperbolic geometry, which require a version of Tian's theorem that allows short curves, and why such a version should hold. This is joint work in progress with Ken Bromberg and Yair Minsky.
 
===Alex Waldron===
 
I'll describe some recent work on 2D harmonic map flow, in which I show that a familiar bound on the blowup rate at a finite-time singularity is sufficient for continuity of the body map. This is relevant to a conjecture of Topping.
 
===Sean Paul===
 
An interesting problem in complex differential geometry seeks to characterize the existence of a constant scalar curvature metric on a Hodge manifold in terms of the algebraic geometry of the underlying variety. The speaker has recently solved this problem for varieties with finite automorphism group. The talk aims to explain why the problem is interesting (and quite rich) and to describe in non-technical language the ideas in the title and how they all fit together.
 
===Tian-Jun Li===
 
(NOTE: This talk will be on zoom)
 
I will describe a project with Mikio Furuta to enhance Gauge theory invariants using various generalized cohomology theories. This was motivated by the Bauer-Furuta stable cohomotopy Seiberg-Witten invariants.
 
===Max Engelstein===
 
Wave maps are harmonic maps from a Lorentzian domain to a
Riemannian target. Like solutions to many energy critical PDE, wave maps
can develop singularities where the energy concentrates on arbitrary
small scales but the norm stays bounded. Zooming in on these
singularities yields a harmonic map (called a soliton or bubble) in the
weak limit. One fundamental question is whether this weak limit is
unique, that is to say, whether different bubbles may appear as the
limit of different sequences of rescalings.
 
We show by example that uniqueness may not hold if the target manifold
is not analytic.  Our construction is heavily inspired by Peter
Topping’s analogous example of a “winding” bubble in harmonic map heat
flow. However, the Hamiltonian nature of the wave maps will occasionally
necessitate different arguments.  This is joint work with Dana Mendelson
(U Chicago).
 
===Matthew Stover===
 
This talk will be about the engine behind my colloquium: a superrigidity theorem. I will start describing what a superrigidity theorem is, and how it relates to proving arithmeticitiy. I will also discuss some other applications of our superrigidity theorem to geometry. For example, if M is a finite-volume hyperbolic 3-manifold obtained by Dehn filling on another hyperbolic 3-manifold N, then only finitely many totally geodesic surfaces on N remain totally geodesic (up to isotopy) under the filling. For the rest of the talk, I will describe the main ingredients going into proving a superrigidity theorem, in particular an elegant formulation due to Bader and Furman.
 
===Aleksander Doan===
 
I will discuss an idea of constructing a category associated with a pair of holomorphic Lagrangian submanifolds in a hyperkahler manifold, or, more generally, a manifold equipped with a triple of almost complex structures I,J,K satisfying the quaternionic relation IJ =-JI= K. This putative category can be seen as an infinite-dimensional version of the Fukaya-Seidel category: a well-known invariant associated with a Lefschetz fibration (i.e. manifold with a complex Morse function). While many analytic aspects of this proposal remain unexplored, I will argue that in the case of the cotangent bundle of a Lefschetz fibration, our construction recovers the Fukaya-Seidel category. This talk is based on joint work with Semon Rezchikov, and builds on earlier ideas of Haydys, Gaiotto-Moore-Witten, and Kapranov-Kontsevich-Soibelman.
 
===McFeely Goodman===
 
We show that the moduli space of nonnegatively curved metrics on each manifold homeomorphic to S^7 has infinitely many path components. The components are distinguished using the Kreck-Stolz s-invariant computed for metrics constructed by Grove and Ziller (for the so called “Milnor” spheres), and  Goette, Kerin and Shankar (for the “non-Milnor” spheres). The invariant is computed by extending each metric to the total space of a disc bundle and applying the Atiyah-Patodi-Singer index theorem for manifolds with boundary.  We will discuss the extension of these methods to the orbifold context, as is necessary to deal with the “non-Milnor” spheres.
 
===Aaron Kennon===
 
Given the successes of the Ricci Flow, it is sensible to look for other settings in which geometric flows may be useful. In the context of G2-Geometry, it is natural to flow the defining three-form by its Hodge Laplacian. This geometric flow of G2-Structures is called the Laplacian Flow. After briefly reviewing G2-Geometry, I'll summarize what has been proven about the Laplacian flow and outline the major open questions. I'll then discuss soliton solutions of this flow, and in particular, present some new results on these structures.
 
== Fall 2021 ==
 
{| cellpadding="8"
!align="left" | date
!align="left" | speaker
!align="left" | title
|-
|Sep. 10
|
|Organizational meeting
|-
|Sep. 17
|Alex Waldron
|Harmonic map flow for almost-holomorphic maps
|-
|Sep. 24
|Sean Paul (Cancelled due to flight delay)
|Geometric Invariant Theory, Stable Pairs, Canonical Kähler metrics & Heights
|-
|Oct. 1
|Andrew Zimmer
|Entropy rigidity old and new
|-
|Oct. 8
|Laurentiu Maxim
|Topology of complex projective hypersurfaces
|-
|Oct. 15
|Gavin Ball
|Introduction to G2 Geometry
|-
|Oct. 22
|Chenxi Wu
|Stable translation lengths on sphere graphs
|-
|Oct. 29
|Brian Hepler (Note: seminar begins at 2:30 in VV B313)
|Vanishing Cycles for Irregular Local Systems
|-
|Nov. 5
|Botong Wang
|Topological methods in combinatorics
|-
|Nov. 12
|Nate Fisher
|Horofunction boundaries of groups and spaces
|-
|Nov. 19
|Sigurd Angenent
|Questions for Topologists about Curve Shortening
|-
|Dec. 3
|Pei-Ken Hung (U Minnesota)
|Toroidal positive mass theorem
|-
|Dec. 10
|Nianzi Li
|Asymptotic metrics on the moduli spaces of Higgs bundles
|-
|}
 
== Fall Abstracts ==
 
===Alex Waldron===
 
I'll describe some history, recent results, and open problems about harmonic map flow, particularly in the 2-dimensional case.
 
===Sean Paul===
 
(See Spring semester)
 
===Andrew Zimmer===
 
Informally, an "entropy rigidity" result characterizes some special geometric object (e.g. a constant curvature metric on a manifold) as a maximizer/minimizer of some function of the objects asymptotic complexity. In this talk I will survey some classical entropy rigidity results in hyperbolic and Riemannian geometry. Then, if time allows, I will discuss some recent joint work with Canary and Zhang. The talk should be accessible to first year graduate students.
 
===Laurentiu Maxim===
 
I will overview old and new results which show how the presence of singularities affects the topology of complex projective hypersurfaces.
 
===Gavin Ball===
 
I will give an introduction to the theory of manifolds with holonomy group G2. I will begin by describing the exceptional Lie group G2 using some special linear algebra in dimension 7. Then I will give an overview of the holonomy group of a Riemannian manifold and describe Berger's classification theorem. The group G2 is one of two exceptional members of Berger's list, and I will explain the interesting properties manifolds with holonomy G2 have and sketch the construction of examples. If time permits, I will describe some of my recent work on manifolds with closed G2-structure.
 
===Chenxi Wu===
 
I will discuss some of my prior works in collaboration with Harry Baik, Dongryul Kim, Hyunshik Shin and Eiko Kin on stable translation lengths on sphere graphs for maps in a fibered cone, and discuss the applications on maps on surfaces, finite graphs and handlebody groups.
 
===Brian Hepler===
 
We give a generalization of the notion of vanishing cycles to the setting of enhanced ind-sheaves on to any complex manifold X and holomorphic function f : X → C. Specifically, we show that there are two distinct (but Verdier-dual) functors, denoted φ+∞ and φ−∞, that deserve the name of “irregular” vanishing cycles associated to such a function f : X → C. Loosely, these functors capture the two distinct ways in which an irregular local system on the complement of the hypersurface V(f) can be extended across that hypersurface.
 
Note: due to teaching conflict, Brian's talk will start at 2:30 in Van Vleck B313.
 
===Botong Wang===
 
We will give a survey of two results from combinatorics: the Heron-Rota-Welsh conjecture about the log-concavity of the coefficients of chromatic polynomials and the Top-heavy conjecture by Dowling-Wilson on the number of subspaces spanned by a finite set of vectors in a vector space. I will explain how topological and algebra-geometric methods can be relevant to such problems and how one can replace geometric arguments by combinatorial ones to extend the conclusions to non-realizable objects.
 
===Nate Fisher===
 
In this talk, I will define and motivate the use of horofunction boundaries in the study of groups. I will go through some examples, discuss how the horofunction boundary is related to other boundary theories, and survey a few applications of horofunction boundary.
 
===Sigurd Angenent===
 
Curve Shortening is the simplest and most easy to visualize of the geometric flows that have been considered in the past few decades.  Nevertheless there are many open questions about the kind of singularities that can appear in CS, and several of these questions probably, hopefully, have topological answers.  I'll give a short overview of what is and what isn't known.  While geometric flows have had success in solving old problems in topology (Poincaré conjecture, etc.) , I would like turn things around in my talk and argue that rather than asking what analysis can do for topology, we should ask what topology can do for analysis.
 
===Pei-Ken Hung===
 
We establish the positive mass theorem for 3-dimensional asymptotically hyperboloidal initial data sets with toroidal infinity. In the umbilic case, a rigidity statement is proven showing that the total mass vanishes precisely when the initial data manifold is isometric to a portion of the canonical slice of the associated Kottler spacetime. Furthermore, we provide a new proof of the recent rigidity theorems of Eichmair-Galloway-Mendes in dimension 3, with weakened hypotheses in certain cases. These results are obtained through an analysis of the level sets of spacetime harmonic functions. This is a joint work with Aghil Alaee and Marcus Khuri.
 
===Nianzi Li===
 
I will introduce the definition of Higgs bundles, discuss some structures and metrics on the moduli spaces of Higgs bundles. Then I will give an overview of the results of Mazzeo-Swoboda-Weiss-Witt and Fredrickson on the exponential decay of the difference between the hyperkähler L^2 metric and the semi-flat metric along a generic ray. Finally, I will briefly talk about Boalch's modularity conjecture, and describe an ongoing work of extending the results to Higgs bundles with irregular singularities on a Riemann sphere, some of the moduli spaces are shown to be ALG gravitational instantons.
 
== Archive of past Geometry seminars ==
2020-2021  [[Geometry_and_Topology_Seminar_2020-2021]]
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2019-2020 [[Geometry_and_Topology_Seminar_2019-2020]]
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2018-2019  [[Geometry_and_Topology_Seminar_2018-2019]]
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2017-2018 [[Geometry_and_Topology_Seminar_2017-2018]]
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2016-2017  [[Geometry_and_Topology_Seminar_2016-2017]]
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2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]
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2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]
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2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]
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2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]
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2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]
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[[Fall-2010-Geometry-Topology]]<br>
[[Dynamics_Seminar_2020-2021]]

Revision as of 14:53, 13 June 2025

Link to Differential Geometry Seminar

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