Spring 2023
Date 
Time (Eastern) 
Speaker 
Title 
January 20

2pm Eastern

Nathan Sagman 
Hitchin representations and minimal surfaces

January 27

2pm Eastern

Ethan Dlugie

The Burau Representation and Shapes of Polyhedra

February 3

2pm Eastern 
Rebekah Palmer 
Totally geodesic surfaces in knot complements

February 10

2pm Eastern 
Lorenzo Ruffoni

Hyperbolization, cubulation, and applications

February 17

2pm Eastern 
Ty Ghaswala

Small covers of big surfaces

February 24

2pm Eastern 
Tam CheethamWest

Distinguishing hyperbolic knots using finite quotients

March 3

2pm Eastern

Marissa Loving

Unmarked simple length spectral rigidity for covers

March 10

2pm Eastern 
Dan Margalit

A new proof of Thurston's theorem

March 17

2pm Eastern

No Seminar

Spring break

March 24

2pm Eastern

Beibei Liu

Complexhyperbolic Kleinian groups of large critical exponents 
March 31

2pm Eastern

No Seminar

Simons Collaboration Meeting in NYC 
April 7 
2pm Eastern

Noelle Sawyer 
The Boundary at Infinity and Geodesic Currents 
April 21

2pm Eastern

Kyle Hayden

A handleholding approach to Walltype stabilization problems

April 28

2pm Eastern 
Giuseppe Martone

Pressure metrics on the space of finite area convex RP^2 surfaces

Abstracts
Nathaniel Sagman
Date: January 20
Title: Hitchin representations and minimal surfaces
Abstract:
Labourie proved that every Hitchin representation into PSL(n,R) gives rise to an equivariant minimal surface in the corresponding symmetric space. He conjectured that uniqueness holds as well (this was known for n=2,3) and explained that if true, then the Hitchin component admits a mapping class group equivariant parametrization as a holomorphic vector bundle over Teichmüller space. After giving the relevant background, we will explain that Labourie’s conjecture fails for n at least 4, and point to some future questions
Ethan Dlugie
Date: January 27
Title: The Burau Representation and Shapes of Polyhedra
Abstract:
The Burau representation of braid groups has been around for almost a century. Still we don't know the full answer to whether this representation is faithful. The only remaining case is for the $n=4$ braid group, and faithfulness here has intimate connections to the question of whether the Jones polynomial detects the unknot. By specializing the $t$ parameter in this representation to certain roots of unity, an interesting connection appears with the moduli space of flat cone metrics on spheres explored by Thurston. Leveraging this connection, I will explain how one can place strong restrictions on the kernel of the $n=4$ Burau representation.
Rebekah Palmer
Date: February 3
Title: Totally geodesic surfaces in knot complements
Abstract:
Studying totally geodesic surfaces has been essential in understanding the geometry and topology of hyperbolic 3manifolds. Recently, BaderFisherMillerStover showed that containing infinitely many such surfaces compels a manifold to be arithmetic. We are hence interested in counting totally geodesic surfaces in hyperbolic 3manifolds in the finite (possibly zero) case. In joint work with Khánh Lê, we expand an obstruction, due to Calegari, to the existence of these surfaces. On the flipside, we prove the uniqueness of known totally geodesic surfaces by considering their behavior in the universal cover. This talk will explore this progress for both the uniqueness and the absence.
Lorenzo Ruffoni
Date: February 10th
Title: Hyperbolization, cubulation, and applications
Abstract:
A hyperbolization procedure is a construction that turns a polyhedron into a space of negative curvature, while retaining some of its topological features. Originally introduced by Gromov, these procedures have been used to construct examples of manifolds that exhibit various pathologies, despite having negative curvature. One may expect to see such pathologies also at the level of the fundamental group. On the other hand, it turns out that the fundamental groups of these hyperbolized spaces are always very wellbehaved: they are linear over the integers, hence residually finite. We obtained this by showing that they admit actions on suitable CAT(0) cubical complexes with controlled stabilizers. This is joint work with J. Lafont.
Ty Ghasawala
Date: February 17th
Title: Small covers of big surfaces
Abstract:
Imagine the plane R^2 where every point with integer coordinates has been removed. Call this surface X. Which surfaces arise as finitesheeted covers of X? Which surfaces can X cover by finitelymany sheets?
I will talk about work Alan McLeay investigating the above seemingly innocent questions, and the more general version: Given two surfaces, when does there admit a finitesheeted cover of one over the other? A complete answer is available if the two surfaces are of finite type. In the infinitetype world, the question is less innocent than one might expect.
Tam CheethamWest
Date: February 24th
Title: Distinguishing hyperbolic knots using finite quotients
Abstract:
The fundamental groups of knot complements have lots of finite quotients. We give a criterion for a hyperbolic knot in the threesphere to be distinguished (up to isotopy and mirroring) from every other knot in the threesphere by the set of finite quotients of its fundamental group, and we use this criterion as well as recent work of BaldwinSivek to show that there are infinitely many hyperbolic knots distinguished (up to isotopy and mirroring) by finite quotients.
Marissa Loving
Date: March 3rd
Title: Unmarked simple length spectral rigidity for covers
Abstract:
A fundamental question in geometry is the extent to which a manifold M is determined by its length spectrum, i.e. the collection of lengths of closed geodesics on M. This has been studied extensively for flat, hyperbolic, and negatively curved metrics. In this talk, we will focus on surfaces equipped with a choice of hyperbolic metric. We will explore the space between (1) work of Otal (resp. Fricke) which asserts that the marked length spectrum (resp. marked simple length spectrum) determines a hyperbolic surface, and (2) celebrated constructions of Vignéras and Sunada, which show that this rigidity fails when we forget the marking. In particular, we will consider the extent to which the unmarked simple length spectrum distinguishes between hyperbolic surfaces arising from Sunada’s construction. This represents joint work with Tarik Aougab, Max Lahn, and Nick Miller.
Dan Margalit
Date: March 10th
Title: A new proof of Thurston's theorem
Abstract:
Thurston's theorem in complex dynamics gives necessary and sufficient conditions for a branched cover of the sphere to be a rational map. After explaining the statement of the theorem, we present a new proof. The proof is an enhancement of the proof of the NielsenThurston classification devised by Bers at Columbia four decades ago.
Beibei Liu
Date: March 24th
Title: Complexhyperbolic Kleinian groups of large critical exponents.
Abstract:
A Kleinian group is a discrete isometry subgroup of hyperbolic spaces and the critical exponent is one important group invariant of Kleinian groups. It is a longterm question of whether there is a gap in the value of the critical exponent in complex hyperbolic spaces. We use complex hyperbolic surfaces constructed by DeligneMostow to prove that there is no gap in the values of critical exponents for the complexhyperbolic Kleinian group. This is joint work with Subhadip Dey.
Noelle Sawyer
Date: April 7
Title: The Boundary at Infinity and Geodesic Currents
Abstract:
I plan to spend the first half talking about the boundary at infinity, geodesic currents (a measure on the space of geodesics), and why they're both interesting tools to help to understand geometric notions. In the second half, I will go more in depth about how I use these tools: Given a surface, the marked length spectrum (MLS) is the collection of the lengths of the closed geodesics, with each length marked by the free homotopy class it belongs to. Under certain conditions, information on most of the MLS is enough to completely determine the metric of the surface; I will give some intuition on how and why geodesic currents tie into the proof.
Kyle Hayden
Date: April 21
Title: A handleholding approach to Walltype stabilization problems
Abstract:
In dimension four, the differences between continuous and differential topology are vast but fundamentally unstable, disappearing when manifolds are enlarged in various ways. I will discuss Wall's stabilization problem and some of its variants, all of which aim to quantify this instability. In particular, I'll outline a simple "atomic" approach to these problems and, as a proof of concept, use it to produce exotic pairs of knotted surfaces (with boundary) in the 4ball that remain exotic after one "internal stabilization". The key obstruction comes from the universal version of Khovanov homology. Time permitting, I will speculate on some potential connections to Floer homology.
Giuseppe Martone
Date: April 28
Title: Pressure metrics on the space of finite area convex RP^2 surfaces
Abstract:
Real convex projective structures on a surface S are a generalization of hyperbolic structures, and they provide prominent examples of representations in a higher rank Teichmuller space.
In this talk, we construct mapping class group invariant metrics on the space of finite area real convex projective structures on S. This construction is motivated by McMullen’s dynamical interpretation of the WeilPetersson metric on the space of hyperbolic surfaces and by Bridgeman, Canary, Labourie, and Sambarino’s pressure metrics for Anosov representations. Our results generalize to the context of cusped Hitchin representations. The key dynamical ingredients in our construction come from the Thermodynamic Formalism of countable Markov shifts.
This talk is based on joint work with Bray, Canary and Kao.
Previous semesters:
Fall 2022,
Spring 2022,
Fall 2021,
Spring 2021, Fall 2020, Spring 2020, Fall 2019, Spring 2019, Fall 2018, Spring 2018, Fall 2017, Spring 2017, Fall 2016, Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, 2010/11, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006.