# Difference between revisions of "Geometry and Topology Seminar"

The Geometry and Topology seminar meets in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:20pm (with some exceptions). For more information, contact Sean Paul or Gavin Ball.

## Fall 2022

date speaker 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.

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

date speaker title
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

date speaker 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

Fall-2010-Geometry-Topology