The Gradute Student Topology and Geometry Conference is an annual conference in its 17th year running. It is organized by and for graduate researchers in geometry and topology. The purpose of this conference is to provide a venue for young researchers to be exposed to new research areas, and even present their own work in a productive and welcoming environment, while having the chance to meet with experts in the field.

For this reason, the majority of the conference talks will be given by graduate students, with nine other talks given by distinguished plenary speakers and young faculty. In the weeks leading up to the conference we will also host an online forum so that attendees can begin interacting with each other, and look forward to meeting in person. One hope we have is that this conference can be the beginning of many future learning seminars, online reading groups, and even research collaborations amonst the participants.

The 2019 GSTGC will take place March 30-31 at the University of Illinois Urbana Champaign

## Invited speakers

Plenary speakers:

• Mike Hill (UCLA)

## Exotic smooth structures on spheres:

How many smooth structures are there on a sphere? Starting with work of Milnor, we know that often the answer is more than 1, and Kervaire and Milnor proved a beautiful theorem connecting the number of smooth structures on spheres to the homotopy groups of spheres, up to a small fudge factor: the Kervaire invariant. This talk will start with these classical theorems before moving into more recent results about when spheres have unique smooth structures.
• Rafe Mazzeo (Stanford)

## Prospects on the Kapustin-Witten equations:

The use of gauge-theoretic equations as a remarkably useful tool in di erential topology is now a venerable subject. I will describe a fairly recent set of equations discovered by Kapustin and Witten and a subsequent proposal by Gaiotto and Witten to use this in knot theory. Progress on this proposal will requiresubstantial new analysis. I will describe the current state of this along with some interesting new directions.
• Amie Wilkinson (University of Chicago)

## Asymmetrical diffeomorphisms:

I will discuss, from the perspective of the $C^1$ topology, the dynamics of a typical" diffeomorphism of a closed manifold, focusing on a question posed by Steve Smale: does the typical diffeomorphism have trivial centralizer?

Young faculty speakers:

• Tarik Aougab (Brown University)

## Covers and Curves:

Given a regular cover $p:S^\prime\rightarrow S$ between surfaces, we attempt to characterize the map $p$ in terms of simple closed curves on $S^\prime$ and on $S$. For example, given two such covers with the same base surface $S$, are they equivalent if for any closed curve gamma on S, gamma lifts to a simple closed curve on one cover if and only if it does to the other? Using Teichmuller theory and the complex of curves, we answer this and a related question in the armative. We'll begin by explaining how the question is motivated by the study of isospectral hyperbolic surfaces and then give some idea of the proof; no specialized background will be assumed. This represents joint work with Max Lahn, Marissa Loving, and Sunny Yang Xiao.
• Agnes Beaudry (University of Colorado Boulder)

## Higher Real K-Theories:

Atiyah's Real K-theory $K_{\mathbb{R}}$ is one of the first example of a $C_2$-equivariant cohomology theory. It has lead to the construction of the Real cobordism spectrum $MU_{\mathbb{R}}$, which is now an essential tool in theory and computations. In this talk, I will discuss $MU_{\mathbb{R}}$ and how to use it to construct generalizations of Atiyah's Real K-theory which are $G$-equivariant cohomology theory for groups $G$ other than $C_2$.
• Michelle Chu (University of California Santa Barbara)

## Counting hyperbolic manifolds which bound geometrically:

A hyperbolic n-manifold is said to bound geometrically if it is isometric to the boundary of a hyperbolic (n + 1)-manifold with totally geodesic boundary. We might expect that most hyperbolic manifolds will not bound geometrically, and if they do then their volumes should be quite big. We will discuss joint work with Kolpakov on how the number of arithmetic hyperbolic manifolds which bound geometrically grows with volume.
• Ben Knudsen (Harvard University)

## Connectivity and growth in the homology of graph braid groups:

I will discuss recent work with An and Drummond-Cole showing that the homology of con guration spaces of graphs exhibits eventual polynomial growth, an analogue of classical homological and representation stability results for manifolds. We compute the degree of this polynomial in terms of an elementary connectivity invariant, in particular verifying an upper bound conjectured by Ramos. Along the way, we uncover a new edge stabilization mechanism and a family of spectral sequences arising from a small chain model first introduced by Swiatkowski.
• Jing Wang (Purdue University)

## Heat Content on the Heisenberg Group:

In this talk we discuss small time asymptotic expansion of the heat content for a smoothly bounded domain with non-characteristic boundary in the Heisenberg group. This expansion captures geometric information of the of the boundary including perimeter and the total horizontal mean curvature.

We will give an intuitive explanation of the probabilistic approach, which takes advantage on the connection between the solution of the Dirichlet boundary problem and the escaping probability of the canonically associated horizontal Brownian motion process on the Heisenberg group. This is a joint work with J. Tyson.

• Xuwen Zhu (University of California Berkeley)

## Deformation of constant curvature conical metrics:

Uniformization with singularities is a subject with a long history, and I will focus on describing constant curvature metrics with conical singularities which has seen a lot of recent development. In particular, in the case of positive curvature with large cone angles, the deformation has obstructions, and the moduli space is singular. I will describe a new geometric construction of compactified configuration spaces, which enables us to tackle the obstructed analytic problem by understanding clustering of cone points. Joint work with Rafe Mazzeo.

## Statement of inclusion

The organizers of GSTGC 2019 are committed to creating an environment that is inclusive, supportive, and safe. We support the rights of all attendees to fully participate in the conference regardless of race, sex, religion, national origin, sexual orientation, gender identity, disability, age, pregnancy, immigration status, or any other aspect of identity.

## Conference Policy on Harassment and Discrimination

All participants at GSTGC 2019 will be treated with dignity and respect, and discrimination or harassment of any form will not be tolerated. Participants who do not comport themselves in keeping with the conference's statement of inclusion run the risk of losing both their travel and housing funding. Since all funding for participants travel and lodging has been made possible by an NSF Conference Grant, we take this opportunity to remind our participants of the NSF's zero tolerance policy on harassment:

“NSF will not tolerate harassment, including sexual harassment or sexual assault within the agency, at awardee organizations, field sites, or anywhere NSF-funded science and education is conducted.”

## Previous conferences

Past editions of this conference were held at:

• 2018 — University of Illinois Chicago
• 2017 — Michigan State University
• 2016 — Indiana University, Bloomington
• 2015 — University of Illinois Urbana Champaign
• 2014 — University of Texas at Austin
• 2013 — University of Notre Dame
• 2012 — Indiana University, Bloomington
• 2011 — Michigan State University
• 2010 — University of Michigan
• 2009 — University of Wisconsin, Madison
• 2008 — University of Illinois at Urbana-Champaign
• 2007 — University of Chicago
• 2006 — Indiana University, Bloomington
• 2005 — Northwestern University
• 2004 — University of Minnesota
• 2003 — University of Notre Dame