# Abstracts

- 2017-02-27 Federico Vigolo (University of Oxford)
**An introduction to expanders and how to construct them**

I will give a soft introduction to expander graphs, trying to motivate them. I will also explain a very geometric way of constructing some family of expanders out of rotations of the sphere.- 2017-03-06 Katie Vokes (University of Warwick)
**Geometry of the separating curve graph**

To each topological surface, we can associate a number of graphs, each of whose vertices is a curve or collection of curves in the surface. These graphs have been important in the study of the geometry of mapping class groups and Teichmüller spaces. I shall introduce some concepts in this area and present a result on the large scale geometry of the separating curve graph.- 2017-03-13 Victor González Moreno (Royal Holloway University of London)
**Classifying spaces for families of subgroups**Classifying spaces for families of subgroups have been widely studied in the case of the families of finite subgroups and virtually cyclic subgroups, due to them being the geometrical objects in the Baum-Connes Conjecture and Farrell-Jones Conjecture, respectively. However, those definitions and the Bredon Cohomology on which the algebraic meaning of this objects relies are stated for all families of subgroups. For that reason, classifying spaces for larger families of subgroups is a hardly explored and rich field.

The aim of this talk is to define and illustrate with some examples and properties the concept of classifying spaces for families of subgroups and present a piece of my work on such spaces. In particular, I will explain the construction of models for the classifying space for the family of subgroups of a polycyclic group

G of Hirsch length less than or equal tor .- 2017-04-24 Elia Fioravanti (University of Oxford)
**An introduction to CAT(0) cube complexes**

I will give a gentle introduction to the geometry of CAT(0) cube complexes, focussing especially on the combinatorics of hyperplanes and the construction of the Roller boundary. If time allows, I will sketch a proof of the Tits Alternative in this context, a result originally due to Sageev and Wise.- 2017-05-02 Nicolaus Heuer (University of Oxford)
**(Bounded) Cohomology of groups**

The bounded cohomology of groups was promoted by Gromov in the 80s to attack rigidity questions. It has very exotic and unexpected behaviour. I will try to make it accessible by comparing the tools and results of bounded cohomology to their well understood counterparts in classical cohomology. Key words are Mayer-Vietoris, product structures, functoriality and group extensions.- 2017-05-08 Ronja Kuhne (University of Warwick)
**Train tracks, curves and efficient position**

Train tracks were introduced by Thurston in the late 1970s as a combinatorial tool for studying surface diffeomorphisms. After giving relevant background material and elaborating on the interplay between train tracks and curves on surfaces, I plan to define the notion of efficient position of curves with respect to train tracks. Efficient position can be understood as some kind of general position for curves on surfaces with respect to train tracks and I intend to address the question of its existence as well as discuss possible applications.- 2017-05-15 Andreas Bode (University of Cambridge)
**Coadmissible D-modules on rigid analytic flag varieties**

The Beilinson-Bernstein localization allows us to study representations of Lie algebras geometrically, as D-modules on the associated flag variety. Ardakov and Wadsley have begun to develop a theory of D-modules on rigid analytic spaces in the sense of Tate, hoping for analogous results in a p-adic locally analytic setting. In this setting, the notion of coherence gets naturally replaced by that of 'coadmissibility'. I will explain the general theory before discussing various versions of a Proper Mapping Theorem for coadmissible D-modules, in particular showing that the functors in our Beilinson-Bernstein equivalence preserve coadmissibility.- 2017-05-22 Alex Wendland (University of Warwick)
**Finiteness conditions in infinite groups**

In this talk we will explore different definitions of finiteness conditions for infinite groups discussing their connections to each other and geometric interpretations. We will go on to talk about a new definition which has arisen from a generalisation of Benjamin-Schram graph convergence and, time allowing, it connections to an old conjecture of Remesselenikov to do with the genus of free groups.- 2017-05-30 George Kenison (University of Warwick)
**Asymptotics comparing length functions on free groups**

Let $F$ be a free group with rank at least 2. We suppose that $F$ is a discrete and convex co-compact group of isometries of $n$-dimensional hyperbolic space or, more generally, a CAT(-1) space $M$. To each $x\in F$ we associate two lengths: the word length of $x$ for a given generating set and the geometric displacement $d_M(o,xo)$ for a prescribed point $o\in M$.In this talk we compare the two length functions asymptotically (ordering the group elements by word length). Time permitting we establish asymptotics when the group elements are restricted to a non-trivial conjugacy class. This is joint work with Richard Sharp.

In this talk we compare the two length functions asymptotically (ordering the group elements by word length). Time permitting we establish asymptotics when the group elements are restricted to a non-trivial conjugacy class. This is joint work with Richard Sharp.

- 2017-06-05 Louis Bonthrone (University of Warwick)
**Ricci flow with cone singularities**In recent years there has been a large amount of work being done to understand metrics with conic singularities along a divisor. We will look at the motivation for studying such objects and some of the key results in complex dimension greater than two. On the other hand in complex dimension metrics with cone singularities have been well understood since the work of Troyanov, Luo-Tian in the 80's and 90's. This theory was developed using variational techniques for the Liouville equation.

In this talk we consider the Ricci flow on surfaces, which, in some sense, is the parabolic version of the Liouville equation. More precisely, we are interested in a recent collection of results allowing one to flow while preserving any cone singularities and their angles. We will then see that Troyanov's elliptic theory yields natural convergence results and how one might hope to generalise this work.

- 2017-06-12 María Cumplido (Université Rennes 1/Universidad de Sevilla)
**On the genericity of pseudo-Anosov elements in the mapping class group of a surface (with Bert Wiest)**

This talk is motivated by a well-known conjecture which claims that "most" elements of the mapping class group G of a surface are pseudo-Anosov. This means that, if we take a ball in the Cayley graph of G, the proportion of vertices in the ball representing pseudo-Anosov elements tends to 1 as the radius of the ball tends to infinity. The aim of the talk is to prove that this proportion is positive. Eventually, this proof will lead us to give a condition, so that if a subgroup H of G fulfills this condition, then H also has a postive proportion of pseudo-Anosov elements.- 2017-06-16 Federica Fanoni (Max Planck Institute for Mathematics, Bonn)
**Mapping class group orbits of non-simple curves**

The number of mapping class group orbits (topological types) of simple closed curves on surfaces is well-known and easy to compute. If we consider non-simple curves instead, counting orbits becomes more complicated. I will talk about this problem and about the ideas to get the asymptotics of the number of orbits of curves with k self-intersections (as the genus goes to infinity). Joint work with Patricia Cahn and Bram Petri.- 2017-06-19 Claudius Zibrowius (University of Cambridge)
**On the Fukaya category of marked surfaces via curved complexes**With an oriented surface (plus some choice of extra data), one can associate a category of curved complexes. I will discuss the construction of this category in some detail and explain why I care about it.

- 2017-10-02 Beatrice Pozzetti (Heidelberg University)
**Symmetric spaces of non-compact type**

A Riemannian symmetric space is a Riemannian manifold X whose group of isometries contains the geodesic involution at any point. If such a space X has no compact factor, it is a CAT(0) space whose isometry group acts transitively. I will introduce the geometric properties of these spaces needed to give the idea of a beautiful proof due to Ballmann-Gromov-Schroeder of a rigidity theorem in higher rank.- 2017-10-09 Alex Wendland (University of Warwick)
**A survey of Topology of finite graphs**I will conduct a review of the methods used in Stalling's Topology of finite graphs. Here he uses maps of finite graphs to give simple proofs for results withing free group theory, such as Howson's theorem (intersection of f.g. subgroups of free groups is f.g.) and M. Hall's theorem (free groups are LERF). The paper has been cited in many further works and time allowing I will mention some work followed up by Gersten.

- 2017-10-16 Louis Bonthrone (University of Warwick)
**J-holomorphic curves from J-anti-invariant forms**

Since the 1980's there has been a well known folklore theorem which says that for a generic Riemannian metric on a 4-manifold the zero set of a self-dual harmonic 2-form is a finite number of embedded circles. We prove that in the almost complex setting the corresponding result holds without a genericity assumption. That is, we show the zero locus of a closed J-anti-invariant 2-form is a J-holomorphic curve in the canonical class. This is based on joint work with Weiyi Zhang.- 2017-10-23 Ben Barrett (University of Cambridge)
**Bestvina and Mess's double-dagger condition**

It is a fundamental tenet of geometric group theory that groups look like the spaces on which they act, at least on a large scale, and so large scale properties of such spaces can be thought of as being intrinsic to the group. One such large scale property is the Gromov boundary of a space with a negative curvature property, which generalises the circular boundary of the hyperbolic plane. Some important connectivity properties of the Gromov boundary of a space are controlled by a so-called double-dagger condition on the space itself. In this talk I will describe this link between the hyperbolic geometry of a space and the "connectivity at infinity" of that space.

- 2017-10-30 Esmee te Winkel (University of Warwick)
**Mostow's rigidity theorem**

Given a closed, connected, oriented 3-manifold that admits a hyperbolic metric, it is a result of Mostow that this metric is unique. More generally, the geometry of a closed, connected, oriented n-manifold is determined by its fundamental group, when n is at least 3. This is awfully false in dimension 2 – actually, there is an entire space of hyperbolic structures on a surface, called Teichmüller space.I will introduce Mostow's theorem, motivate its relevance and, if time permits, sketch a proof.

- 2017-11-06 Paul Colognese (University of Warwick)
**An introduction to rational billiards and translation surfaces**Consider a game of billiards/pool/snooker. If we assume that the ball is a moving point and that there is zero friction, we can consider the long term dynamics of a trajectory. One way of studying this problem is by unfolding the table to get a closed surface known as a translation surface. In this talk, I'll provide a very brief introduction to the subject, focusing on the basic geometry as well hopefully providing some insight into how this perspective can be fruitful when solving problems about billiards.

- 2017-11-13 Sophie Stevens (University of Bristol)
**Point-Line Incidences in Arbitrary Fields**

Points and lines are simple-sounding sets of objects, and to help us out, we'll talk only about finite sets of both. We can ask simple-sounding questions about them, such as "how often do they intersect?" or "if they intersect lots, do they have special structure?". Answering these types of questions is an active area of mathematics, with strong links to additive combinatorics. I will talk about the situation in arbitrary fields, presenting two incidence theorems and some of their applications.- 2017-11-20 Stephen Cantrell (University of Warwick)
**Counting with Quasimorphisms on Hyperbolic Groups**

Let $G$ be a hyperbolic group. A map $\phi : G \to \mathbb{R}$ is called a quasimorphism if it is a group homomorphism up to some bounded error.In this talk we introduce a counting problem related to quasimorphisms. We discuss how to tackle this problem using ideas from both geometry and ergodic theory. We will examine the interplay between these two areas of maths and will explore how they can be used together to solve the counting problem in the case that $G$ is a surface group. We will then discuss the difficulties in extending this result to the general case of any hyperbolic group $G$.

- 2017-11-27 Alex Evetts (Heriot-Watt University)
**Aspects of Growth in Groups**

Elements of a finitely generated group have a natural notion of length. Namely the length of a shortest word over the generators which represents the element. This allows us to see such groups as metric spaces, and in particular to study their growth by looking at the sizes of spheres centred at the identity. This idea of growth can be generalised in various ways. In this talk I will describe some of the important results in the area, and try to give an idea of the tools used to study growth.

- 2017-12-04 Ana Claudia Lopes Onorio (University of Southampton)
**Ends of spaces and groups**

The theory of ends of topological spaces and ends of groups has its beginnings in the work of Freudenthal (1931) and Hopf (1944). In my talk, I would like to introduce these concepts and some interesting facts concerning such a group invariant. I will assume basic knowledge of algebraic topology and group theory from the audience. I will try not to give too many algebraic details and focus on the main ideas, so that it can be understandable by 4th-year undergrads and first year PhD students. I hope the end justifies the means! This is closely related to the subject of my research at the moment, which I might comment a bit about at the end of the talk if time permits.- 2018-01-08 Giles Gardam (Technion)
**The geometry of the word problem**The word problem is the task of deciding, given a word in a fixed generating set of a group, whether it represents the identity element of the group or not. While this sounds very algebraic, it has real geometric meaning. In this talk, I will introduce Dehn functions, which provide a geometric quantification of the difficulty of the word problem. After treating classical examples, I will give new examples from joint work with Daniel Woodhouse showing that one-relator groups have a rich collection of polynomial Dehn functions.

- 2018-01-15 Néstor León Delgado (Max Planck Institute for Mathematics, Bonn)
**Jet bundles and local maps**When talking about maps between spaces of functions, the term locality often comes up. This could mean a map of sheaves, a map depending only on the value of the function at a point, or on some of its derivatives. In the later case we say that the map descends to a map from a jet bundle. In this talk we will learn what jet bundles are and how are they related to locality.

- 2018-01-22 Anna Parlak (University of Warwick)
**Roots of Dehn twists**

It is well known that the mapping class group $\mathrm{Mod}(S_{g})$ of the orientable surface of genus $g$ is generated by a finite number of Dehn twists. Quite recently (2009) Margalit and Schleimer showed that, surprisingly, these elements are not primitive in $\mathrm{Mod}(S_g)$. They proved that every Dehn twist about a nonseparating circle in $\mathrm{Mod}(S_{g+1})$, $g \geq 1$, has a root of degree $2g+1$. Natural questions that arose were:

- what other degrees of roots are possible, apart from divisors of $2g+1$?

- if a root of degree $n$ of a Dehn twist about a nonseparating circle exists, is it unique up to conjugation in $\mathrm{Mod}(S_{g+1})$?

These questions were answered by McCullough and Rajeevsarathy (2011) who derived numerical equations whose solutions are in bijective correspondence with the conjugacy classes of roots of Dehn twists about nonseparating circles. Later, using similar techniques, this work was extended to the case of Dehn twists about separating circles (Rajeevsarathy, 2013), multicurves (Rajeevsarathy, Vaidyanathan, 2017) and Dehn twists in the mapping class group of a nonorientable surface.During the talk I will be primarily focused on presenting the results of McCullough and Rajeevsarathy from 2011. If time permits, I will also give some remarks about the analogous investigation in the nonorientable case.

- 2018-01-29 Selim Ghazouani (University of Warwick)
**Affine structures on (closed) manifolds**

An affine structure on a manifold is a geometric structure that is modelled on the standard affine space R^n through the group of affine transformations. These structures are somewhat reminiscent of Euclidean structures and if one can draw a certain number of analogies with the Euclidean case (that led to a certain number of beautiful theorems), many questions about these structures remain widely open. In particular, it is very hard to tell whether a given manifold carries an affine structure.I will try to give an insight to this mysterious world by giving a vaguely historical account of developments in the field and by stating a number of fairly simple questions that are still open.

- 2018-02-05 Lucas Ambrozio (University of Warwick)
**Critical points of the area functional: where to find them, and how to use them**

In this talk we will explain a few ideas involved in the variational approach to the construction of minimal surfaces. Moreover, we will show a few instances where the information about the index of instability of the constructed minimal surface allows the derivation of beautiful geometric applications.- 2018-02-12 Abigail Linton (University of Southampton)
**Massey products in toric topology**

With a history stemming from symplectic and algebraic geometry, toric topology began as the study of topological spaces with m-torus actions. One notable object of study in toric topology is the moment-angle complex, whose cohomology can actually be described combinatorially. In particular, this combinatorial structure provides an avenue for studying higher cohomology operations, such as Massey products. The goal of this talk is to give an introduction to these objects, and to discuss some combinatorial descriptions of Massey products in moment-angle complexes.- 2018-02-19 Richard Birkett (University of Cambridge)
**Critical Recurrence in the Mandelbrot Set**

We will introduce the essentials of holomorphic dynamics over the complex numbers, in particular using f_c(z) = z^2 + c as our archetypical function inducing the famous picture of the Mandelbrot Set. Having set up the general long term dynamical problem, we will turn our attention to a short term recurrence problem of our so-called critical orbit and get an overview of my recent research. We will finish by talking about the great open problems in the field with interesting relations to number theory and more.- 2018-02-26 Agnese Barbensi (University of Oxford)
**The Reidemeister graphs**We describe a locally finite graph naturally associated to each knot type K, called the Reidemeister graph. We determine several local and global properties of this graph and prove that the graph-isomorphism type is a complete knot invariant up to mirroring. Lastly (time permitting), we introduce another object, relating the Reidemeister and Gordian graphs, and briefly present an application to the study of DNA. Joint work with Daniele Celoria.

- 2018-03-05 Marissa Loving (University of Illinois at Urbana-Champaign)
**Least dilatation of pure surface braids**The n-stranded pure surface braid group of a genus g surface can be described as the subgroup of the pure mapping class group of a surface of genus g with n-punctures which becomes trivial on the closed surface. I am interested in the least dilatation of pseudo-Anosov pure surface braids. For the n=1 case, upper and lower bounds on the least dilatation were proved by Dowdall and Aougab—Taylor, respectively. In this talk, I will describe the upper and lower bounds I have proved as a function of g and n.

- 2018-03-05 Marko Berghoff (Humboldt University of Berlin)
**Why you shouldn't be scared of integrals**... especially if they are given to you by physicists and seem to not make any sense. This is the case for Feynman integrals, a class of integrals encountered in quantum field theory. The study of those (including the art of finding a sensible interpretation of their divergences) is not only important for perturbative calculations in high-energy physics but also provides a rich playing field for mathematics as there are numerous connections to problems in algebra, number theory, geometry and topology. In this talk, I will give an informal introduction and a broad overview of the field and discuss some of these connections in detail.

- 2018-03-12 Gerrit Herrmann (University of Regensburg)
**Thurston norm and L^2 -Betti numbers**In my talk, I will define the Thurston norm of a closed irreducible 3-manifold M. This is a semi-norm on the second homology H_2(M; Z). Morally speaking it measures the complexity of a class by embedded surfaces representing this class. I will then characterize embeddings of surfaces which release the Thurston norm by certain L^2 -Betti numbers.

- 2018-04-23 Benjamin Brück (University of Bielefeld)
**Buildings and the free factor complex**The core idea of geometric group theory is to study groups acting nicely on beautiful spaces, where both the definitions of "nicely" and "beautiful" can vary. In my talk, I will present two such spaces equipped with group actions: The building of type A_{n-1} which is associated to GL_n(Q) and the free factor complex which comes with an action of Aut(F_n), the automorphism group of the free group. I will give different definitions of these simplicial complexes, show why the descriptions are in fact equivalent and try to give an idea of basic similarities and differences between the two complexes.

- 2018-04-30 Marco Barberis (University of Warwick)
**Quasi-isometric rigidity for hyperbolic lattices**

An interesting problem in Geometric Group Theory is quasi-isometric rigidity, which is a way to deduce (weak) algebraic properties from the coarse geometry of groups given by their natural metric.I will try to set the stage of quasi-isometric rigidity problems, and then talk in detail about the results regarding hyperbolic lattices. In this case I'll try to give an idea of the tools used for the proofs, which turn out to effectively be the study of the coarse geometry of hyperbolic spaces.- 2018-05-08 Davide Spriano (ETH)
**Morse subsets in hierarchically hyperbolic spaces**

When dealing with geometric structures one natural question that arises is "when does a subset inherit the geometry of the ambient space"? In the case of hyperbolic space, the concept of quasi-convexity provides answer to this question. However, for a general metric space, being quasi-convex is not a quasi-isometry invariant.This motivates the notion of Morse subsets. In this talk we will motivate the definition and introduce some examples. Then we will introduce the class of hierarchically hyperbolic groups (HHG), and furnish a complete characterization of Morse subgroups of HHG. If time allows, we will discuss the relationship between Morse subgroups and hyperbolically-embedded subgroups. This is a joint work with Hung C. Tran and Jacob Russell.

- 2018-05-14 Katie Vokes (University of Warwick)
**Curve graphs, disc graphs and the topology of 3-manifolds**Given any closed, orientable 3-manifold M, we can always decompose M into a union of two handlebodies of the same genus, glued along their boundary surfaces by a homeomorphism. This is called a Heegaard splitting, and can be described by sets of curves in the common boundary surface which bound discs in one or other of the handlebodies. The set of curves in the surface of a handlebody which bound essential discs in the handlebody gives a subgraph of the curve graph called the disc graph, and Hempel defined a distance for a Heegaard splitting using this inclusion. We will give some background on Heegaard splittings and Hempel distance, and, time permitting, present a result on how the disc graph sits in the curve graph.

- 2018-05-21 Rachael Boyd (University of Aberdeen)
**Homological stability for Artin monoids**

Many sequences of groups satisfy a phenomenon known as homological stability. In my talk, I will report on recent work proving a homological stability result for sequences of Artin monoids, which are monoids related to Artin and Coxeter groups. From this, one can conclude homological stability for the corresponding sequences of Artin groups, assuming a well-known conjecture in geometric group theory called the K(\pi,1)-conjecture. This extends the known cases of homological stability for the braid groups and other classical examples. No familiarity with Coxeter and Artin groups, homological stability or the K(\pi,1)-conjecture will be assumed.- 2018-05-29 Samuel Colvin (University of Bristol)
**Boundaries of Hyperbolic Groups**The jungle of infinite groups is vast and unwieldly, but with the machete of geometry and the bug-spray of topology, we can attempt to explore some of its tamer wilderness. In other words, given an infinite group, we can associate to it an infinite graph which gives us a notion of ‘the geometry of a group’. Through this we can ask what kind of groups have hyperbolic geometry, or at least an approximation of it called Gromov hyperbolicity. Hyperbolic groups are quite a nice class of groups but a large one, so we introduce the Gromov boundary of a hyperbolic group and explain how it can be used to distinguish groups in this class.

Key words: Cayley graph, quasi-isometry invariants, Hyperbolic group, Gromov, boundaries, Conformal dimension.- 2018-06-04 Joe Scull (University of Warwick)
**An Introduction to Seifert Fibred Spaces**A core problem in the study of manifolds and their topology is that of telling them apart. That is, when can we say whether or not two manifolds are homeomorphic? In two dimensions, the situation is simple, the Classification Theorem for Surfaces allows us to differentiate between any two closed surfaces. In three dimensions, the problem is a lot harder, as the century long search for a proof of the Poincaré Conjecture demonstrates, and is still an active area of study today.

As an early pioneer in the area of 3-manifolds Seifert carved out his own corner of the landscape instead of attempting to tackle the entire problem. By reducing his scope to the subclass of 3-manifolds which are today known as Seifert fibred spaces, Seifert was able to use our knowledge of 2-manifolds and produce a classification theorem of his own.

In this talk I will define Seifert fibred spaces, explain what makes them so much easier to understand than the rest of the pack, and give some insight on why we still care about them today.

- 2018-06-07 Senja Dominique Barthel (EPFL)
**Spatial graphs and minimal knottedness**Spatial graph theory investigates embeddings of graphs in R3. We will define some properties of spatial graphs that can be considered as generalisation of unknottedness and see relations between them along examples. Finally, we show that there exist no minimally knotted planar spatial graphs on the torus.

- 2018-06-18 Alex Margolis (University of Oxford)
**QI rigidity of commensurator subgroups**One of the main themes in geometric group theory is Gromov's program to classify finitely generated groups up to quasi-isometry. We show that under certain situations, a quasi-isometry preserves commensurator subgroups. We will focus on the case where a finitely generated group G contains a coarse Poincaré duality subgroup H such that G=Comm(H). Such groups can be thought of as coarse fibrations whose fibres are cosets of H; quasi-isometries of G coarsely preserve these fibres. This generalises work of Whyte and Mosher--Sageev--Whyte.

- 2018-06-25 Harry Petyt (University of Warwick)
**Sphere Packings, Kissing Numbers, and Integers**In its original form the sphere packing problem asks: "What is the most efficient way to stack cannonballs?" This turns out to be unreasonably difficult to answer, even if we ask for our pile to look the same everywhere, so we might try to change the question a bit and ask: "Okay, how many cannonballs can touch (or kiss) a single cannonball at the same time?" This isn't much better, and it took over 200 years for these questions to be answered.

By the mid 19th century the complex numbers were mostly accepted by mathematicians, and motivated by their utility Hamilton was led to discover the quaternions, their four dimensional brother. Quaternions share a lot of properties with complex numbers, so even though they don't commute we can still think of them as numbers. Not long after this an eight dimensional cousin was found, which we now call the octonions, and this completes the family.

The aim of this talk is to describe a link between "integers" for these number systems and solutions to both the packing and kissing number problems in the relevant number of dimensions.

- 2018-10-01 Marco Barberis (University of Warwick)
**Introduction to Curve Complexes.**The Curve Complex, a structure which encodes information about curves on a surface, is one of the most important construction in the field since its introduction in 1981, thanks to W. J. Harvey. This complex has both applications to the study of other geometric objects and a very interesting geometric structure by itself. We will introduce the definition of the Curve Complex, with as many examples as possible, along with some application and properties. In particular we will seize the opportunity to introduce the tremendously important concept of Gromov hyperbolicity, and talk about how this is one of the main features of the Curve Complex.

- 2018-10-09 Benedict Sewell (University of Warwick)
The mystery of triviality of the prehomogeneous fibre bundle.

Ongoing work with David Mond.

A fibre bundle is a more general version of a covering space. Heuristically, it can be seen as a continuous map that locally looks like a projection (i.e. like U x F projects down to U). Perhaps the simplest question is that of triviality: is this map globally just a projection (if you move things around a bit) or not?

We’ll investigate some cool examples arising from the theory of prehomogeneous vector spaces, which will start to veer dangerously in the direction of algebraic geometry and character theory, and then at the last minute save the day with some friends from basic algebraic topology.

This talk will hopefully require no more of you than an understanding of the fundamental group and of covering spaces, so should be quite accessible!

- 2018-10-18 Stephen Cantrell (University of Warwick)
**Comparison theorems for actions on CAT(-1) spaces**

A CAT(-1) space is a metric space with a concept of negative curvature. Suppose that a group G acts nicely on a CAT(-1) space X. Such an action gives rise to two natural real valued functions on G. These are the word length function and the displacement function. In this talk we discuss these two quantities and explore different ways in which to compare them. This will be a gentle introduction to the topic – no prior knowledge required!- 2018-10-22 Joseph MacColl (LSGNT)
**Ordering free groups and the Hanna Neumann Conjecture**Given finitely-generated subgroups H and K of a free group F, Hanna Neumann conjectured the existence of a bound on the rank of their intersection coming from the individual ranks of H and K. After giving some context for the conjecture, I will describe how Mineyev proved it using properties of an ordering of the elements of F which are reflected in how the subgroups act on its Cayley graph. If time permits I will show how the same argument in fact gives a strengthened version of the result, which maximises the information obtained from the correspondence between free groups and the topology of graphs

- 2018-10-29 Michal Buran (University of Cambridge)
**Alternating quotients of RAAGs, RACGs and surface groups**

We say that a group*has many alternating quotients*if for every finite set of group elements there exists a surjection onto an alternating group, which is injective on this finite set. Right-angled Artin groups (RAAGs) are interpolation between free groups and free abelian groups. I will show that every RAAG satisfies exactly one of the following:

1. it is infinite cyclic.

2. it is a direct product of RAAGs.

3. it has many alternating quotients.

Therefore, every RAAG is a direct product of groups with many alternating quotients and infinite cyclic groups. Along the way I will prove a similar result for right-angled Coxeter groups and I will show that the fundamental groups of hyperbolic, closed, orientable surfaces have many alternating quotients.- 2018-11-05 Simon Baker (University of Warwick)
**An introduction to Fractal Geometry**The purpose of this talk is to give a gentle introduction to some topics from Fractal Geometry. I will discuss two notions of dimension that are well suited to fractal sets. We will also see how one can generate many fractals using an object called an iterated function system. If time permits I will discuss a recent paper of mine and Nikita Sidorov where we construct a fractal generated by an iterated function system that has empty interior and positive Lebesgue measure.- 2018-11-12 Luca Pol (University of Sheffield)
**What are spectra?**

This talk will be a brief and gentle introduction to stable homotopy theory: the study of those topological phenomena that they occur in essentially the same way independent of dimension. In particular, I will explain what spectra are, what they are good for and why do we care about them.- 2018-11-19 Divya Sharma (University of Münster)
**The two descriptions of tangent space(s) to Teichmüller space**

The description of tangent space(s) to Teichmüller space (of a compact Riemann surface of genus ≥ 2) comes in two different flavours: Analytic description and Cohomological description. The main goal of my talk is to introduce the notion of a harmonic vector field on the upper half plane which is an important tool to make “a bridge” between the above-mentioned descriptions. This is based on my research work.- 2018-11-26 Annette Karrer (Karlsruhe Institute of Technology)
**The contracting boundary of a CAT(0) group**To every CAT(0) group one can associate a topological space, the so-called contracting boundary. The contracting boundary measures how similar the associated group is to a hyperbolic group. A strong motivation of studying the contracting boundary is that it is a quasi-isometry invariant.

In this talk we shortly introduce quasi-isometry invariants. We define and compare hyperbolic and CAT(0) groups and discuss associated boundaries. This leads us to the definition of the contracting boundary. As an application, we look at contracting boundaries of right-angled Coxeter groups, which I study in my PhD thesis. Right-angled Coxeter groups are special CAT(0) groups, which are defined using graphs.

- 2018-12-03 Ronja Kuhne (University of Warwick)
**Polynomial-time efficient position**In this talk, we start with the very basics: we introduce surfaces, their homeomorphisms and discuss Thurston’s classification result for the latter. Naturally, this leads to the following question: given a surface homeomorphism, how (quickly) can we determine its type? In 2016, Bell and Webb gave a polynomial-time algorithm to answer this question. Their approach relies on ideal triangulations and edge coordinates. We discuss how train tracks might be used to present an alternative proof. One of the main ingredients will be a concept called efficient position. In this talk, we define what it means for a curve to be in efficient position with respect to a train track and explain how efficient position can be obtained in polynomial time.- 2019-01-14 Paul Colognese (University of Warwick)
**Volume growth on translation surfaces**

For Riemannian manifolds with negative sectional curvature, Margulis proved that there exist simple asymptotic formulae for the growth of various quantities on the manifold, including the growth of the volume of a ball in the manifold as the ball's radius tends to infinity. In recent work with my supervisor, Mark Pollicott, we turn our attention to translation surfaces which are surfaces equipped with a flat metric, except at a finite number of points which are cone-type singularities. Following the intuition that these cone-type singularities are points with high negative curvature, our work and the recent work of Alex Eskin and Kasra Rafi, proves that Margulis' asymptotic formulae extend to translation surfaces. In this talk, I aim to give some context to our work, a brief introduction to translation surfaces and an overview of how we used their geometry to prove the asymptotic results. If time permits, I'll talk about some far-out further research directions.- 2019-01-21 Matteo Barucco (University of Warwick)
**Homological Instability**A sequence of groups is said to satisfy homological stability, if the induced maps on the k^th homology groups are isomorphisms after a certain index increasing with k. In 2014 Nathalie Wahl and Oscar Randall-Williams improved a classical technique to study homological stability problems for sequences of groups admitting a braided monoidal structure. After a slight introduction to the homology of a group, I will give an idea of the categorical context of the Homological Stability Theorem that they proved. In the second part, we will play a bit with some interesting examples of sequences of groups that look like they should stabilize (in the sense that they fit perfectly in these categories), but do not, and indeed the connectivity axiom fails. This will allow us to point out how deeply Homological stability for sequences of groups of this kind seems related to the connectivity of the associated spaces, suggesting that also the opposite direction of the theorem could be true.

- 2019-01-28 Carlo Collari (Alfréd Renyi Institute Budapest)
**Bracket polynomial: applications and generalisations.**his talk is meant as an introduction to the Kauffman bracket (or bracket polynomial), which is a useful tool to define (and compute)

quantum invariants for knots, links, graphs, and 3-manifolds. The talk

is organised as follows.

First, I will introduce the notions of knots and links, and their diagrams.

Afterwards, I will define the bracket polynomial associated to a

(framed) link in S^3, which was introduced by Louis Kauffman in the

late 80's, and describe some of its properties. A suitable re-scaling

of the bracket polynomials yields the Jones polynomial, which is a

famous invariant of links, while certain evaluations of the bracket

recover the so called SU(2)-Rashitikitin-Turaev-Witten (RTW)

invariants.

Subsequently, I will describe one among the earlier applications of

the bracket polynomial: the solution of the Tait conjectures. These

conjectures were formulated by the Scottish physicist Peter Tait when

he was producing the first tabulation of knots (which were believed to

be related to atoms). I will outline how we can use the bracket to

prove some of these conjectures.

To conclude, I will discuss some generalisations of the Kauffman

bracket (namely skein modules, and possibly Khovanov homology), and

some open problems concerning them. Time permitting, I will also spend

a few words on the related invariants for 3-manifolds.- 2019-02-04 Francesca Iezzi (University of Edinburgh)
**Graphs of curves, arcs, and spheres, and connections between them**Given a surface S, the curve graph of S is defined as the graph whose vertices are simple closed curves on S up to isotopy, where two vertices are adjacent if the two corresponding curves can be realised disjointly. This object was defined by Harvey in the 80’s, and has been an extremely useful tool in the study of surface mapping class groups.

Similarly one can define the arc graph of a surface with boundary, and the sphere graph of a 3-manifold.

In this talk I will introduce all these objects, describe some of their properties and some maps between these objects. Time permitting, I will describe some joint work with Brian Bowditch, where we prove that, under particular hypothesis, there exists a retraction between the sphere graph of a 3-manifold and the arc graph of a surface.

- 2019-02-11 Elia Fioravanti (University of Oxford)
**Cross ratios on cube complexes and length-spectrum rigidity**Given a Riemannian metric on a closed manifold $M$, we can associate to every element of $\pi_1(M)$ the length of its shortest geodesic representative. Does this function determine the manifold $M$ and its metric? An open conjecture from the '80s claims that the answer should be yes if the sectional curvature is negative. The special case of hyperbolic metrics on surfaces follows immediately from the classical (9g-9)-theorem, but already the situation of more general metrics on surfaces requires a deep theorem of Otal (1990). Not much is known in higher dimension.We consider the corresponding question in the world of non-positively curved cube complexes, providing a positive general answer. Our approach relies on a new notion of 'cross ratio' for points in the Roller boundary. Joint work with J. Beyrer and M. Incerti-Medici.- 2019-02-25 Marco Linton (University of Warwick)
**Dehn Filling in Relatively Hyperbolic Groups**It is well known that the fundamental group of a closed hyperbolic manifold is a hyperbolic group. For a general hyperbolic manifold this might not be true, but our fundamental group may have a relatively hyperbolic structure. Thurstons hyperbolic Dehn filling Theorem tells us that if our manifold has some toral cusps, we can `fill' them in and obtain a new hyperbolic manifold. A natural question arises, what are the group theoretic consequences of this theorem and to what extent can this be phrased purely in terms of groups?In this talk I will introduce the cusped space of a relatively hyperbolic group. This is a 2-complex constructed from a group presentation which possesses some strong geometric properties when the group in question is relatively hyperbolic. I will discuss some properties of this space and show how Groves and Manning (2008) answered the above question for torsion free relatively hyperbolic groups. If there is time, I will also discuss how their techniques can be extended when the torsion free assumption is dropped.- 2019-03-04 Jone Lopez de Gamiz (University of Warwick)
**Coherence of right-angled Artin groups**Free groups and free abelian groups have been deeply studied and so many important properties are well-known. In this talk, I will define a class of groups that 'generalize' them, known as right-angled Artin groups. We know that subgroups of free groups or free abelian groups are again of the same type, and trivially they are coherent groups (meaning that all the finitely generated subgroups are finitely presented). I will explain that right-angled Artin groups keep these properties under certain conditions. For that, a brief introduction to Bass-Serre Theory will be necessary.

- 2019-03-11 Fiona Torzewska (University of Leeds)
**A Homotopy type Topological Quantum Field Theory**

This talk is intended to be an introduction to Topological Quantum Field Theories (TQFTs). Although initially motivated by Physics as a zero energy version of 'real' Quantum Field Theories, we will consider a purely mathematical approach, following Atiyah.Roughly speaking, by a TQFT we mean a functor from some category of cobordisms to the category of vector spaces and linear maps. I will give a specific example of a homotopy TQFT using the fundamental groupoid of the cobordism. The construction relies heavily on a groupoid version of the Van Kampen theorem. Throughout the talk I will work with a category of 1+1D cobordisms to demonstrate some calculations. If there is time I will briefly touch on the equivalence between 1+1D TQFTs and commutative Frobenius Algebras and how the construction fits into the wider class of TQFTs.

- 2019-04-29 Joe Scull (University of Oxford)
**A Beginner's guide to 3-manifolds and the Poincaré Conjecture**The Poincaré Conjecture was first formulated over a century ago and states that there is only one closed simply connected 3-manifold, hinting at a link between 3-manifolds and their fundamental groups. This seemingly basic fact went unproven until the early 2000s when Perelmann proved Thurston's much more powerful Geometrisation Conjecture, providing us with a powerful structure theorem for understanding all closed 3-manifolds.

In this talk I will introduce the results developed throughout the 20th century that lead to Thurston and Perelmann's work. Then, using Geometrisation as a black box, I will present a proof of the Poincaré Conjecture. Throughout we shall follow the crucial role that the fundamental group and hopefully demonstrate the geometric and group theoretical nature of much of the modern study of 3-manifolds. As the title suggests, no prior understanding of 3-manifolds will be expected.

- 2019-05-13 Marta Maggioni (Universiteit Leiden)
**Natural extensions for continued fractions**

Continued fractions offer a representation of real numbers that is in many ways more natural that the canonical decimal representation. The first studies date back at the end of the seventeenth century, and since then continued fractions have become more and more common, with applications in Diophantine equations and in approximations of real numbers by rationals. From a dynamical point of view, continued fractions are obtained through the iterates of a map of the unit interval into itself. In the last fifty years, metric and ergodic properties of these underlying dynamical systems have been studied for different classes of continued fractions. In this talk we will focus on the canonical planar natural extension for continued fractions, and we will use it to derive invariant measures and to compute the entropy.- 2019-05-20 Jordan Williamson (University of Sheffield)
**Constructing Algebraic Models for Equivariant Cohomology**Constructing algebraic models for topological objects is a handy technique in algebraic topology. For example, Quillen and Sullivan pioneered the study of (simply connected) spaces up to rational homotopy via an equivalence with certain commutative differential graded algebras. In this talk, we’ll focus on modelling (equivariant) cohomology theories. There are several tricks one can use to try to construct algebraic models. I’ll describe some of these methods such as Morita theory, localizations and cellularizations and demonstrate these techniques through many examples, such as modular representation theory, rational cohomology theories and rational equivariant cohomology theories on spaces with free group action.

- 2019-05-29 Anna Parlak (University of Warwick)
**Fibrations of a 3-manifold carried by the same veering triangulation**

The main topic of the talk will be pseudo-Anosov mapping tori of oriented hyperbolic surfaces. These 3-manifolds naturally come equipped with a fibration over the circle by surface fibres. However, typically they fibre in infinitely many distinct ways.Given a fibration of a fibred oriented hyperbolic 3-manifold we can build a canonical triangulation which is veering and whose 2-skeleton carries the foliation by fibres. This is an ideal triangulation of the cusped 3-manifold obtained from the initial one by drilling out the singular orbits of the suspension flow. A crucial fact for this talk is that a veering triangulation is canonical not only for a fibration, but really an invariant of a fibred face of the Thurston norm ball in the second homology group of the manifold.

I will explain how from a single veering triangulation which carries a fibration can we derive data on all fibrations lying over the corresponding fibred face. The obtained information includes the dilatation factors of their monodromies as well as the half-translation structures on the fibres with respect to which the monodromies act affinely.

- 2019-06-03 Marco Moraschini (University of Regensburg)
**Simplicial volume and amenable covers***Simplicial volume*is a homotopy invariant of compact manifolds introduced by Gromov in the early ’80s. It measures the complexity of manifolds in terms of (real) singular chains. Despite its topological meaning, simplicial volume has many applications in geometry. For instance it provides useful information about the Riemannian volume of negatively curved manifolds. However, as soon as we consider non-compact manifolds its geometric meaning is much more mysterious. Indeed, one may extend the notion of simplicial volume to non-compact manifolds by considering locally finite homology, but its behaviour is not yet well understood. Among the key ingredients for studying the simplicial volume of (non-)compact manifolds,amenable groups play a fundamental role. Recall that amenable groups are groups carrying invariant means.The aim of this talk is to investigate the relation between simplicial volume and amenable groups. More precisely, after having introduced the notion of amenable cover of compact manifolds, we will discuss a classical vanishing result for the simplicial volume. Later we will construct special amenable covers of non-compact manifolds. This will allow us to obtain the corresponding vanishing result in this setting. If there will be enough time, we will discuss a striking application of these results: the simplicial volume of the product of at least three non-compact manifolds always vanishes.Some results presented in this talk are part of a joint work with Roberto Frigerio.

- 2019-06-10 Lawk Mineh (University of Warwick)
**Boundaries of Relatively Hyperbolic Groups**

Hyperbolic groups play a central role in geometric group theory. Broadly speaking, they model fundamental groups of compact hyperbolic manifolds. Relatively hyperbolic groups form a natural extension of this theory, approximating the fundamental group of a noncompact hyperbolic manifold with finitely many cusps. There is a natural way we can assign a topological boundary to hyperbolic and relatively hyperbolic groups.It happens that some key aspects of the algebraic structure of a relatively hyperbolic are reflected in topological properties of its boundary. In particular, connectedness features like cut points have a strong relation to splittings of relatively hyperbolic groups as graphs of groups. In this talk we will discuss the idea of relatively hyperbolic groups, learn how to define their boundaries, and explore this splitting phenomenon with the help of a few examples.

- 2019-06-17 Selim Ghazouani (University of Warwick)
**On the preciousness of linear representations**I will try to convince you, through examples and an overview of important rigidity theorems, that non-trivial representations of a given abstract group into GL(n,C) are very rare objects which deserve to be treasured.

- 2019-06-24 MurphyKate Montee (University of Chicago)
**Random Groups and Cube Complexes**Random groups in the Gromov density model are known to be cubulated for d<1/6 (Olivier-Wise), admit a non-trivial cocompact action by isometries on a CAT(0) cube complex for d<5/24 (Mackay-Przytycki), and have Property (T) for d>1/3 (Zuk, Kotowski-Kotowski). Since acting on a cube complex and having Property (T) are mutually exclusive, this raises the question: What happens for densities between 5/24 and 1/3? This talk will present some of the necessary background on random groups and cube complexes, and then discuss work in progress showing that random groups at density d < 3/14 act non-trivially on a CAT(0) cube complex.- 2019-09-30 Thomas Richards (University of Warwick)
**Homotopy Shadowing and the Dynamics of Complex Hénon Maps**The homotopy shadowing framework of Ishii-Smillie allows us to construct conjugacies between dynamical systems which are not ‘close’ in the sense of structural stability. We do this by considering more general multivalued dynamical systems. Complex Hénon maps are polynomial diffeomorphisms of $\mathbb{C}^2$ and a result of Friedland and Milnor tells us that these maps are interesting to study in terms of the dynamics. We will present an alternative proof, in the homotopy shadowing framework of Ishii-Smillie, of the well known theorem that for Hénon maps $H_{p,a}$ which are small perturbations of hyperbolic polynomials $p$, the topology of the Julia set of $p$ determines the Julia set of $H_{p,a}$

- 2019-10-07 Harry Petyt (University of Bristol)
**What are hierarchically hyperbolic groups?**The mapping class group of a surface is a group of homeomorphisms of that surface, and these groups have been very well studied in the last 50 years. Most of the talk will be focused on a way to understand such groups by looking at the subsurfaces of the corresponding surface; this is the so-called "Masur-Minsky hierarchy machinery". This will lead to a non-technical discussion of hierarchically hyperbolic groups, which are a popular area of current research, and of which mapping class groups are important motivating examples. No prior knowledge of the objects involved will be assumed.- 2019-10-14 Solly Coles (University of Warwick)
**Symbolic Dynamics for Hyperbolic Flows**Smale's Axiom A flows have been studied in the field of dynamical systems and ergodic theory since their introduction in 1967. They are in some sense a generalisation of geodesic flow on a Riemannian manifold with negative sectional curvature. The work of Bowen in the early 1970s shows that they can be effectively modelled using the much more well-understood suspended flows. We introduce Axiom A and the approach of Bowen, before describing a specific application to the growth of closed geodesics on these manifolds.

- 2019-10-21 Luke Peachey (University of Warwick)
**Introduction to Comparison Theorems in Geometry**Given a Riemannian manifold, we may compare its geometric quantities with those of a suitably nice model space (e.g hyperbolic space). If these quantities are reasonably similar, a typical comparison theorem would show that the manifold retains geometric properties of the model space. In this talk we will motivate and introduce one such comparison theorem, the Bishop-Gromov inequality, and its consequence to the manifolds underlying topology.

- 2019-10-28 Catherine Bruce (University of Manchester)
**Projections of fractals; the theory of a digital sundial.**When we orthogonally project an object onto lower dimensional Euclidean space, for example from the plane onto the real line, it’s easy to spot an ‘expected’ value for the dimension of the projection. For example, we expect the projection of a curve in the plane onto the real line to have dimension one. It turns out this ‘expected’ value is correct for almost every projection, even when we extend our definition of dimension so that it can take non-integer values, due to a theorem of Marstrand in the 50s. We will talk about projections and fractal dimension until we are ready to state Marstrand’s theorem. The second part of the talk will concern what these projections actually look like. A theorem by Falconer tells us that we can construct planar sets which can be projected to pretty much whatever we please which, when extended to higher dimensions, gives us a theoretical construction of a digital sundial.

- 2019-11-04 Benedict Sewell (University of Warwick)
**α-Kakutani sequences: taking it ad infinitum**Start with the unit interval [0, 1] and fix your favourite 0 < α < 1. Starting with the trivial partition (into one interval) we define the α-Kakutani sequence recursively: at each stage, take all the intervals which have maximal length and split them into two, in the ratio α : 1 - α.Given this setup, a natural question is as follows: does the increasing set of endpoints of our partition (starting with {0, 1} for our trivial partition) become dense in a uniform way?This simple question captured the attention of a surprising number of mathematicians. We'll discuss this original problem, and move onto the "infinitely more exciting" version cooked up by me and Mark Pollicott.- 2019-11-11 Jone Lopez de Gamiz Zearra (University of Warwick)
**Geometric and Algebraic Properties of RAAGs**Right-angled Artin groups, also known as RAAGs, are a class of groups generalising free groups and free abelian groups. In this talk, I will introduce RAAGs and discuss some of the interesting properties. For example, their homology, their ends, and their relation with three manifolds. Finally, I will talk about research concerning the subgroups of direct products of RAAGs.

- 2019-11-25 Marco Linton (University of Warwick)
**Hierarchies for one-relator groups**A group splits as an HNN-extension if and only if the rank of its abelianisation is strictly positive. If we fix a class of groups one may ask a few questions about these splittings: How distorted are the vertex and edge groups? What form can the vertex and edge groups take? If they remain in our fixed class, do they also split? If so, under iteration will we terminate at something nice? In this talk we will answer all these questions for the class of one-relator groups and go through an example or two. Time permitting, we will also discuss possible generalisations to groups with staggered presentations.

- 2020-01-13 David Sheard (LSGNT)
**Cutting up surfaces and commutators in free groups**When is an element $g\in G$ of a group a product of commutators $[a,b]$? What is the minimal number of commutators $n$ such that $g=[a_1,b_1]\cdots[a_n,b_n]$? What are all possible solutions $(a_1,b_1,\dots,a_m,b_m)$ to the equation $g=[a_1,b_1]\cdots[a_m,b_m]$? Very difficult, yet important, questions --- but ones whose answers seem to lie in the darkest recesses of combinatorial group theory.

Not so! At least for $G$ a free group, these questions can all be answered elegantly and beautifully by cutting up and colouring surfaces. In this talk I shall present solutions to these problems with an emphasis on drawing nice pictures.

- 2020-01-20 Nick Bell (University of Bristol)
**Generalising Mirzakhani’s curve counting result**On any hyperbolic surface, the number of curves of length at most

*L*is finite. However, it is not immediately clear how quickly this number grows with*L*. We will discuss Mirzakhani’s breakthrough result regarding the asymptotic behaviour of this number, along with recent efforts to generalise her result using currents.- 2020-01-27 Jordan Frost (University of Bristol)
**Surfaces in Groups and how to find them**Hyperbolic groups are supposedly everywhere. After free groups, the next ‘simplest’ case is surface groups \$\pi_1(S_g)\$. Playing Ping-Pong lets us find free subgroups, what about surfaces? In this talk I will discuss some combinatorial methods to find surfaces using fatgraphs and discuss Calegari and Walker’s methods in showing a random group almost surely contains many surfaces. I will also explain what that sentence means.

- 2020-02-10 Irene Pasquinelli (Sorbonne Université)
**Cutting sequences on Veech surfaces**Consider the dynamical system given by the geodesic flow on a flat surface. Given a polygonal representation for a surface, one can code the trajectory using the sides of the polygons and thus obtain a cutting sequence.

A natural question to ask then, is whether any sequence one picks can come from a certain trajectory. In other words, can we characterise the set of cutting sequences in the set of all sequences in the alphabet? And when the answer is yes, can we recover the direction of the trajectory?

In this talk we will give an overview of the cases where these questions have been answered.

- 2020-02-17 Alice Kerr (University of Oxford)
**Product set growth in acylindrically hyperbolic groups**Finding the growth of a finitely generated group involves studying the behaviour of balls of radius n as n tends to infinity. A more general question is to ask how any finite subset of a group grows when you take its nth product. This question no longer requires the group to be finitely generated, and is usually much harder to answer. Despite this, some remarkably strong results have been found. We will discuss Safin’s result for free groups, and how it was generalised by Delzant and Steenbock for hyperbolic and acylindrically hyperbolic groups. Time permitting, we will also mention how quasi-trees can be used to improve their theorem for acylindrically hyperbolic groups.

- 2020-02-24 Tom Holt (University of Warwick)
**Harmonic Analysis on the Heisenberg manifold**PDE's can sometimes be studied by decomposing the function in question into the sum of simpler functions. This is the motivation behind Fourier analysis, which can be used to decompose functions on tori. Can a similar technique be applied to a wider class of manifolds? We'll discuss a method for decomposing functions on the Heisenberg manifold and explore how it may be applied more generally to manifolds given by the quotient of a Lie group.

- 2020-03-02 Joe Thomas (University of Manchester)
**The Laplacian on Large Genus Random Surfaces**In mathematical physics, eigenfunctions of the Laplacian on a Riemannian manifold are thought to exhibit features depending upon the geometry of the manifold. Deterministic results to demonstrate this relationship can often be somewhat weaker than what one would expect for a typical manifold. In the case of Riemann surfaces, Mirzakhani amongst others provided a collection of beautiful tools with which one can meaningfully determine probabilities of surfaces possessing certain geometric features. In this talk, I will highlight some of the key features of this theory and in particular mention how it has been recently used to understand the geometric dependence of some spectral properties possessed by eigenfunctions.

- 2020-03-09 Tom Ferragut (Université de Montpellier)
**Horocyclic product of Gromov hyperbolic spaces.**The Gromov hyperbolicity is a property to metric spaces that generalises the notion of negative curvature for manifolds.

After an introduction about these spaces, we will explain the construction of horocyclic products related to lamplighter groups, Baumslag-Solitar groups and the Sol geometry.We will describe the shape of geodesics in them, and present rigidity results on their quasi-isometries due to Farb, Mosher, Eskin, Fisher and Whyte.

- 2020-10-26 Selim Ghazouani (Orsay)
**Universality in dynamics**I will introduce a simple model for the evolution over time of the fish population in a lake. Simple computer experiments indicate that it exhibits some geometric features that are universally observed in nature (i.e do not really depend of the animal nor the environmental parameters). I will try to give a conceptual explanation for this phenomenon, which is based on a fashionable philosophy inspired from statistical mechanics called 'renormalisation group'.

- 2020-11-02 Ian Runnels (University of Virginia)
**Right-Angled Artin Subgroups of Mapping Class Groups**Using the "hierarchical" structure of the curve graph, a combinatorial gadget associated to a surface, we will show that mapping class groups contain many free, and more generally right-angled Artin, subgroups. The subgroups in question are built out of large powers of (mostly) arbitrary collections of elements, and can be made to have nice geometric and dynamical properties.

- 2020-11-09 Harry Petyt (University of Bristol)
**Injective metric spaces and groups acting on them**Injective (or hyperconvex) spaces are nice metric spaces that can be defined by how balls intersect, and it turns out that you can say a surprising amount about groups that have a good action on an injective space. The game is then to try to show that your favourite group has such an action. In this talk, I'll describe and give examples of injective spaces, discuss a cool tool for building them and actions on them, and mention some recent joint work with Thomas Haettel and Nima Hoda, in which we applied this tool to mapping class groups.- 2020-11-16 Natalia Jurga (University of St Andrews)
**Box dimensions of (×m,×n) invariant sets**We study the box dimensions of sets which are invariant under the toral endormorphism (x,y)→(mx mod 1, ny mod1) for integers n>m≥2. This is a fundamental example of an expanding, nonconformal dynamical system, and invariant sets have many subtle properties. The basic examples of such invariant sets are Bedford-McMullen carpets and, more generally, invariant sets are modelled by subshifts on the associated symbolic space. When this subshift is topologically mixing and sofic, the situation is well-understood by results of Kenyon and Peres, in particular the box dimension satisfies a natural formula in terms of entropy and the expansion coefficients m,n. In this talk we will discuss recent results with Jonathan Fraser on what happens beyond the sofic and mixing case.

- 2020-11-30 Marc Homs Dones (University of Warwick)
**Iterative periodic functions and a generalization of the Babbage functional equation.**Can you find iteratively periodic continuous functions, i.e. f^n =id? It is easy to see that rational rotations and symmetries are periodic but can you find any other examples? The classical Kerékjártó’s theorem asserts that in R^2 these are all the periodic functions (up to topological conjugacy). After covering the periodic case we will move on to study a generalization, namely f^n=f^k. We will show how with certain assumptions of regularity we can get similar results to Kerékjártó’s theorem, and how without these assumptions, such classifications are impossible.

- 2020-12-07 Valentin Huguin (Université de Toulouse)
**Unicritical polynomial maps with rational multipliers**In his paper on Lattès maps, Milnor conjectured that power maps,

Chebyshev maps and flexible Lattès maps are the only rational maps that

have an integer multiplier at each periodic point. We provide here a

partial positive answer to his question, proving that every unicritical

polynomial map that has only rational multipliers is either a power map

or a Chebyshev map.- 2021-01-11 Jone Lopez de Gamiz Zearra (University of Warwick)
**Subgroups of the direct product of graphs of groups with free abelian vertex groups**In the first part of the talk, I will motivate the study of subgroups of the direct products of limit groups over free groups and I will state some known results.I will then introduce the class of right-angled Artin groups (RAAGs) and some of its subclasses, such as the class of Droms RAAGs, tree groups and coherent RAAGs. We will see that the results that hold for limit groups over free groups are also true for limit groups over Droms RAAGs.In the second part of the talk, we will focus on finitely presented subgroups of the direct product of two tree groups (more generally, graphs of groups with free abelian vertex groups and cyclic edge groups). We will see that they are virtually H-by-(free abelian), where H is the direct product of two subgroups of tree groups. Thanks to this simple structure, the conjugacy and the membership problem are decidable in this class.

- 2021-01-18 Paul Colognese (University of Warwick)
**Flat surfaces are negatively curved**On negatively curved surfaces, the number of geodesics of a bounded length on the surface grows exponentially as the bound goes to infinity. Translation surfaces are genus g>1 surfaces which admit a flat metric except at finitely many singular points. The Gauss-Bonnet theorem tells us that these "flat surfaces" should have negative curvature on average which hints at the idea that the singularities behave as points of concentrated negative curvature. In this talk I'll explain why we can think of these flat surfaces as behaving like surfaces of negative curvature when it comes to geometric growth. I'll present some results which bolster this analogy and explore related results if I have time.

- 2021-02-01 Naomi Andrew (University of Southampton)
**Free-by-cyclic groups and their automorphisms**Free-by-cyclic groups are easy to define – all you need is an automorphism of F_n. But they are surprisingly tricky to pin down: it is difficult though by no means impossible to extract properties of the group from that automorphism. We’ll introduce these groups and some of their properties, and see how they connect to properties of the defining automorphism. Then we’ll move on to look at their automorphism groups and how we can study them by constructing useful actions on trees.

- 2021-02-08 Solly Coles (University of Warwick)
**Helicity and tangled orbits of Anosov flows**The helicity of a volume-preserving flow is one of the basic invariants studied in fluid dynamics. Its diffeomorphism invariance makes it useful for solving certain variational problems, particularly in magnetohydrodynamics. Arnold proposed that one can characterise helicity in terms of the linking numbers of knots constructed by closing up trajectories of the flow with geodesic arcs. In this talk we will describe Arnold's characterisation, followed by a new characterisation for the case of Anosov flows, in terms of the linking of periodic trajectories. We will also discuss an application of this characterisation to a particular class of (almost) geodesic flows.

- 2021-02-15 Jean Pierre Mutanguha (Max Planck Institute)
**When invariants are equivalent.**A surface homeomorphism can be thought of as a dynamical system; its mapping torus is a 3-manifold that fibers of a circle; and a Riemannian metric on this 3-manifold determines a path metric on its universal cover. I will discuss how, in some cases, dynamical invariants of the surface homeomorphism, topological invariants of the mapping torus, and geometric invariants of the universal cover are all equivalent. For example, assuming the surface is closed and hyperbolic, then the following are equivalent:1) the surface homeomorphism has finite order up to isotopy;

2) the mapping torus is finitely covered by a product of the surface and the circle; and

3) the universal cover "is" the product of the hyperbolic plane and the real line.This is mostly an expository talk of "classical"/old results but, if time permits, I will end the talk with a discussion on the few results that have been translated into the setting of free group automorphisms.

- 2021-02-22 Argyrios Christodoulou (Queen Mary)
**The Hausdorff dimension of self-projective fractals**In this talk we discuss iterated function systems (IFS) on real projective space. We present a formula for the Hausdorff dimension of the attractor of an IFS that involves the minimal root of its pressure function. Moreover, we discuss the necessary contraction properties of projective IFS that make such a formula work. This is joint work with Natalia Jurga that generalises a recent result of Solomyak and Takahashi.- 2021-03-01 Amlan Banaji (University of St Andrews)
**Intermediate dimensions and infinite conformal iterated function systems**The intermediate dimensions, introduced by Falconer, Fraser and Kempton, are a family of dimensions which interpolate between two well-studied notions of fractal dimension, namely the Hausdorff and box dimensions. In this talk we will summarise some key properties of the intermediate dimensions, discuss general bounds, and demonstrate how the interpolation between the Hausdorff and box dimensions can be recovered for sets whose intermediate dimensions are discontinuous at 0. We will then discuss joint work in progress with Jonathan Fraser in which we give a formula for the intermediate dimensions of limit sets of infinite conformal iterated function systems. Time permitting, we will consider applications to sets of irrational numbers whose continued fraction expansions have restricted entries.

- 2021-03-08 Bernhard Reinke (Aix-Marseille Université)
**The Weierstrass root finder is not generally convergent**Finding roots of univariate polynomials is one of the fundamental tasks ofnumerics, and there is still a wide gap between root finders that are well understood intheory and those that perform well in practice.We will give an overview of root-finding methods andtheir interpretation as complex dynamical systems. The main focuswill be the Weierstrass/Durand-Kerner method and its similarities and differencesto the Newton and the Ehrlich-Aberth methods.In particular, we show how to use methods from computer algebra to investigate (and/or establish) the existence ofattracting periodic cycles, as well as diverging orbits, and present explicit examples of both phenomena for the Weierstrass method.We thereby settle the conjecture by Smale that the Weierstrass root finder is not generally convergent.This talk is based on joint work with Dierk Schleicher and Michael Stoll.- 2021-03-15 Ben Sewell (University of Warwick)
**The Hausdorff dimension of the Rauzy gasket—who cares?**The Rauzy gasket is, on the face of it, a dodgy version of the Sierpinski gasket—it doesn’t even have a Wikipedia page! It falls to me to convince you of its interestingness, with its myriad connections to

· 2D continued fraction algorithms,

· triangular tiling billiards,

· monocrystal models, and

· periodic surfaces.

Moreover, nobody knows its fractal dimension, and that’s probably important, or at least interesting. Vast papers and preprints involving a Fields medalist and a Marseillaise have carved an upper bound of 1.825.

If our potatoes are good, we’ve done better and with less effort. It all boils down to a dynamical threshold, some Markov stuff, and my favourite, renewal theory. Crucially, there will be good pictures.

This work is done under the guidance of the benevolent Mark Pollicott.

- 2021-04-26 Caroline Davis (Indiana University)
**I’m Rubber and You’re Glue: Elementary methods for complex maps**Come back to primary school with me, and let’s play with rubber bands and paste together funny shapes.

We’ll look at the nice case of “post critically finite hyperbolic” rational maps, where a recent result of Dylan Thurston shows how thinking in terms of rubbery “graph spines“ can detect when certain topological maps on the sphere are secretly rational maps in disguise. After discussing this theorem, we’ll apply it to the question of when can the Julia sets of two quadratic polynomials be glued together so that the resulting object is the Julia set of a (quadratic) rational map?

- 2021-05-10 Sam Hughes (University of Southampton)
**Lattices in non-positive curvature**In this talk I will introduce the study of lattices in locally compact groups through their actions on CAT(0) spaces. This is an extremely rich class of groups including S-arithmetic groups acting on products of symmetric spaces and buildings, right angled Artin and Coxeter groups acting on polyhedral complexes, Burger-Mozes simple groups acting on products of trees, and the recent CAT(0) but non biautomatic groups of Leary and Minasyan. If time permits I will discuss some of my recent work related to the Leary-Minasyan groups.

- 2021-05-17 Rylee Lyman (Rutgers University)
**Outer Space for free groups and free products of finite and****cyclic groups**Outer Space was introduced by Culler and Vogtmann to perform a similar function for the outer automorphism group of a free group that the Teichmüller Space of a surface performs for the mapping class group of that surface. Later Guirardel and Levitt defined an Outer Space for a free product; its properties are most similar to Culler and Vogtmann's Outer Space when the factors in the free product are finite or cyclic. In this expository talk we'll meet these Outer Spaces and their spines, calculate their dimension and learn how to move around in them. If time permits, I'll say a little about my current interest in the behavior of the spine of Guirardel–Levitt Outer Space at infinity.

- 2021-05-24 Alex Evetts (Erwin Schrödinger Institute)
**Equations and growth in virtually abelian groups**A familiar notion from algebraic geometry is an algebraic set (or variety), the zero-locus of a finite system of polynomials. In this talk I will discuss a group-theoretic analogue – solution sets to systems of equations in (finitely generated) groups. I will briefly talk about different approaches for understanding these sets, and then focus on their relative growth series. I will explain Benson’s 1983 structure results for virtually abelian groups, and how they can be used to study the growth of solution sets to equations in this class of groups. This is joint work with Alex Levine (Heriot-Watt).

- 2021-05-31 Ioannis Iakovoglou (Université de Bourgogne)
**On the relations between equilibrium states of stable systems**Dynamical systems even in small dimensions and on manifolds of modest topology, like the circle, include a plenitude of different behaviours: systems with an abundance of periodic orbits, systems with no periodic orbits, systems resisting to small perturbations (known as stable systems), systems where a small perturbation changes everything (known as unstable systems)... By a theorem of S.Smale a stable system contains a finite number of equilibrium states (known as basic pieces). Any orbit that is close to an equilibrium will try in the future to get closer and closer to a perfect equilibrium state. Equilibrium states can be periodic orbits, but can also contain much more chaotic and complex dynamics. In next week's talk, we will discuss the question of what kind of relations (known as Smale orders) can exist between the equilibrium states of a 2 dimensional stable system. Does Nature impose restrictions on the set of possible relations or is every Smale order realisable?- 2021-06-07 Sami Douba (McGill University)
**Virtually unipotent elements of finitely generated groups**

Suppose a group G contains an infinite-order element g such that every finite-dimensional linear representation of G maps some nontrivial power of g to a unipotent matrix. As observed by Button, since unitary matrices are diagonalizable, and since a unipotent matrix is torsion if its entries lie in a field of positive characteristic, such a group G does not admit a faithful finite-dimensional unitary representation, nor is G linear over a field of positive characteristic. We discuss manifestations of the above phenomenon in various finitely generated groups.- 2021-06-14 Lindsay Dever (Bryn Mawr College)
**Counting geodesics on compact hyperbolic 3-manifolds**The study of hyperbolic 3-manifolds draws deep connections between number theory, geometry, topology, and quantum mechanics. An important geometric invariant of a hyperbolic 3-manifold is the set of its closed geodesics, which are parametrized by their length and holonomy. Sarnak and Wakayama showed in 1999 that holonomies of geodesics of increasing lengths become equidistributed throughout the circle. In this talk, I will present new results including a refined count of length and holonomy which implies equidistribution in shrinking intervals. In addition, I will introduce the Selberg trace formula, which relates geometric information to spectral information, and is the primary tool for geodesic counting problems.

- 2021-06-28 David Parmenter (University of Warwick)
**Gibbs measures from an unstable's point of view**Gibbs measures are important invariant measures in ergodic theory and have been since their introduction in the 60s. We introduce the relevant ideas of thermodynamic formalism and hyperbolic dynamical systems through classic examples such as hyperbolic toral automorphisms. Intuitively, a system is hyperbolic if the tangent space at each point can be split into expanding (unstable) and contracting (stable) directions. In line with this intuition, the unstable manifold at a point $x$ is defined to be the set of points whose backwards orbit is close to that of $x$ (and so the forward orbits essentially expand.) It is well known that looking at the unstable direction produces interesting dynamics, for example, the introduction of the SRB-measure, and in this talk we will present a construction of Gibbs measures as the limit of push forward measures supported on pieces of unstable manifold.- 2021-10-11 Marco Linton (University of Warwick)
**Dynamics and isoperimetric inequalities of double mapping tori**A folklore conjecture of Gromov says that discrete groups that admit finite classifying spaces are hyperbolic if and only if they contain no Baumslag--Solitar subgroups. Given an automorphism of a free group f:F -> F, we can construct a new group called the mapping torus M(f) of f. By relating the dynamics of f to the geometry of M(f), Brinkmann showed that this conjecture holds for all M(f). I will first go over the necessary background. Then, I will discuss the advancements that have occurred since Brinkmanns result. Finally, I will introduce a more general construction, the double mapping torus of a pair of homomorphisms f, g:F -> G, and see how much structure can still be recovered.- 2021-10-18 Kiho Park (Korea Institute for Advanced Study)
**Transfer operators and limit laws for typical cocycles**We show that generic matrix cocycles (those called “typical" cocycles) over irreducible subshifts of finite type obey several limit laws with respect to the unique equilibrium states for Hölder potentials. These include the central limit theorem and the large deviation principle. The transfer operator and its spectral properties play key roles in establishing these limit laws. This is joint work with Mark Piraino.- 2021-10-25 Leticia Pardo Simón (University of Manchester)
**The maximum modulus set of an entire function**The set of points where an entire function achieves its maximum modulus is known as the maximum modulus set, and usually consists of a collection of disjoint analytic curves. In this talk, we discuss recent progress on the description of the features that this set might exhibit. Namely, on the existence of discontinuities, singleton components, and on its structure near the origin. This is based on joint work with D. Sixsmith and V. Evdoridou.