Schedule

All lectures will be in room B3.03 (Zeeman Building). All other activities will be in the Mathematics Common Room (Zeeman Building).

Time \ Day Monday Tuesday Wednesday Thursday Friday 9:30-10:30 Tiozzo Martelli Demonstrations de Mesmay Series 10:30-11:00 Coffee Coffee Coffee Coffee Coffee 11:00-12:00 Bell Hironaka Problem session Samperton Sivek 12:00-14:30 Lunch Lunch Lunch Lunch Lunch 14:30-15:30 Delecroix Culler Working groups Worden Burton 15:30-16:00 Tea Tea Tea Tea Tea 16:00-17:00 Purcell Guilloux Working groups Owens Chéritat 18:00 Dinner Dinner Dinner

Titles and abstracts

Mark Bell (Warwick)

The conjugacy problem for Mod(S) (YouTube) (Code)

Abstract: We will discuss a new polynomial-time solution to the conjugacy problem for the mapping class group of a surface. This relies on recently developed tools for finding tight geodesics in the curve complex of which we will highlight some key details. This is joint work with Richard Webb.

Benjamin Burton (Queensland)

Topology on tiny machines: logspace and finite-state complexity (YouTube)

Abstract: Instead of the usual focus on time complexity, here we discuss the space complexity of topological algorithms. We explain how to test 2-manifolds for homeomorphism in logspace (joint with Elder, Kalka and Tillmann), and we prove a variety of positive and negative results for 3-dimensional problems on finite state tree automata (joint with Fellows).

Benjamin Burton (Queensland)

Regina demonstration (YouTube) (Code)

Arnaud Chéritat (Toulouse)

Raster rendering for kleinian groups (YouTube) (Slides)

Abstract: Limit sets of quasifuchsian groups have been beautifully illustrated in the book Indra's Pearls by clever methods of group element enumeration.

On the other hand, raster methods have been quite successful for drawing Julia sets of polynomials. We explore the use of raster methods for limit sets. In some situations, one or the other approach can be preferred.

Marc Culler (UIC)

Computing $A$-polynomials (YouTube) (Code)

Abstract: I will describe an old algorithm for computing $A$-polynomials as well as some new additions which have significantly increased the number of once-cusped manifolds for which it is feasible to compute $A$-polynomials.

Arnaud de Mesmay (CNRS, Gipsa-Lab)

Hard embedding problems in three dimensions (YouTube) (Slides)

Abstract: While the search for efficient algorithms to solve three-dimensional topological problems has been very active in the past decades, results on computational hardness have been comparatively very scarce. In this talk, we will present two NP-hardness proofs for problems on embeddings into three-manifolds. The first one is about finding an embedded non-orientable surface of Euler genus $g$ in a triangulated three-manifold, and the second one is about deciding whether a two- or three-dimensional complex embeds into $\mathbb{R}^3$. This hardness stands in stark contrast with the lower dimensional cases which can be solved in linear time, and a variety of computational problems in $\mathbb{S}^3$ like unknot or three-sphere recognition which are in NP and co-NP (assuming the generalized Riemann hypothesis). This is based on joint work with Benjamin Burton, Yo'av Rieck, Eric Sedgwick, Martin Tancer and Uli Wagner.

Vincent Delecroix (Bordeaux)

Is there an algorithm for proving minimality of surface foliations? (YouTube) (Code)

Abstract: A foliation on an orientable surface can be defined via a train-track and some lengths data. It is well known since the work of M. Keane that the only obstruction to minimality is the presence of loops. The first consequence of Keane work is that enough irrationality between the lengths implies minimality. Though, interesting examples of foliations are often defined over number fields. We will explain M. Boshernitzan idea for his algorithm over quadratic field and how it can be extended to a partial algorithm for higher field degrees. Sadly the algorithm is only partial due to the evil minimal "SAF 0" examples.

All the algorithms will be illustrated with the "surface dynamics" package for SageMath (linked to above).

Nathan Dunfield (UIUC)

Snappy demonstration (YouTube) (Code)

Antonin Guilloux (Pierre et Marie Curie)

Volume function on character varieties (YouTube) (Sildes) (Volume over M-plane) (Volumes over M-circle)

Abstract: For a cusped hyperbolic manifold $M$, any representation of its fundamental group in $\text{SL}(2,C)$ has a well-defined volume. This fascinating function volume may be approached from several paths - theoritical or experimental - and is a key tool for the study of character varieties. I will describe this landscape: description of the volume, of the computational aspects, and what can be done with it.

Eriko Hironaka (Florida, AMS)

Directed train tracks for fibered hyperbolic manifolds (YouTube)

Abstract: We define the notion of a directed train track that carries the semi-flow associated to a fibered hyperbolic three-manifold. This directed train track is a convenient object from which one can derive the multi-variable Alexander polynomial of the 3-manifold, and the Teichmüller polynomial of the associated fibered face. In certain cases, this gives useful information about the relation between the homological and geometric dilatations in a flow-equivalence class. We give some applications to the minimum dilatation problem for pseudo-Anosov mapping classes.

Patrick Hooper (CCNY)

Flat surfaces with FlatSurf (YouTube) (Slides/Notebook) (Slides/HTML) (Slides/PDF)

Abstract: I'll demonstrate the FlatSurf module for SAGE which Vincent and I have been working on. The module can work with Euclidean cone surfaces, translation surfaces, and even similarity surfaces defined over algebraic number fields. I'll demonstrate some of the features of the module. One recently added feature is the ability to work with polyhedral surfaces in 3-space.

Bruno Martelli (Pisa)

Shadow complexity of smooth closed four-manifolds (YouTube) (Slides)

Abstract: We use Turaev shadows to define a complexity on any smooth closed four-manifold. This "shadow complexity" is a natural number that measures how complicated the two-skeleton of a four-manifold is. This complexity has the advantage of being tightly related to well-studied problems in three-dimensional topology: decomposing a three-manifold along spheres and tori, and classifying the exceptional Dehn fillings of a (multi-cusped) hyperbolic three-manifold. These problems can be attacked using SnapPy and by solving them we understand (or guess...) how the four-manifolds of low complexity are organized. This is partly joint work with Koda and Naoe.

Brendan Owens (Glasgow)

A computer search for ribbon alternating links (YouTube) (Slides)

Abstract: This is joint work with Frank Swenton. Our goal is to develop an algorithm to determine whether an alternating knot is ribbon. We cannot do this yet but we have an algorithm that has been remarkably, and indeed mysteriously, successful in finding new slice knots. The main theoretical input is Donaldson’s diagonalisation theorem, and in some sense we are looking for slice disks that resemble those described by Casson-Harer and Lisca for 2-bridge knots.

Jessica Purcell (Monash)

Effective geometry and thin tubes ﻿(YouTube)

Abstract: A tool in the study of hyperbolic three-manifolds is the thick-thin decomposition: any finite volume hyperbolic three-manifold can be decomposed into a compact thick part, and finitely many cusps and tubes that make up the thin part. Since tubes and cusps are quotients of hyperbolic three-space by elementary groups, it would seem that thin parts should be simple to analyse. However, in practice thin parts can be difficult to control. For example, the radius of a thin tube depends not just on a constant used to define "thin", but also on complex translation parameters, and the radius is not a smooth function of these parameters. Nevertheless, to compute examples, and to control their geometry, we need effective bounds on the geometry of thin parts. In this talk, I will describe work to give universal bounds on distances between thin tubes, and applications. This is joint with D. Futer and S. Schleimer.

Eric Samperton (UC Davis)

Computational complexity and three-manifolds and zombies (YouTube) ﻿(Slides)

Abstract: Let $G$ be a finite group, and let $M$ be a three-manifold. I’ll discuss the computational complexity of the problem of counting homomorphisms $\pi_1(M) \to G$. When $G$ is nonabelian simple, we show that the problem is $\#\mathbf{P}$-complete. This conclusion holds even if we only consider integer homology three-spheres, or knot complements. The structure of the proof is inspired by topological quantum computing, except nothing is quantum. The main tools and ideas are $\text{Aut}(G)$-equivariant reversible circuits, joint surjectivity lemmas in group theory, mapping class group actions on $G$-representation sets, and stabilization results for $G$-covers a la Livingston and Dunfield-Thurston.

Caroline Series (Warwick, LMS)

The suggestive power of pictures (YouTube) (Slides)

Abstract: Since the 1980s, computer graphics have played a large role in kleinian group theory, and in particular, in exploring parameter spaces of discrete groups. They have been a crucial element in formulating new ideas and have given impetus to major results including the ending lamination theorem and the bending measure conjecture. In this talk we will briefly outline how computer graphics led to the discovery of Keen-Series pleating rays and hence accurate pictures of such parameter spaces, and then look at some recent pictures which not only answer some interesting questions but which are also suggestive of new ideas, open problems, conjectures and theorems.

Steven Sivek (Imperial)

On the complexity of torus knot recognition (YouTube)

Abstract: The problem of recognizing that a knot diagram represents the unknot is known to be in the complexity class $\mathbf{NP} \cap \text{co-}\mathbf{NP}$. The membership in co-$\mathbf{NP}$ was proved by Kuperberg, assuming the generalized Riemann hypothesis (GRH), and then unconditionally by Lackenby. By combining techniques from both of their approaches, we will show that torus knot recognition is also in $\mathbf{NP}$ unconditionally and in co-$\mathbf{NP}$ assuming GRH. This is joint work with John Baldwin.

Giulio Tiozzo (Toronto)

Counting loxodromic elements (YouTube) (Slides)

Abstract: In a group of isometries of a delta-hyperbolic space, a loxodromic element is one which acts with positive translation length: it has north-south dynamics. Under general combinatorial conditions, we prove that loxodromic elements are generic (they have full density with respect to counting in

balls for the word metric) and their translation lengths grow linearly. We provide applications to relatively hyperbolic groups and graph products,

including right-angled Artin groups and right-angled Coxeter groups. This is joint work with I. Gekhtman and S. Taylor

William Worden (Temple)

Experimental statistics of veering triangulations ﻿(YouTube) (Slides)

Abstract: Certain fibered hyperbolic 3-manifolds admit a layered veering triangulation, which can be constructed algorithmically given the stable lamination of the monodromy. These triangulations were introduced by Agol [2011], and have been further studied by several others since then. We will present some experimental results, and outline a proof that random layered veering triangulations (those coming from a simple random walk on $\mathrm{Mod}(\Sigma)$) are non-geometric with probability approaching 1 as the length of the walk goes to infinity. This is joint work with Dave Futer and Sam Taylor.