# Geometry and Topology Seminar

The Geometry and Topology seminar meets in room **901 of Van Vleck Hall** on **Fridays** from **1pm - 2:20pm**. For more information, contact Alex Waldron.

## Contents

## Fall 2021

date | speaker | title |
---|---|---|

Sep. 10 | Organizational meeting | |

Sep. 17 | Alex Waldron | Harmonic map flow for almost-holomorphic maps |

Sep. 24 | Sean Paul | 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 | |

Nov. 19 | Sigurd Angenent |

## 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

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.

Note: this talk will provide some background for Sean's colloquium later in the afternoon.

### 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.

## 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