# Abstracts

- 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.

- 2021-11-01 Alex Levine (Heriot-Watt University)
**Groups, equations and languages**Ever since Makanin proved that the satisfiability of systems of equations in free groups was decidable, there has been significant interest in the study of equations in groups. There have been a few attempts to describe solutions to specific equations using languages. Since then, however, Ciobanu, Diekert and Elder made a leap by using EDT0L languages to describe the sets of solutions to systems of equations in free groups. Following this there have been several attempts use languages for systems of equations in other classes of groups. We give a brief introduction to this area.

- 2021-11-15 James Everitt (University of Warwick)
**Transitivity Properties of Dynamical Systems on Covers**A dynamical system $(X, f)$ is topologically transitive if there exists some point $x\in X$ with a dense forward orbit. Given a manifold $M$ with a regular covering space, we can lift a flow on the manifold to a flow on the covering space. If the flow on $M$ is topologically transitive, it is interesting to ask when the lifted flow will also be transitive. A result of Eberlein, which generalised work of Hedlund, showed that in the case of geodesic flows on compact manifolds with negative sectional curvature, transitivity only fails on the lift to the universal cover of the manifold $M$. If $(M, \phi)$ is a transitive Anosov flow, recent work of Gogolev and Rodriguez Hertz showed that the lift to the universal abelian cover with covering group $H_1(M,\mathbb{Z})$ will be transitive if and only if the Anosov flow $\phi$ is homologically full (every element of $H_1(M,\mathbb{Z})$ can be represented by a periodic $\phi$-orbit). We will discuss these results and suggest ways they might be extended.

- 2021-11-22 Alex Kapiamba (University of Michigan)
**The Yoccoz inequality**Understanding the geometry of the Mandelbrot set is a central part of complex dynamics. A key result in this area is the Yoccoz inequality, which bounds the diameter of the limbs of the Mandelbrot set. In this talk, we will give an introduction to the Yoccoz inequality, its proof, and its applications. At the end, we will discuss new results which improve the Yoccoz inequality in a few special cases.

- 2021-11-29 Kevin Li (University of Southampton)
**Generalised Lusternik--Schnirelmann category of groups**Abstract: For a space $X$, the idea of generalised categorical invariants is as follows: Consider covers of $X$ by ``small" subsets $U_0, U_1,\ldots,U_n$ and determine the minimal possible cardinality $n$ of such a cover. Here a subset $U_i$ is considered to be small if the image of $\pi_1(U_i)\to \pi_1(X)$ lies in a prescribed family of subgroups of $\pi_1(X)$.

One obtains interesting group invariants by considering $X=BG$ and various families of subgroups of $G$. Important special cases are: cohomological dimension, virtual cohomological dimension, amenable category, and Farber's topological complexity. My leading question is to find an algebraic characterisation of these invariants in general. I will present an approach via classifying spaces for families of subgroups that yields answers in certain cases.

- 2021-12-06 Mario Shannon (Penn State University)
**Birkhoff sections on suspension Anosov and affine structures**Despite the good comprehension that we have nowadays about the asymptotic dynamical behaviour of a general Anosov flow, classification of the different orbital equivalent classes rest a major subject nowadays. In the particular case of dimension three, a lot of different examples of Anosov flows can be constructed using

*surgery methods*, which shows that the set of different equivalence classes is not at all simple to describe.If we restrict to the special subfamily of transitive 3-dimensional Anosov flows, each flow has (many) associated open book decompositions of the 3-manifold with pseudo-Anosov monodromy, constructed via an immersed Birkhoff section. Since pseudo-Anosov homeomorphisms can be classified by terms of a combinatorial invariant, there is a hope of producing combinatorial invariants of the orbital equivalence class of these Anosov flows in terms of those available for pseudo-Anosov. The problem to solve is :

*Given two Birkhoff sections with**pseudo-Anosov monodromy, how to determine if both correspond to the same**flow?*We study a simpler question related to the previous one:

*Given an open book decomposition with pseudo-Anosov monodromy, can we determine whether or not the corresponding Anosov flow is a suspension Anosov flow ?*In this talk we will explain how to translate this question into a problem of existence of some particular*singular affine structures*associated with the pseudo-Anosov monodromy. In turn, we can provide a natural bijection between the set of genus one Birkhoff sections of a suspension Anosov flow and these affine structures.- 2022-01-17 Xabier Legaspi Juanatey (Instituto de Ciencias Matemáticas)
**On growth rate of quasi-convex subgroups and constricting elements**Let $G$ be a group acting properly on a geodesic metric space $X$. Let $H$ be a subgroup of $G$. Have you ever wondered how big is $H$ inside $G$ with respect to the geometry induced by $X$? It turns out that the exponential growth rate $\omega(H,X)$ is a real number that gives an estimation of this ``size''. In this talk, we will see that if $G$ acts on $X$ with a constricting element -- namely, an element with ``hyperbolic-like'' manners --, then we have the strict inequality $\omega(H,X)<\omega(G,X)$ provided that $H$ is quasi-convex in $X$ and of infinite index. If time permits so, we will see that another invariant $\omega(H\backslash G,X)$ can be defined for the quotient space $H\backslash G$ -- thought as a Schreier graph -- and that, under the same hypothesis, we have the equality $\omega(H\backslash G,X)=\omega(G,X)$. Applications include relatively hyperbolic groups, $CAT(0)$ groups and hierarchically hyperbolic groups containing a Morse element.

- 2022-01-31 Gustavo Rodrigues Ferreira (Open University)
**Internal dynamics of wandering domains: a quick guide**We discuss the different behaviours for the hyperbolic metric in wandering domains of meromorphic functions. We start with a brief overview of Benini et al.’s classification of simply connected wandering domains in 2019, but focus on how the multiply connected setting offers at the same time more rigid and more intricate possibilities. Finally, we explore conditions under which multiply connected wandering domains can mimic their simply connected counterparts.

- 2022-02-07 Macarena Arenas (University of Cambridge)
**A cubical Rips construction**The Rips exact sequence is a useful tool for producing examples of groups satisfying combinations of properties that are not obviously compatible. It works by taking as an input an arbitrary finitely presented group Q, and producing as an output a hyperbolic group G that maps onto Q with finitely generated kernel. The ``output group" G is crafted by adding generators and relations to a presentation of Q, in such a way that these relations create enough ``noise" in the presentation to ensure hyperbolicity. One can then lift pathological properties of Q to (some subgroup of) G. Among other things, Rips used his construction to produce the first examples of incoherent hyperbolic groups, and of hyperbolic groups with unsolvable generalised word problem.

In this talk, I will explain Rips’ result, mention some of its variations, and survey some tools and concepts related to these constructions, including small cancellation theory, cubulated groups, and asphericity. Time permitting, I will describe a variation of the Rips construction that produces cubulated hyperbolic groups of any desired cohomological dimension.

- 2022-02-14 Luke Jeffreys (University of Bristol)
**Ratio-optimising pseudo-Anosovs in the Johnson filtration**Pseudo-Anosov homeomorphisms are a special class of surface homeomorphism. Such a homeomorphism is said to be ‘ratio-optimising’ if it further satisfies a condition relating its actions on two important spaces associated to the surface – the Teichmüller space and the curve graph. The Johnson filtration is a well-studied sequence of subgroups of the mapping class group of a surface of algebraic and geometric interest. In this talk, I will discuss how to construct ratio-optimising pseudo-Anosovs in the Johnson filtration and how such constructions are connected to combinatorial objects called meanders.

- 2022-02-28 Will Hide (Durham University)
**Spectral gaps for random hyperbolic surfaces with cusps.**We shall study the discrete spectrum of the Laplacian on finite-area hyperbolic surfaces, focusing on the size of the first non-zero eigenvalue i.e. the spectral gap. The spectral gap of a surface contains information about its connectivity and the dynamics of its geodesic flow. We are interested in the size of the spectral gap for random surfaces.First we shall introduce a random model, arising from the Weil-Petersson metric on moduli space. Then we shall discuss some recent results in this model for compact surfaces and their extension to the non-compact case. In particular, we prove the existence of a positive uniform spectral gap of explicit size for random large genus non-compact surfaces.- 2022-03-07 Will ORegan (University of Warwick)
**Dimension drop for projected self-similar measures**Let G be a finite group of isometries of Euclidean space which fix the unit cube. Consider an iterated function system consisting of compositions of an element of G, a contraction scaling, and a translation. It is known that if G is the trivial group then any invariant measure (of which self-similar measures are a subset) on K will have a rational projection for which the entropy dimension will drop from 1. In this talk we will discuss the generalisation for these more general carpets and show that for every self-similar measure on K one can obtain the same result. Time permitting we will discuss progress on the conjecture that these carpets are tube-null.

- 2022-03-14 Thomas Richards (University of Warwick)
**External rays, monodromy, and complex Hénon maps**In this talk I will give an introduction to complex dynamics in one-dimension and the monodromy problem considered by Blanchard, Devaney, and Keen. I will then discuss the analogous problem for complex Hénon maps, the difficulties arising in this setting, and experimental work, building on that of Lipa, that we are using to explore this problem.

- 2022-04-25 Selim Ghazouani (Imperial)
**Are dynamicists of any use to physicists?**I will discuss an open problem that puts dynamicists to shame. Loosely speaking, we will consider the following widely open question : is your typical dynamical system chaotic or predictable?

- 2022-05-09 Tania Gricel Benitez (University of Liverpool)
**Transcendental entire functions with pseudo-arcs Julia continua**A transcendental entire function that is hyperbolic with connected Fatou set is said to be of disjoint type. It is known that the Julia set of such entire functions may contain topological objects which could be considered ''pathological''. In this sense, we may ask how pathological the Julia set could become. In this presentation, we discuss how to construct a disjoint type function whose Julia set is a ''bouquet of pseudo-arcs''.- 2022-05-16 Leonidas Daskalakis (Rutgers University)
**Quantitative forms of pointwise convergence of Ergodic Averages**In this talk, I will describe a standard two-step procedure for establishing pointwise convergence of Ergodic Averages. We will apply that procedure for a wide class of non-conventional ergodic averages and we will use quantitative forms of pointwise convergence to carry out those two steps and establish pointwise convergence on L^1 (as a bonus this disproves a conjecture of Rosenblatt–Wierdl).

- 2022-05-28 Amlan Banaji (St. Andrews)
**Dimensions of infinitely generated self-conformal sets**

If an iterated function system consists of a countably infinite number of contractions then the Hausdorff, box and Assouad dimensions of the limit set can all differ, even if the contractions are assumed to be conformal and well-separated. After explaining what is known about the Hausdorff, box and intermediate dimensions of these limit sets, we will focus on the Assouad-type dimensions, which quantify the `thickest’ part of the set. In particular, we present bounds for the Assouad spectrum which are sharp in general. The Assouad spectrum of the class of examples which we use to show that these bounds are sharp can display interesting behaviour, such as having two phase transitions. This is based on joint work with Jonathan Fraser..

- 2022-05-30 Amlan Banaji (St. Andrews)
**Dimensions of infinitely generated self-conformal sets**If an iterated function system consists of a countably infinite number of contractions then the Hausdorff, box and Assouad dimensions of the limit set can all differ, even if the contractions are assumed to be conformal and well-separated. After explaining what is known about the Hausdorff, box and intermediate dimensions of these limit sets, we will focus on the Assouad-type dimensions, which quantify the `thickest’ part of the set. In particular, we present bounds for the Assouad spectrum which are sharp in general. The Assouad spectrum of the class of examples which we use to show that these bounds are sharp can display interesting behaviour, such as having two phase transitions. This is based on joint work with Jonathan Fraser.

- 2022-06-13 Vilma Orgoványi (Budapest University of Technology & Economics)
**Orthogonal projections of the random Menger sponge**To model turbulence, Mandelbrot introduced a family of statistically self-similar random sets E which is now called fractal percolation or Mandelbrot percolation. This is a two-parameter (M, p) family of random sets in R^d, where M>1 is an integer and 0<p<1 is a probability. The inductive construction of E is as follows. The closed unit cube of R^d is divided into M^d congruent cubes. Each of them is retained with probability p and discarded with probability 1-p.

In the retained cubes, we repeat this division and retaining/discarding process independently of everything ad infinitum or until there are no retained cubes left. The random set E that remains after infinitely many steps (formally: the intersection of the unions of retained cubes on all construction stages) is the fractal percolation set.

Using an analogous method it is possible to define more general random fractals for example the random Menger sponge. In this talk, I will study the orthogonal projections of the random Menger sponge to straight lines from the point of the positivity of the Lebesgue measure, and the existence of interior points.

This is based on joint work with Károly Simon.

- 2022-06-20 Marc Homs Dones (University of Warwick)
**Simplest bifurcation diagram of vector fields on a torus**In this talk we will explore the bifurcation diagram of a family of ODEs on a torus. This family is of interest as we can proof it exhibits the simplest global bifurcation diagram of any "monotone" family on the torus. In doing so I will present classical bifurcations, such as a saddle-node, as well as more specific ones that only come up on the torus.

- 2022-06-27 Nicholas Fleming Vázquez (University of Warwick)
**Optimal iterated moment bounds for nonuniformly hyperbolic maps**Let (M,T,mu) be a measure-preserving dynamical system. Moment bounds for iterated sums of the form

\sum_{0≤i<j≤n} v∘T^i w∘T^j arise when proving deterministic homogenisation (convergence of a fast-slow system to a stochastic differential equation).

In the first part of the talk, we introduce the deterministic homogenisation problem and motivate its connections

with statistical properties of the dynamical system T. In particular, we give examples of systems for which it is now possible to prove deterministic homogenisation. Time permitting, we will also discuss the Functional Correlation Bound, which is the main technical tool used in our proofs.- 2022-10-07 Marco Linton (University of Oxford)
**Poison subgroups for hyperbolic groups**

It is a well-known result that hyperbolic groups cannot contain certain `poison' subgroups. A lot of progress has been made towards understanding when the converse to this statement also holds. This includes several positive results, but also several negative results. In this talk, I will introduce hyperbolic groups, discuss some of these results and present the current state of the art for the class of one-relator groups.- 2022-10-17 Alexander Baumgartner (University of Warwick)
**Coding Flows on Homogenous spaces**

The idea of coding geodesics on the modular surface was first due to Emil Artin (1924). In this talk I consider a related flow on a fibration of this modular surface with torus fibres. This has applications for inhomogeneous Diophantine equations, Teichmüller flows and the distribution of gaps in √n mod 1.- 2022-10-24 Marley Young (University of Cambridge)
**Effective bounds on S-integral preperiodic points for polynomials**Through an analogy between torsion points on abelian varieties and preperiodic points of rational functions, problems in unlikely intersections motivated S. Ih to conjecture the finiteness of preperiodic points for a rational function on P^1, defined over a number field, which satisfy a certain integrality condition.This conjecture remains open, but equidistribution theorems, which provide a deep connection between geometry and dynamics, have led to partial results. Namely, the conjecture has been proved in the case of very special maps, and under certain local conditions. I will discuss how to obtain effective results in the latter context (following work of Petsche), and how to compute explicit bounds in the case of a unicritical polynomial. This involves studying the geometry of Julia sets, and making p-adic analytic arguments at non-archimedean places.- 2022-10-31 Samuel Kittle (University of Cambridge)
**Absolutely continuous self-similar measures with exponential separation**We prove that a self-similar measure is absolutely continuous providing it satisfies a condition depending on its Garsia entropy, contraction ratio, and the separation between different points in approximations to the self-similar measure. In the special case of Bernoulli convolutions, we show that the Bernoulli convolution with parameter lambda is absolutely continuous providing lambda satisfies a simple condition in terms of lambda and its Mahler measure.

- 2022-10-3 Aleksi Pyörälä (University of Oulu)
**Normal numbers in self-conformal sets**During recent years, the prevalence of normal numbers in natural subsets of the reals has been an active research topic in fractal geometry. The general idea is that in the absence of any special arithmetic structure, almost all numbers in a given set should be normal, in every base. In our recent joint work with Balázs Bárány, Antti Käenmäki and Meng Wu we verify this for self-conformal sets on the line. The result is a corollary of a uniform scaling property of self-conformal measures: roughly speaking, a measure is said to be uniformly scaling if the sequence of successive magnifications of the measure equidistributes, at almost every point, for a common distribution supported on the space of measures. Dynamical properties of these distributions often give information on the geometry of the uniformly scaling measure.

- 2022-11-07 Matteo Tabaro (Imperial College London)
**Existence Conditions of ACIPs for Real Multimodal Maps**In the logistic family of real quadratic polynomials there is an open and dense subset of maps that exhibit hyperbolic behaviour. For these maps, Lebesgue almost every initial point tends to an attracting periodic cycle. The remaining maps, which form a sizeable set in a measure theoretical sense, showcase seemingly chaotic dynamics. In this talk we will explore the concepts of physical and absolutely continuous invariant probability measures (ACIPs) and how they help us order such chaos. We will discuss sufficient metric and topological criteria to establish the existence of such measures and focus on the difference between all such conditions.

- 2022-11-14 Ainoa Murillo (University of Barcelona)
**Periodic perturbation of a 3D conservative flow with a heteroclinic connection to saddle-foci.**We shall consider a 3D volume-preserving flow with a 2-dimensional heteroclinic connection to saddle-foci, obtained by truncation of the normal form of a generic unfolding of a Hopf-Zero bifurcation. First, we describe the phase space of the system. Then we consider the effect of a non-autonomous periodic perturbation on it. The perturbation introduces an additional frequency, which interacts with the intrinsic frequency (which is related to the multiplier of the foci). In particular, we will focus on the splitting of the 2D stable and unstable invariant manifolds and illustrate its quasi-periodic asymptotic behaviour.

- 2022-11-21 Dimitrios Charamaras (EPFL)
**Joint ergodicity of additive and multiplicative measure-preserving systems**In this work, we study a fundamental principle of multiplicative number theory in the context of ergodic theory. According to this, multiplicative structures of the integers should behave independently of additive structures. In BR, Bergelson and Richter explored this concept in the context of topological systems. Motivated by their work, we apply this to measure-preserving systems. In this setting, the ``independence'' between two systems or actions (on the same probability space) can be captured by the notion of joint ergodicity. Therefore, the main goal is to give necessary and sufficient conditions to determine when additive and multiplicative systems are jointly ergodic. Furthermore, our results and methods can be employed to provide combinatorial applications regarding large sets of integers that contain configurations with both additive and multiplicative objects.

- 2022-11-28 David Martí-Pete (University of Liverpool)
**Wandering domains in transcendental dynamics: topology and dynamics**For a transcendental entire or meromorphic function, the Fatou set is the largest open set on which its iterates are defined and form a normal family. A wandering domain is a connected component of the Fatou set which is not eventually periodic. The first example of a transcendental entire function with a wandering domain was constructed by Baker in the 1970s.

Wandering domains, which do not exist for rational maps, play an important role in transcendental dynamics and in the last decade there has been a resurgence in their interest. For example, Bishop proved that the Julia sets of transcendental entire functions can have Hausdorff dimension 1 by constructing a function with wandering domains.

Wandering domains are very diverse in terms of both their topology (simply connected or multiply connected) and their dynamics (escaping, oscillating or, perhaps, even have bounded orbit). Recently, Boc Thaler proved the surprising result that every bounded regular domain such that its closure has a connected complement is the wandering domain of some transcendental entire function. Inspired by this result, together with Rempe and Waterman, we were able to obtain wandering domains that form Lakes of Wada.

In this talk, I will describe the main topological and dynamical properties of wandering domains (and their boundaries) and give an overview of the current open questions.- 2022-12-05 Ignacio del Amo (University of Exeter)
**Transition times of a particle driven by Lévy noise in a double well potential**- 2022-12-05 Vasiliki Evdoridou (University of Liverpool)
**Simply connected wandering domains**Let f be a transcendental entire function. Considering the iterates of f we divide the complex plane into two sets: the Fatou, or "stable", set, and the Julia, or "chaotic", set. The connected components of the Fatou set of f that are not eventually periodic are called wandering domains. Although Sullivan's celebrated result showed that rational maps have no wandering domains, transcendental entire functions can have wandering domains. In this talk we will discuss a classification of simply connected wandering domains and the construction of examples of different types of such domains. Finally, we will present a new result on the existence of unbounded simply connected wandering domains where points escape to infinity fast. Based on joint work with A. M. Benini, N. Fagella, P. Rippon, G.Stallard and A. Glücksam, L. Pardo-Simón.- 2023-01-07 Lawk Mineh (University of Southampton)
**Combining quasiconvex subgroups in hyperbolic groups**Hyperbolic groups form a class of groups of central importance in geometric group theory. They include many interesting examples of groups while still being well-behaved enough to admit analysis by a large suite of tools. Quasiconvex subgroups are, geometrically speaking, the "natural" subgroups of hyperbolic groups. After giving a brief introduction to these groups and some of the tools available to study them, we will see that under fairly general circumstances, pairs of quasiconvex subgroups can be combined to create other quasiconvex subgroups. Time permitting, I will discuss a generalisation to the class of "relatively hyperbolic" groups.- 2023-01-16 Andrew Mitchell (University of Birmingham)
**A new class of measures satisfying the multifractal formalism**

In this talk I will introduce a new class of measures satisfying the multifractal formalism, which exhibit properties not previously witnessed. These measures arise from random substitutions, which are a generalisation of (deterministic) substitutions where the substituted image of a letter is determined by a Markov process. I will provide an introduction to random substitutions and show how, to a given random substitution, one can associate a dynamical system and ergodic measure in a natural way, before moving on to present the key multifractal properties of these measures. This talk is based on joint work with A. Rutar (University of St Andrews).- 2023-01-23 Damian Dabrowski (University of Jyväskylä)
**Vitushkin’s conjecture and sets with plenty of big projections**In this talk I will describe recent progress made on Vitushkin’s conjecture, an old problem lying at the intersection of geometric measure theory, harmonic analysis, and complex analysis. Based on joint work with Michele Villa.

- 2023-01-30 Borys Kuca (University of Crete)
**Multiple ergodic averages along polynomials and joint ergodicity**Furstenberg’s dynamical proof of the Szemerédi theorem initiated a thorough examination of multiple ergodic averages, laying the grounds for a new subfield within ergodic theory. Of special interest are averages of commuting transformations with polynomial iterates, which play central role in Bergelson and Leibman’s proof of the polynomial Szemerédi theorem. Their norm convergence has been established in a celebrated paper of Walsh, but for a long time, little more has been known due to obstacles encountered by existing methods. Recently, there has been an outburst of research activity which sheds new light on their limiting behaviour. I will discuss a number of novel results, including new seminorm estimates and limit formulas for these averages. Additionally, I will talk about new criteria for joint ergodicity of general families of integer sequences whose potential utility reaches far beyond polynomial sequences. The talk will be based on recent papers written jointly with Nikos Frantzikinakis.

- 2023-02-06 Fanni Sélley
**Real and complex continued fractions**In the first part of the talk, I will recall the standard definition of the continued fraction expansion of a real number, then introduce a family of more general $\alpha$-continued fraction expansions. Finite convergents of these expansions give good rational approximations. I will review some well-known results on the goodness of these approximations, and ergodic properties of the interval maps generating the expansions.

In the second part, I will talk about Hurwitz's complex continued fractions and their $(\alpha,\beta)$-variant. I will show that the complex map generating the digits of the expansion has a unique (Lebesgue) absolutely continuous invariant measure and give a semi-explicit expression for the density. Finally, I will briefly discuss approximation properties of this family of complex continued fraction algorithms. The second part is a joint work with C. Kalle (Leiden) and J. Thuswaldner (Leoben).

- 2023-02-13 Lauritz Streck (University of Cambridge)
**Conditions for absolute continuity and Fourier decay of Bernoulli convolutions**We consider power series X_b in some 0<b<1 with random iid coefficients - one example of these are Bernoulli convolutions. Questions about absolute continuity and the type of Fourier decay on R of X_b go back to Erdös and are notoriously hard to answer. The main result of our talk is then that if one considers a natural extension of X_b to a more general space and asks about absolute continuity and power Fourier decay there, one can completely classify when their occurrences. This also gives a breadth of explicit examples for which X_b is absolute continuous and has power Fourier decay on R.

- 2023-02-20 James Everitt (University of Warwick)
**Counting periodic orbits under homological constraints**Given a transitive Anosov flow on a smooth, compact, Riemannian manifold, then there exist countably many prime periodic orbits of the flow of a given length. The asymptotic growth of this number, \(\pi(T)\), was given by Margulis in the late 60s. We can restrict our counting to periodic orbits with a given homology class. Fixing a basis so that \(H_1(M, \mathbb{Z})=\mathbb{Z}^k\) for some \(k\geq 1\), then we can take a subset \(A\subset \mathbb{Z}^k\) and \(\pi(T, A)\) counts the number of prime periodic orbits \(\gamma\) with length \(l(\gamma)\) less than or equal to \(T\) and homology class \([\gamma]\in A\). Collier and Sharp showed that if the winding cycle of the flow disappear, then \(\lim_{T\to\infty}\pi(T,A)/\pi(T)=d(A)\) where \(d(A)\) is the density of \(A\) in \(\mathbb{Z}^k\) with respect to a certain norm. We will consider what happens if the density of \(A\) is 0 and when the winding cycle for the flow introduces some linear drift.

- 2023-03-06 Nikolai Edeko (University of Zürich)
**Betti numbers and eigenvalues of dynamical systems on compact manifolds**A dynamical system lives from the interaction between the underlying space and the dynamics that acts on it: the richer the dynamics, the richer the space must be, and, conversely, the more restrictive the space, the more obstructions it poses for dynamical systems on it. But how do specific geometric properties of a space translate into concrete obstructions for dynamical systems on it?

We will consider this question for compact manifolds by understanding how, under appropriate assumptions, the eigenvalues of a dynamical system are related to an obstruction in terms of the first Betti number of the state space. To do this, we will switch our viewpoint from functional analysis (spectral theory) to a geometric point of view and encounter a peculiar topological property of quotient maps that implies a representation theorem for isometric quotients and thereby relates geometry and spectral theory.

- 2023-03-12 Sakshi Jain (University of Rome "Tor Vergata")
**Discontinuities cause essential spectrum**We study transfer operators associated to piecewise monotone interval transformations and show that the essential spectrum is large whenever the Banach space bounds L∞ and the transformation fails to be Markov. Constructing a family of Banach spaces we show that the lower bound on the essential spectral radius is optimal. Indeed, these Banach spaces realise an essential spectral radius as close as desired to the theoretical best possible case.

- 2023-05-08 Chris Bruce (University of Glasgow)
**From algebraic actions to C*-algebras and back again**Each algebraic action of a semigroup gives rise to a concrete C*-algebra generated by the left regular C*-algebra of the group being acted on together with the Koopman representation for the semigroup action. I will explain this construction and then give an overview of recent results on structure and rigidity for such C*-algebras. For certain actions from number theory, the dynamics can be completely encoded by C*-algebraic, or, equivalently, groupoid-theoretic data. This is joint work with Xin Li.

- 2023-05-15 Slade Sanderson (Utrecht University)
**Frequencies of digits of symmetric golden maps via matching**In 2020, Dajani and Kalle determined absolutely continuous invariant measures and frequencies of digits of number expansions corresponding to a family of piecewise-linear interval maps of slope two. Central to their results was a property called ‘matching,’ which occurs when the orbits of the left and right limits of discontinuity points eventually coincide. We obtain analogous results for a family of ‘symmetric golden maps’ of constant slope equal to the golden mean. We characterize the (more delicate) matching phenomenon in our setting, present explicit absolutely continuous invariant measures and determine frequencies of digits of the corresponding number expansions. (Joint with Karma Dajani, arXiv:2301.08623Link opens in a new window)

- 2023-05-22 Jérôme Carrand (Sorbonne University)
**Existence of equilibrium states and MME for Sinai billiard flows**A Gibbs measure is an ergodic invariant probability measure determined by a potential on cylinders. Such a measure maximize a quantity involving the metric entropy and its potential. Maximizers of this quantity are called equilibrium states. In this talk I will present recent results about existence and uniqueness of equilibrium states in the context of Sinai billiards -- that is, dispersive billiards on the two-torus -- for H\"older potentials, some of their properties (Bernoulli, full support, adapted). I will explain their construction by pairing maximal eigenfunctions of a weighted transfer operator acting on anisotropic Banach spaces. In a joint work with Viviane Baladi and Mark Demers, we bootstrap from the existence and uniqueness result to obtain a measure of maximal entropy for the billiard flow.

- 2023-05-22 Joris De Moor (Friedrich-Alexander-Universität)
**Footprint of a topological phase transition on the density of states**For a one-dimensional random discrete Schrödinger operator, the energies at which all transfer matrices commute and have their spectrum off the unit circle are called critical hyperbolic. Disorder driven topological phase transitions in such models are characterized by a vanishing Lyapunov exponent at the critical energy. It is shown that the density of states away from a transition has pseudogap with an explicitly computable Hölder exponent, while it has a logarithmic divergence (Dyson spike) at the transition points. The proof is based on renewal theory for the Prüfer phase dynamics and the optional stopping theorem for suitably constructed comparison martingales.

- 2023-06-05 Dániel Prokaj (Budapest University of Technology and Economics)
**Fractal dimensions of continuous piecewise linear iterated function systems on the line**In the last four decades, considerable attention has been given to self-similar Iterated Function Systems (IFS). In this talk, we consider a more general family of IFSs on the line. Namely, systems consisting of continuous piecewise linear functions whose slopes are different from zero and smaller than one in absolute value. We call these systems Continuous Piecewise Linear Iterated Function Systems (CPLIFS). I will show that the Hausdorff and box dimensions of the attractor of a typical CPLIFS are equal to the minimum of one and the exponent obtained from the most natural system of covers of the attractor. The new results presented are joint with Peter Raith and Károly Simon.

- 2023-06-19 Gaetan Leclerc (Paris Rive Gauche)
**On oscillatory integrals with Hölder phase**

Oscillatory integrals appear naturally in analysis. Various theorems ensure decay properties under some non-concentration hypothesis on the phase: this is for example the content of the non-stationary phase lemma, and of the Van der Corput lemma. All these results ask for a regularity condition on the phase, which may seem unnatural. We will discuss some existing results in the case where the phase is only Holder regular, first in a random setting (we will discuss results about the Brownian motion), and then in a deterministic setting (we will discuss results about conjugacy between expanding maps).- 2023-06-26 Frank Trujillo (Universität Zürich)
**Hausdorff Dimension of the Invariant Measures of Circle Homeomorphisms**Under quite general assumptions, a circle homeomorphism without periodic points is topologically conjugate to an irrational rotation on the circle, and, in particular, it possesses a unique invariant probability measure.

In this talk, we will discuss the dimensional properties of these measures in different settings, namely, for smooth circle diffeomorphisms, critical circle maps, and circle maps with breaks. Furthermore, we will explore how the presence and nature of singularities relate to the Hausdorff dimension of these measures. If time permits, we will discuss a link of this question with the renormalization theory for interval exchange transformations.

- 2023-10-02 Konstantinos Tsinas (University of Crete)
**Multiple ergodic theorems for sequences of polynomial growth**We investigate convergence and multiple recurrence for multiple ergodic averages along sequences that arise from smooth functions with certain regularity properties and which grow polynomially. We establish the convergence of these multiple averages in the -sense under some simple conditions on their growth rates. Our methods rely on some seminorm estimates and a theorem of Frantzikinakis (in the "jointly ergodic" setting), or some equidistribution results in nilmanifolds (in the more general setting).- 2023-10-09 Constantin Kogler (University of Oxford)
**Ergodic actions, random walks and additive combinatorics**Recall that a probability measure preserving group action is called ergodic if every G-invariant function is constant almost surely. We will survey what can be said in general for random walks of such actions, discuss why spectral gap is interesting and mention concrete examples. If time permits, we will touch on our recent joint work with Wooyeon Kim establishing effective density of random walks on homogeneous spaces. All of the results discussed rely on additive combinatorics.

- 2023-10-16 Thomas Richards (Kyushu University)
**Pseudo-monodromy and the Mandelbrot Set**A non-trivial loop in the complement of the Mandelbrot set induces the non-trivial automorphism of the one-sided 2-shift. We consider certain*pseudo-loops*in quadratic parameter space and describe the induced*pseudo-monodromy*. This is closely related to the monodromy problem in Hénon parameter space.This talk is based on joint work with Yutaka Ishii.- 2023-11-06 Alejandro Rodriguez Sponheimer (Lund University)
**A Strong Borel-Cantelli Lemma for Recurrence**Although dynamical Borel-Cantelli lemmas for shrinking targets have been well-studied, it is only recently that versions for recurrence have been considered. In this talk, I will focus on a quantitative description - in the form of a strong dynamical Borel-Cantelli lemma - of the rate of recurrence in measure-preserving systems that satisfy exponential decay of 3-fold correlations. More precisely, we consider shrinking balls around points and find that if the measures of the balls do not shrink too quickly, then the rate of recurrence is asymptotically equivalent to the sum of the measures of the balls. I will give an overview of the proof, highlighting the difficulties that arise in the recurrence setting compared to the shrinking targets case, as well as how the decay of 3-fold correlations plays a role. I will conclude with an application to hitting times and the pointwise dimension of the measure.

- 2023-11-13 Filippo Baroni (University of Oxford)
**Navigating the curve graph with train tracks**

It is a truth universally acknowledged, that an infinite group in possession of a good algebraic structure, must be in want of a hyperbolic space to act on. For the mapping class group of a surface, one of the most popular choices is the curve graph. This is a combinatorial object, built from curves on the surface and intersection patterns between them. Hyperbolicity of the curve graph was proved by Masur and Minsky in a celebrated paper in 1999. In the same article, they showed that elements of the mapping class group act qualitatively differently on this graph depending on their dynamical/topological properties. In light of this, one would like to better understand distances in the curve graph. The graph is locally infinite, and finding a shortest path between two vertices is highly non-trivial. In this talk, we will see how to use the machinery of train tracks to overcome this issue and compute (approximate) distances in the curve graph. If time permits -- which, somehow, it never does -- we will also analyse this construction from an algorithmic perspective.- 2023-11-20 Peej Ingarfield (University of Manchester)
**Dimension Drop in Rational Projections of Sierpinski's Gasket**Sierpinski's Triangle is one of the most famous fractals and partly as a result of this it is very well understood. It's dimension and construction are known as are many other properties such as the natural measure that it supports. In this talk I will discuss projecting this measure at a rational angle. I restrict my attention to rational angles for the irrational case is already well understood. I will discuss the use of automata and shift spaces to encode exact overlaps, a phenomenon conjectured to be responsible for the behaviour of rational projections. Beyond this I will discuss how thermodynamic formalism and slices through the torus allow us to consider rational projections collectively.

- 2023-11-27 Julia Münch (University of Liverpool)
**Extension of Thurston maps and quasisymmetric Uniformisation**Uniformly quasi-regular mappings are a suitable class to study in order to understand which results from holomorphic dynamics can hold for non-holomorphic functions. The dynamics of quasi-regular mappings is particularly interesting in for at least 3 - where we either don't have holomorophic maps or many classical results from complex analysis in the plane don't hold. It is not trivial to find examples of such maps and in the talk I will present a construction that works for Lattès maps - a special subfamily of Thurston maps on the sphere. Then I will explain how one can parametrise a fractal sphere with a quasisymmetric map and explain how these two topics are related.

- 2023-12-04 Minsung Kim (Scuola Normale Superiore di Pisa)
**New phenomena on the deviation spectrum of Birkhoff integrals for locally Hamiltonian flows**The deviation of the Birkhoff integral for area-preserving flow on compact surfaces was first studied by Forni. He proved that the deviation spectrum was determined by the Lyapunov exponents of a renormalization cocycle so-called Kontsevitch-Zorich cocycle. This deviation result was later proved again by Bufetov and Frączek-Ulcigrai for translation flows and locally Hamiltonian flows for non-degenerate types.In this talk, we study the spectrum for deviations of Birkhoff integrals for locally Hamiltonian flows beyond the case of Forni where the observable vanishes at the singularities. Our new developments include a better understanding of the asymptotics at singularities (degenerate type) and the appearance of a new deviation spectrum. This is a joint work with Krzysztof Frączek.- 2024-01-08 Mattias Byléhn (Chalmers University of Technology)
**A spectral approach to long-range order**A stationary random point set in R^n is

*hyperuniform*in the sense of Torquato and Stillinger- 2024-01-15 Daniel Devine (Trinity College Dublin)
**Convergence Results for a system of PDE**In this talk, we will discuss a system of PDE which has its origins in the study of the dynamics of viscous, heat-conducting fluids. PDE systems have received much attention over the last four decades, but they are still relatively poorly understood. The system we will discuss contains source terms with a nonlinear gradient dependence, which presents considerable theoretical challenges. By restricting attention to solutions which are radially symmetric, the problem becomes far more mathematically tractable. To begin, an outline of some of the progress made since the early 2000’s will be given, followed by more recent results based on joint work with Paschalis Karageorgis. In particular we will discuss the convergence of solutions to an explicit solution which we can easily calculate.

- 2024-01-22 Jossy Russell (Imperial College London)
**Bi-critical irrationally indifferent fixed points**For unicritical maps, the development of an arithmetic geometric model for the renormalisation of irrationally indifferent fixed points has proved immensely fruitful, for example in describing the topology of invariant subsets. In this talk, I will introduce some of the difficulties in extending these techniques to the bicritical case.

- 2024-01-29 Zonglin Li (University of Bristol)
**Geometry and distribution of roots of quadratic congruences**The recent paper by Marklof and Welsh established limit laws for the fine-scale distribution of the roots of the quadratic congruence $\mu^2 \equiv D \pmod m$, ordered by the modulus $m$, where $D$ is a square-free positive integer and $D \not\equiv 1 \pmod 4$. This is achieved by relating the roots (when $D > 0$) to the tops of certain geodesics in the Poincare upper half-plane and (when $D<0$) to orbits of points, under the action of the modular group.

In this talk, we will investigate the remaining case when $D>1$ is a square-free integer and $D \equiv 1 \pmod 4$. We will understand how the roots can be related to the tops of the geodesics by considering ideals of the ring $\mathbb{Z}[\sqrt{D}]$ and the ring of integers of the quadratic number field $\mathbb{Q}(\sqrt{D})$. This is joint work with Matthew Welsh.

- 2024-02-26 Fernando Argentieri
**Reducibility without KAM**In the talk I will focus on smooth quasi-periodic SL(2,R) cocycles in one frequency that are close to a constant rotation. The main source of examples of cocycles of these types are the Schrodinger quasi-periodic cocycles, where reducibility results give a better understanding of the spectrum of the respective Schrodinger operator.The aim of the talk is to prove reducibility results for SL(2,R) valued cocycles without any arithmetic assumption on the rotation on the base of the skew-product and with finite regularity assumption on the cocycle.In contrast with the usual KAM schemes, where the finite regularity setting is treated with Nash-Moser techniques by analytic approximations, the proof in this case will be rather different.This is joint work with Bassam Fayad.- 2024-05-02 Axel Péneau (University of Rennes)
**Products of independent matrices without moment conditions**We consider a random sequence**(M^n)**of real valued square matrices of size**d****≥ 2**. We assume that the sequence is constructed as**M^n := X_1 ... X_n**for**(X_k)_{k ≥ 1}**a sequence of independent random matrices of distribution law**ν**. The first observation one can make is that for all integer**n**, one has**||M^{n+1}|| ≤ ||M^n||.||X_{n+1}||.**A result by Furstenberg and Kesten published in 1956 and refined by Kingmann in 1968 tells us that under some**L^1**moment assumptions, the sequence**||M^n||^1/n**converges almost surely to a non-random limit. The issue is that this method does not apply to the study of the coefficients. Using a more constructive method one can show that when**ν**is strongly irreducible and proximal and under some**L^1**moment assumptions, the sequence**|M^n_{i,j}|^1/n**converges to the same non-random limit.This approach is based on the notion of qualitative alignment is semi-group and on contraction properties of matrices. The idea is to find a sequence of "pivotal" times**(p{n})**such that the random words**X^p_k := (X_{p{k}}, ... , X_{p{k+1}-1})**are i.i.d for**k ≥1**, such that the steps**p{k+1} - p{k}**have finite exponential moment for all**k ≥ 0**and such that the partial products**X^p_k :=****X_{p{k}}...X_{p{k+1}-1}**are aligned and proximal in the sense that: there is a constant**ε > 0**such that for all**k,**we have almost surely**||X^p_k X^p_{k+1}****|| ≥ ε ||X^p_k||.||X^p_{k+1}||**and**X^p_k**can be written as the sum of a rank one matrix plus a matrix of norm at most**||X^p_k||ε²****/16**.We will see how this Markovian decomposition allows us to show that the action of**M^n**on the projective space is exponentially mixing and how the lines generated by the columns of**M^n**all converge exponentially fast to a random line. If time permits we will also show how this allows us to prove the convergence result for coefficients.- 2024-2-12 Ethan M. Ackelsberg (EPFL)
**Spectral theory of nilsystems**A nilmanifold is a compact manifold obtained as a quotient of a nilpotent Lie group by a discrete co-compact subgroup. The dynamics of translations on nilmanifolds play a central role in the study of multiple ergodic averages, additive combinatorics, number theory, and higher order Fourier analysis.In joint work with Florian K. Richter and Or Shalom, we improve the understanding of the dynamical behavior of nilsystems by giving a complete description of the spectral decomposition of Koopman operators associated to ergodic translations on nilmanifolds. I will describe this spectral decomposition, explain some of the key ingredients in the proof, and discuss a few natural questions that arise from our work.- 2024-2-19 Stephen Cantrell (University of Warwick)
**The ergodic theory of hyperbolic groups and geometries**In this talk we will discuss how to use ideas and techniques from ergodic theory to study negatively curved groups and geometries. This will be a gentle introduction to the subject with no prior knowledge of hyperbolic geometries or dynamics assumed.

- 2024-3-4 Alex Rutar (University of St Andrews)
**Multifractal analysis via Lagrange duality**In probability theory, fractal geometry, and dynamical systems theory, one is often interested in typical or almost-sure behaviour. But what can be said about exceptional behaviour? The study of such exceptional behaviour in the context of fractal geometry and dynamical systems theory often falls under the domain of multifractal analysis. I will introduce some of the fundamental ideas from multifractal analysis with an emphasis on the (fractal geometric version of the) probabilistic duality between "higher moments" and "large deviations estimates". This talk will touch on ideas from a variety of other fields including (but not limited to!) probability theory, non-convex optimization, information theory, and classical geometric measure theory.