# Geometry and Topology Seminar

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

For more information, contact Sean Paul or Alex Waldron.

## 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 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

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.

### 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

date | speaker | 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

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

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