# 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.