# 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 Alex Waldron.

## 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) | |

Mar. 25 | Max Engelstein (U Minnesota) | |

Apr. 8 | Matthew Stover (Temple) | |

Apr. 15 | Aleksander Doan (Columbia) | |

Apr. 22 | McFeely Goodman (Berkeley) |

## 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 An unpublished theorem of Tian says that if the curvature is very close to 1, the injectivity radius is bounded below, and a certain weighted L^2-norm of the traceless Ricci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.

## 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