# Schedule, Titles and Abstracts

##### Schedule

All talks will be in room MS.02 (Zeeman Building). All other activities will be in the Mathematics Common Room (Zeeman Building).

Monday | Tuesday | Wednesday | Thursday | Friday | |

09:30-10:30 | Dunfield | Boileau | Greene | Wilton | Friedl |

11:00-12:00 | Purcell | Wilkes | Yazdi | Kent | Rasmussen |

14:00-15:00 | Martelli | Futer | Schleimer | Brock | |

15:30-16:30 | Gabai | Kerckhoff | Souto |

##### Titles and abstracts

**Michel Boileau** (Aix Marseille)

*Finite covers of 3-manifolds and Thurston norm*

Abstract: We show that the Thurston seminorms of all finite covers of an aspherical compact orientable 3-manifold, with empty or toroidal boundary, determine whether it is a graph manifold, a mixed 3-manifold or hyperbolic. This is a joint work with Stefan Friedl.

**Jeffrey Brock** (Brown)

*Renormalized volume of hyperbolic 3-manifolds and the Weil-Petersson geometry of moduli space.*

Abstract: In this talk I will describe recent developments that provide a satisfying analytic explanation for the connection between volume of quasi-Fuchsian convex cores and Weil-Petersson distance, as well as volumes of hyperbolic 3-manifolds that fiber over the circle and Weil-Petersson translation distance of pseudo-Anosov automorphisms of Teichmüller space. I'll discuss an array of applications to Weil-Petersson geometry as well as some new results. This is joint work with Ken Bromberg, with new work adding Martin Bridgeman to the collaboration.

**Nathan Dunfield** (UIUC)

*An SL(2, R) Casson-Lin invariant and applications*

Abstract: When M is the exterior of a knot K in the 3-sphere, Lin showed that the signature of K can be viewed as a Casson-style signed count of the SU(2) representations of $\pi_1(M)$ where the meridian has trace 0. This was later generalized to the fact that signature function of K on the unit circle counts SU(2) representations as a function of the trace of the meridan. I will define the SL(2, R) analog of these Casson-Lin invariants, and explain how it interacts with the original SU(2) version via a new kind of smooth resolution of the real points of certain SL(2, C) character varieties in which both kinds of representations live. I will use the new invariant to study left-orderability of Dehn fillings on M using the translation extension locus I introduced with Marc Culler, and also give a new proof of a recent theorem of Gordon’s on parabolic SL(2, R) representations of two-bridge knot groups. This is joint work with Jake Rasmussen (Cambridge).

**Stefan Friedl** (Regensburg)

*Linking forms and Blanchfield forms of 3-manifolds*

Abstract: We will discuss the definition and basic properties of linking forms and Blanchfield forms of links and 3-manifolds. We will also talk about several recent computations and applications. This talk is based on work with Anthony Conway, Gerrit Herrmann, Allison Miller, Mark Powell and Enrico Toffoli.

**David Futer** (Temple)

*Ubiquitous quasifuchsian surfaces in cusped hyperbolic 3-manifolds*

Abstract: I will discuss the theorem that a cusped hyperbolic 3-manifold M contains a ubiquitous collection of closed, immersed, undistorted surfaces. The boundaries of these surfaces are Jordan curves that are ubiquitous in the sense that they separate any pair of round circles on the sphere at infinity. As a corollary, we recover Wise’s theorem that the fundamental group of M is cubulated. This is joint work with Daryl Cooper.

**David Gabai** (Princeton)

*The Two Eyes Lemma
*

Abstract: We will show that subject to certain geometric constraints, a cycle of eight tangent horoballs in hyperbolic 3-space can link two horoballs with disjoint interiors only when they lie in a certain $S^1$ family of configurations. This is joint work with Robert Meyerhoff and Andrew Yarmola. It and it's proof are crucial towards showing that certain cusped hyperbolic $3$-manifolds are obtained by filling explicit cusped hyperbolic 3-manifolds.

**Joshua Greene** (Boston College)

*Fibered simple knots*

Abstract: I will discuss joint work with John Luecke in which we characterize which simple knots in lens spaces fiber.

**Autumn Kent** (UWM)

*Skinning maps along thick rays*.

Abstract. I'll discuss work in progress with K. Bromberg and Y. Minsky. We show that the diameter of the skinning map of an acylindrical $3$-manifold along a thick ray in the Teichmueller space is bounded by constants depending only on the injectivity radius and genus of the boundary.

**Steven Kerckhoff** (Stanford)

*Geometric Transitions: From Rigidity to Flexibility*

Abstract: Hyperbolic structures on 3-manifolds tend to be rigid, relative to certain boundary data. Families of such structures, with varying boundary data, can degenerate to other types of geometric structures that are much more flexible. The particular limit structures typically are solutions to extremal problems. This lecture will discuss several examples of this phenomenon.

**Bruno Martelli** (Pisa)

*Hyperbolic Dehn filling in dimension four*

Abstract: We introduce some deformations of complete finite-volume hyperbolic four-manifolds that may be interpreted as four-dimensional analogues of Thurston's hyperbolic Dehn filling. More precisely, we exhibit an analytic path of complete finite-volume cone four-manifolds that interpolates between two non-isometric hyperbolic four-manifolds by drilling and filling some of their cusps. The cone four-manifolds have singularities along a geodesically immersed surface, with varying cone angles. This is joint work with Stefano Riolo.

**Jessica Purcell** (Monash)

*Decomposing 3-manifolds and the geometry of alternating links on surfaces*

Abstract: A useful technique for studying a 3-manifold is to decompose it into simpler pieces, such as polyhedra, and to examine normal surfaces within the pieces. If the pieces admit additional data, e.g. an angle structure, then there are concrete geometric consequences for the manifold and the surfaces it contains. In this talk, we describe a way to generalise decompositions, and to extend results to broader families of 3-manifolds. For example we allow pieces that are not simply connected, glued along faces that are not disks, and we define normal surfaces in these cases. We give examples of manifolds with these structures (families of knot complements) and show that the existence of such structures ensures geometric consequences (hyperbolicity, surface geometry, volume bounds, etc). This is joint work with Josh Howie.

**Sarah Rasmussen** (Cambridge) L-space surgeries and 3-manifold structures

Abstract: A closed 3-manifold is called an $L$-space when its reduced Heegaard Floer homology vanishes, but this characterization is largely irrelevant to mathematicians outside the Floer-invariant community. I will discuss how some new tools for studying the behaviour of Heegaard Floer homology under Dehn surgery and/or toroidal gluing has led to further progress in relating Heegaard Floer homology, and particularly its vanishing, to structures that other geometers and topologists might care about. Parts of this work are joint with Jake Rasmussen.

**Saul Schleimer** (Warwick)

*From veering triangulations to pseudo-Anosov flows*

Abstract: Veering triangulations, introduced by Agol, are a specialisation of Lackenby's taut ideal triangulations. Agol used these to study families of pseudo-Anosov mappings. Agol and Gueritaud extended work of Gueritaud to show that if $N$ is a closed manifold, equipped with a pseudo-Anosov flow $\Phi$ without perfect fits, then $N_\Phi$ admits an associated veering triangulation. Here $N_\Phi$ is the cusped manifold obtained from $N$ by drilling the singular orbits of $\Phi$.

We prove a partial converse to their result: if $M$ is a cusped manifold equipped with a veering triangulation (with a certain mild side condition) then $M(r)$ admits an associated pseudo-Anosov flow without perfect fits. Here $M(r)$ is a Dehn filling of $M$ where the slope $r$ avoids a finite collection of lines. This is joint work with Henry Segerman.

**Juan Souto** (Rennes)

*Geodesic currents and counting problems*

**Gareth Wilkes** (Oxford)

*Profinite rigidity of graph manifolds*

Abstract: There has been recent interest in the circumstances under which two non-homeomorphic 3-manifolds may have fundamental groups with isomorphic profinite completions. I will describe the classification of those pairs of graph manifolds with this property, and give some examples. I will also discuss the implications of this result for the mapping class group.

**Henry Wilton** (Cambridge)

*Surface subgroups of graphs of free groups*

Abstract: A well known question, usually attributed to Gromov, asks whether every hyperbolic group is either virtually free or contains a surface subgroup. I’ll discuss the answer to this problem for the class of groups in the title when the edge groups are cyclic. The main theorem is a result about a free group F which is of interest in its own right: whether of not an element w of F is primitive can be detected in the abelianizations of finite-index subgroups of F.

**Mehdi Yazdi** (Oxford)

*On Thurston's Euler class one conjecture*

Abstract: In 1976, Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any Euler class with norm equal to one is Euler class of a taut foliation. We construct counterexamples to this conjecture and suggest an alternative conjecture.