pure math colloquium @ st andrews

Birational geometry studies maps between algebraic varieties defined by rational functions. Recently, derived categories, stability conditions and wall-crossing have led to an entirely new approach to fundamental open questions in birational geometry. I will survey these developments, with an emphasis on Hyperkaehler varieties and cubic fourfolds.

The matroid Grassmannian is the space of oriented matroids. 30 years ago MacPherson showed us that understanding the homotopy type of this space can have significant implications in manifold topology. In some easy cases, the matroid Grassmannian is homotopy equivalent to the ordinary real Grassmannian, but in most cases we have no idea whether or not they are equivalent. This is known as MacPherson's conjecture. I'll show that one of the important homotopical structures of the ordinary Grassmannians has an analogue on the matroid Grassmannian, namely the direct sum monoidal product that is commutative up to all higher homotopies.

The talk will describe the state of the art of the Erdös-Szemerédi sum-product conjecture, a central open question in arithmetic combinatorics. It will also sketch the proof of a new result showing that for a set $A$ of $N$ integers, each of which has a small number of prime factors (roughly, at most $log log N$ of them), either the product set $AA$ is *big* or there is a large subset, with *small* additive energy (alias the number of additive quadruples, or the second moment of the convolution of $A$ with itself).Quantitatively, what stands for *big* and *small* is optimal, up to sub-polynomial factors of $N$.

I will explore how the configuration space of points on the Riemann sphere produces a rich interplay between geometry and combinatorics. I’ll first discuss a certain bipartite graph invariant called the "cross-ratio degree”, defined via a simple counting problem in algebraic geometry. I will mention some contexts in which cross-ratio degrees arise, and will then present a perhaps-surprising upper bound in terms of counting perfect matchings. I will also discuss additional combinatorial structures that arise from studying compactifications of configuration spaces, and time permitting, some connections to tropical/polyhedral geometry.

Recent years have seen a rapid development of rough analytic methods in the study of stochastic partial differential equations. In this talk, I will describe an application of these developments to the classical problem of constructing Euclidean field theories. In particular, I will present recent results on the stochastic quantisation of 2- and 3-dimensional Yang-Mills measures, which includes a proof of universality for the exactly solvable 2-dimensional measure. A key feature of these results is an interaction between gauge symmetries and renormalisation. Based on joint works with Ajay Chandra, Martin Hairer and Hao Shen.

We all know that computers can grind out computations, and that sometimes the way to finish a proof is to grind out a computation. A famous example is the proof of the four colour theorem. These so-called "computer-assisted proofs" are powerful tools in some areas of mathematics. But what if you study objects which cannot be usefully stored in a computer? It turns out that computers are coming for you anyway! Modern AI techniques, and so-called computer proof assistants, are ways of using computers to generate not data but *mathematical ideas*. Computers cannot yet replace pure mathematicians, but I believe that we are moving towards a future where computers will be helping mathematicians not just to compute, but to reason. I'll give an overview of the area as it stands, suitable for pure mathematicians. No computer background will be assumed.

Kirigami, an ancient Japanese art of paper cutting, inspires new ways to tailor the morphology and the mechanics of thin elastic sheets. It has been found to be of special relevance to applications in reconfigurable structures (e.g., large deployable structures) and microstructures (e.g., stretchable graphene sheets). Indeed, careful tailoring of cut patterns results in structures with interesting non-linear macroscopic responses emerging from local out-of-plane buckling. In this presentation, we will discuss local effects in Kirigami by focusing our attention on the study of the deformation of a thin sheet with a single cut—i.e., the most basic and fundamental geometric building block of Kirigami. We will also discuss a new phenomenon that arises when kirigami sheets interact with a liquid substrate, namely elastocapillary kirigami. We study the effects of a liquid foundation and how it changes the nature of the instabilities. Our analysis reveals that post-buckling configurations displays two types of a phase transitions: continuous (second order phase transition), which suggests a uniform phase; and a discontinuous transition (first order), leading to a phase propagation through kirigami structures. Our theoretical analysis is supported by a body of experiential work confirming the existence of buckling localisation and its subsequent propagation in linear-cut kirigami.

An abstract polytope is a combinatorial object abstracted from the notions of polygons and polyhedra; we keep data about the incidence of objects (vertices, edges, faces, ... ) but discard all metric data. Thus it has objects of all possible dimensions from $0$ to $r − 1$, where $r$ is the rank.A polytope is regular if its automorphism group acts transitively (and hence regularly) on the set of maximal flags (sets of mutually incident objects). For example, the $r$-dimensional simplex can be represented by taking the $i$-dimensional objects to be the $(i + 1)$- element sets of an $(r+1)$-set; its automorphism group is the symmetric group of degree $n = r + 1$. It can be shown that a polytope with automorphism group $S_n$ has rank at most $n − 1$, with equality if and only if it is the simplex.After a sequence of results over the last decade, Maria Elisa Fernandes (Averio), Dimitri Leemans (Brussels) and I have proved:__Theorem__: For any positive integer $k$, there is a positive integer $c_k$ such that the number of regular polytopes with automorphism group $S_n$ and rank $n − k$ (up to isomorphism and duality) is $c_k$ (independent of $n$) for $n\ge 2k+3$.The sequence $(c_k)$ begins $1, 1, 7, 9, 35, 48, \ldots$. (No further terms are currently known.)I will talk mostly about the background and history of this problem, with only a very brief sketch of the proof; the paper is over 40 pages long.

A framework is a geometric realisation into Euclidean $d$-space of a graph with edges represented by stiff bars and vertices by universal joints. The framework is globally rigid if every other realisation, in the same dimension, with the same edge lengths arises from an isometry of the space. I will try to give a gentle introduction to the theory of global rigidity in the case that the realisation is generic. Specifically, I will describe that generic global rigidity depends only on the underlying graph and the characterisation of such graphs when $d=2$. The corresponding situation when $d>2$ is open. However, in $3$-dimensions, if we restrict to triangulations then a recent result of Cruickshank, Jackson and Tanigawa gives a combinatorial characterisation. In current joint work with Sean Dewar and Eleftherios Kastis, we prove a similar characterisation for all projective planar graphs. I will describe these results and then conclude with an application of our work to maximum likelihood estimation in Gaussian graphical models.

Record events occur in many situations, such as in temperature records within weather, financial asset price records, and in sporting events, e.g. the 100m sprint. Within probability and random processes, the study of records can be formalised and their limit distributions studied. If we take a sequence of random variables $X_1,\ldots, X_n$, a record time corresponds to the time $t$ event where we have $X_t>\max\left(X_1,\ldots, X_{t-1}\right)$, i.e. $X_t$ exceeds all values occurring before time $t$. A topic of interest is the distribution of such record times, and corresponding record values. In the talk, we review classical results which are part of a wider extreme value theory. We consider first processes that are independent, and identically distributed. Then we mention recent progress when the process $(X_n)$ is generated by a dynamical system.

Hypercontractive inequalities for functions on product-spaces have found many applications over the last 50 years, in such fields as analysis, theoretical physics, combinatorics and theoretical computer science. Hypercontractive inequalities are known for such product-spaces as $n$-dimensional Gaussian space and the $n$-dimensional discrete cube, as well as some closely-related non-product spaces such as the $n$-dimensional sphere (for all $n$). Recently, useful hypercontractive inequalities have been obtained for a much wider variety of non-product-spaces (sometimes under necessary restrictions on the functions concerned): these include a (conditional) hypercontractive inequality for functions on spaces of linear maps, and (unconditional) hypercontractive inequalities for various compact Lie groups. We survey some of these hypercontractive inequalities, and give some applications in group theory and in extremal combinatorics. We expect there to be further applications, for example in theoretical physics.Based on joint works with Guy Kindler (HUJI), Noam Lifshitz (HUJI) and Dor Minzer (MIT).

Splines are piecewise polynomial functions defined over a real domain which are continuously differentiable to some order $r$. For a fixed integer $d$, the space of splines of degree at most $d$ and smoothness $r$ is a finite dimensional vector space, and a largely open problem in numerical analysis is to determine its dimension. While considerable attention has been given to this problem in the bivariate setting, the literature on trivariate splines is less conclusive. In particular, the dimension of generic trivariate splines is not known even in large degree when $r > 1$. It is particularly difficult to compute the dimension of splines on partitions in which a vertex is completely surrounded by tetrahedra – we call these domains vertex stars.In the talk, I will present a lower bound formula on the dimension of splines on vertex stars and how that leads to prove a lower bound on the dimension of splines defined over general tetrahedral partitions in large degree. The proofs use apolarity, some results from rigidity theory, and the so-called Waldschmidt constant of the set of points dual to the interior faces of the vertex star. I will show some examples and open problems in the area.

An omega-categorical structure $M$ is a countably-infinite first-order structure (such as a graph) whose automorphism group $G$ has finitely many orbits on $M^n$, for all finite $n$. Such structures can arise as limits of classes of finite structures in various ways. We are interested in $G$-invariant, finitely-additive probability measures on the definable subsets of $M$ (or $M^n$). The questions we are looking at are partly motivated by questions of Elwes and Macpherson on MS-measurability of omega-categorical structures. In this talk I will focus on some tools from representation theory (work of Tsankov and Jahel-Tsankov) and applications of these by my PhD student Paolo Marimon to proving non-existence of invariant measures on certain `sparse’ graphs which are key examples in the questions of Elwes and Macpherson.

Tropical geometry is a modern branch of algebraic geometry. Its starting point is a process called tropicalization, which replaces complicated algebro-geometric objects by much easier to understand piecewise linear tropical objects. Ultimately this lead to a rich theory of geometry in its own right whose study is motivated by the great promise that learning more about tropical geometry will give us new insights into algebraic geometry. To make this come true we have to understand which tropical objects are actually algebraically meaningful, i.e. which arise by tropicalization. This is the realizability problem.

Complex dynamics studies the behaviour under iteration of holomorphic and meromorphic functions. The first ones to develop this theory were P. Fatou and G. Julia in the 1920s. The area has attracted much attention after the 1980s when the use of computers allowed for the first pictures of the Mandelbrot set to be produced. In this talk we shall focus on the dynamics of transcendental entire functions. Through a series of examples, we will revise the main properties of the Fatou and Julia sets of these maps, present some of the latest developments in the field, and discuss open questions in the area.

A lot of physical evidence and numerical experiments strongly suggests that the typical complex system is chaotic, in a sense that will be discussed. In this talk I will try to present some theoretical aspects of the question, and try to convince you that from a mathematical perspective, ubiquity of chaos is far from certain (and to this date completely open).

By a classical theorem of Jordan, every nontrivial finite group acting transitively on a set contains a derangement (an element with no fixed points). Much more is now known, especially in the context of (almost) simple groups. In this talk, I'll discuss some strong versions of this result and some applications. I'll then focus on the question of whether there exist almost simple groups with elements that are derangements in every faithful primitive action, and I'll show how this solves a question of Garzoni about invariable generating sets for simple groups.

Initially studied by Markov around 1880, the Lagrange and Markov spectra arecomplicated subsets of the real line that play a crucial role in the study of Diophantineapproximation and the study of binary quadratic forms. Perron's description of thespectra in terms of continued fractions allowed the machinery of Gauss-Cantor sets(Cantor sets related to the dynamics of the Gauss map on continued fractions) to cometo bear on many problems. In the 1960s, Freiman demonstrated that the Lagrange spectrumis a strict subset of the Markov spectrum. It still remains a difficult task to find pointsin the complement of the Lagrange spectrum within the Markov spectrum and modern researchis focussed on further developing our understanding of this complement. In this talk, I will introduce these spectra, discussing the historical results above,and speak about recent work with Carlos Matheus and Carlos Gustavo Moreira finding newpoints in the complement and obtaining better heuristic lower bounds for its Hausdorffdimension.

In the twentieth century, a particular image of mathematics as a "young man's game" came to dominate popular images of mathematicians and many mathematicians' own ideas of who can do mathematics and how. This model of the modern mathematician is both more recent and more contingent than commonly supposed. I will identify specific historical circumstances and developments that made mathematics appear to be a "young man's game" in the context of the politics and institutions of an internationalizing discipline. These circumstances converge in the quadrennial International Congresses of Mathematics and the history of the Fields Medal, which has become an accidental symbol of the preemenince of young men in modern mathematics. Recognizing the history, contingency, and politics of this dominant mathematical identity and image can offer a means of understanding and confronting present and future challenges around identity and diversity that continue to matter for mathematics and mathematicians.

Tarski's finite basis problem asks for a given finite algebraic structure $A$ whether the equational theory of $A$ is finitely axiomatizable. More explicitely, does there exist some finite set of identities true in $A$ that implies all identities true in $A$? McKenzie showed that this question is not decidable in general in 1996. Still partial results are known for many classes of algebras. I will review some of the long history of the problem, state open questions and present a recent approach on the finite basis problem for nilpotent loops.

Throughout the years there have been introduced several notions of entropy to express the topological complexity of a certain dynamical system, group action or even an invariant foliation. A general goal in ergodic theory is to obtain variational principles that relate such topologicalentropies with the entropy of (invariant) probability measures and, if possible, to construct probability measures which attain the maximum. In this talk I will talk about the recent use of convex analysis to establish general variational principles for pressure functions and to construct some generalized equilibrium states (joint work with A. Bis, M. Carvalho and M. Mendes)

Finite reductive groups hold a special place amongst all finite groups, partly due to their role in the classification of finite simple groups but also for their natural interactions with geometric objects. Examples of finite reductive groups are the matrix groups $\mathrm{GL}_n(q)$, $\mathrm{SL}_n(q)$, $\mathrm{SU}_n(q)$, $\mathrm{Sp}_{2n}(q)$, ..., defined over a finite field of $q$ elements as well as more exotic examples like $G_2(q)$ and $E_8(q)$. The project of trying to classify and compute the irreducible representations of these groups started in earnest in 1907 with the independent work of Jordan and Schur who computed the ordinary character table of $\mathrm{SL}_2(q)$. Much has happened in the following 115 years with outstanding progress being made due to the enormous contributions of Lusztig. In this talk I’ll survey some of the key historical results on representations of finite reductive groups, indicate some of the remaining challenges, and present some more recent work that contributes to the classification problem in the modular setting.

Fourier analysis has proved a fundamental tool in analytic and combinatorial number theory, usually in the guise of the Hardy--Littlewood circle method. When applicable, this method allows one to asymptotically estimate the number of solutions to a given Diophantine equation with variables constrained to a given finite set of integers. I will survey recent work extending the range of applicability of Fourier methods, with a focus on nonlinear problems in additive combinatorics.

A countable subdirect power of a finite algebra $A$ is a subalgebra of the countable cartesian power $A^N$ consisting of countably many elements, for which the natural projection maps onto $A$ are surjections. In 1982, McKenzie proved that for any finite non-abelian group $G$, the number of non-isomorphic countable subdirect powers was uncountable, and is otherwise countable for finite abelian groups. In this talk, we take a tour through the case for some other algebras such as finite commutative semigroups with the aim of giving a McKenzie-like result, determining precisely those $S$ which have countably many non-isomorphic subdirect powers.

Let $f:I \mapsto I$ be a map of an interval equipped with an ergodic measure $\mu$. In this talk we will introduce and discuss the cover time of the system $(f,\mu)$. This roughly speaking describes the asymptotic rate at which orbits become dense in the state space $I$, or in other words, what is the expected amount of time one has to wait for an orbit to reach a given density in the state space. We will discuss how one can combine probabilistic tools and operator theoretic methods in order to estimate the expected cover time in terms of the local scaling properties of the measure $\mu$. This is based on joint work with Mike Todd.

We begin with definitions, examples and motivation for various numerical invariants of permutation groups, such as base size, maximal irredundant base size and relational complexity. We then give upper bounds on these numerical invariants for certain families of groups.

Inverse semigroups are a natural generalisation of groups, with actions of inverse semigroups modelling partial automorphisms of algebraic structures. One and two sided congruences take up the role for semigroups that subgroups and normal subgroups take on for groups. The set of congruences on any semigroup is partially ordered by inclusion, and this ordering forms a complete lattice. In this talk I will discuss how congruence relations on inverse semigroups may be described in terms of pairs consisting of a subsemigroup and a congruence on the semilattice of idempotents.

Given $n$ discs with randomly selected radii, it was shown by Connelly, Gortler and Theran that there can be at most $2n-3$ contacts in any packing involving the $n$ discs. They also conjectured that the opposite is true; given a planar graph $G$ where every subgraph on $m > 1$ vertices has at most $2m-3$ edges, $G$ can be realised as the contact graph of a random disc packing with non-zero probability. I will discuss how these ideas can be extended to homothetic packings of a centrally symmetric (c.s.) convex bodies. The main results are (i) for any strictly convex and smooth c.s. convex body $\mathcal{C}$, every random homothetic packing of $\mathcal{C}$ has a $(2,2)$-sparse contact graph (i.e. all subgraphs of $G$ on $n$ vertices have at most $2n-2$ edges); (ii) for almost every c.s. convex body $\mathcal{C}$, we can realise all $(2,2)$-sparse planar graphs as the contact graph of a random homothetic packing with positiveprobability.

Recursive random partitions in dimension one are staples of Bayesian statistics, with various applications to topics modeling and clustering. Over the last five years, recursive random partitions in higher dimensions have yielded a new class of random forests that are efficient, easy to compute, with good theoretical properties. In particular, there are new, powerful ways to think about recursive random partitions in higher dimensions using stochastic geometry, which promises a lot of future interactions between this field and machine learning.This talk gives a flavor of the problems, techniques and open questions of interest in this area.

The group of automorphisms of an infinite locally finite graph can be given a totally disconnected locally compact topology with respect to which multiplication and inversion are continuous operations. In other words, it is a totally disconnected locally compact group. If this graph is, for instance, a regular tree, then its group of automorphisms is moreover (almost) simple and generated by a compact set. In order to understand general locally compact groups, there has been a push in recent years to try to understand, and build more examples of, locally compact groups that are totally disconnected, compactly generated, simple and not discrete. As well as the automorphism group of an infinite regular tree, another typical example of this sort is the group of almost automorphisms of that tree (a.k.a. Neretin's group). This last group turns out to also be an example of a piecewise full group (a.k.a topological full group) of homeomorphisms of the Cantor set (the boundary of the tree). These piecewise full groups have been a source of new examples of finitely generated infinite simple groups. They are usually built out of certain groupoids, but in this context it is much easier to see them as coming from certain inverse semigroups of partial homeomorphisms of the Cantor set. After introducing the necessary notions, I will report on joint ongoing work with Colin Reid and David Robertson in which we show how to build compactly generated, simple, totally disconnected locally compact groups out of certain groups, or inverse semigroups, acting on the Cantor set.

"Hyperbolic dynamics" are defined by local expansion and contraction. These properties lead to "random looking" orbits, and indeed these systems satisfy various interesting statistical limit laws. Included in this class are some beautiful geometric examples such as the geodesic flow for a compact manifold of negative sectional curvatures. I will describe ways in which these dynamics capture some structure of the phase space. In particular, how dynamical growth characterises certain information about the fundamental group of the phase space and of the deck transformations given by coverings.

If $A,B,C$ are three finite sets in a field, under what assumptions can we find a point $c$ in $C$ such that $A + cB$ is much bigger than $A$? This is a variant of the Erdős-Szemerédi sum-product problem. I will briefly survey the problem in a number of fields. Then I will focus on the $\delta$-discretised version of the problem on the real line. This version is the most relevant one for fractal geometers.

Let $S$ be a closed orientable surface of genus $g$. The mapping class group $\operatorname{MCG}(S)$ of $S$ is the group of isotopy classes of homeomorphisms of $S$. In the 1970s, Thurston revolutionized the way we think about mapping class groups. Generalizing the notion of Anosov maps of the torus, he defined pseudo-Anosov maps of higher genus surfaces. He then showed every element of a mapping class group is one of three types: finite order, reducible, or pseudo-Anosov. This is reminiscent of the classification of elements of $\operatorname{SL}(2,\mathbb{Z})$ into finite order, reducible, or irreducible. While there are these three types, it is natural to wonder which type is more prevalent. In any reasonable way to sample matrices in $\operatorname{SL}(2,\mathbb{Z})$, irreducible matrices should be generic. One expects something similar for pseudo-Anosov elements in $\operatorname{MCG}(S)$. In joint work with Erlandsson and Souto, we define a notion of genericity and show that pseudo-Anosov elements are indeed generic. More precisely, we consider several "norms" on $\operatorname{MCG}(S)$, and show that the proportion of pseudo-Anosov elements in a ball of radius $r$ tends to 1 as $r$ tends to infinity. The norms we consider have the commonality that they reflect that $\operatorname{MCG}(S)$ come from homeomorphisms of S, and can be thought of as the natural analogues of matrix norms on $\operatorname{SL}(2,\mathbb{Z})$.

The word problem for one-relation monoids has been called one of the most fundamental open problems in all combinatorial algebra. It has been open for over a century, and was described by P. S. Novikov as "containing something transcendental". In this talk, I will give an overview of this problem (including all definitions!) and some important reduction results proved by S. I. Adian and his students. I will then focus on certain special classes, and show some recent results forming part of a program to understand the formal language theory of the word problem for one-relation monoids. This latter topic connects with the famous Muller-Schupp theorem, which identifies groups with context-free word problem as those which are virtually free. I will show that the Muller-Schupp theorem can be generalised to rather broad classes of one-relation monoids. As one consequence, I will show that it is decidable whether a one-relation monoid containing a non-trivial idempotent has context-free word problem. As another consequence, the difficult rational subset membership problem is solved for many classes of one-relation monoids.

Open Court began publishing The Monist in 1890 as a journal “devoted to the philosophy of science” and regularly included mathematical contributions. The audience was understood to be “cultured people who have not a technical mathematical training” but nevertheless “have a mathematical penchant.” With these constraints, the mathematical content varied from recreations to logical foundations, but everyone had something to say about so-called modern geometry. While debates around non-Euclidean geometry ranged from psychology to semantics, the focus here will be on the contested value of mathematical expertise in legitimating what should be considered as mathematics. While some mathematicians urged The Monist to uphold disciplinary standards of geometrical reasoning, other authors opposed to non-Euclidean geometry aligned their reasoning with practical applications, universal know-how, and non-hierarchical democracy. As one contributor inquired “how is the professional expert better fitted to see more lucidly in dealing with the elements of geometry than any other person of good geometric faculty?”

The equaliser of a pair of maps $g, h:S\to T$ is the set of points where they agree. Equalisers of free monoid homomorphisms have been studied in Computer Science since the 1940s, when Post proved that their triviality is undecidable (this is Post's Correspondence Problem). In this talk, I'll explain links between equalisers of free group homomorphisms and certain long-standing problems in (Geometric) Group Theory, and explain how we can use "Stallings' foldings", or "Stallings' automata", to study these objects. Based on joint work with Laura Ciobanu.

One of the classical problems in computational semigroup theory is to compute the ideal structure of a finite submonoid defined by a generating set. Many other questions can be solved more efficiently once this structure is known. Since submonoids defined by modestly sized generating sets can be extremely large, algorithms that rely on exhaustively enumerating elements of the submonoid quickly become impractical. However, the ideal structure of the containing semigroup is often known to be characterised by efficiently computable properties of the elements, and such characterisations may allow us to non-exhaustively compute the ideal structure of submonoids. I will give an overview of previous research in this area, before describing the current state-of-the-art algorithms.

Elekes and Rónyai have characterized real bivariate polynomials which have a small image over large Cartesian products. Besides being an interesting problem in itself, the Elekes-Rónyai setup, and certain generalizations thereof (such as those considered by Elekes and Szabó), arise in many Erdős-type problems in combinatorial geometry and additive combinatorics.In the talk I will give some overview of this topic, and then tell about a recent result (joint with J. Zahl) in which cardinality of a finite set is replaced by either the $delta$-covering number of a set, or its Hausdorff dimension.

There is a wealth of number systems, some of which are more classical than others. These number systems give different representations of real numbers, for example binary, non-integer base and continued fraction. The associated theory has led to a number of solutions to practical problems, for instance in signal processing, coding theory and modeling of quasicrystals. In this talk we will discuss various aspects of non-integer base expansions, and highlight some connections to the set of badly approximable numbers, concluding with a number of open questions.

Making sense of mid-twentieth century mathematical abstraction posed problems for both new and ongoing practitioners. To historicize aspects of processes of generalization and abstraction can be tricky as it is easy to be anachronistic. Hilbert's *Grundlagen*, for example, indicates the recognition of several possible positions on the nature of axioms, for example as "self-evident", as idealizations of experience, or as rules. Since the axioms interact with definitions, this variation in ideas about axioms is accompanied by different ideas about definitions, ranging from definitions as descriptions to definitions as prescriptions. Description, though, is an equivocal term, since one can be describing an object one thinks of as existing, or as one that we are in a sense designing.A historical question arises in what ways, and in what terms, do researchers attempt to justify their particular approaches to abstraction and generalization? How do these justifications function? In the first half of the 20th c. they were not merely conventional in my view. In what follows, we discuss some research papers and look at explicit or implicit efforts to explain the value of the approach. Such justifications are so familiar now from textbook and other writing that they are easy to overlook. These various ways of justifying one's approach serve as a kind of guide to how the main models of innovation in twentieth century mathematics became standard.We look in particular at a set of examples around the "Lebesgue-Nikodym" Decomposition theorem in analysis. This is work in progress.

Recent results and open questions will be presented, related with some graphs encoding generating properties of finite and profinite groups. In particular, given a family $\mathcal{F}$ of groups, we will consider the graph whose vertices are the elements of a group $G$ and where two vertices $x, y$ in $G$ are adjacent if and only if the subgroup generated by them does not belong to $\mathcal{F}$.

Rigidity theory asks and answers questions about how a given mechanical structure can deform. This area extends back into the nineteenth century with work of Cauchy and Maxwell, and continues to be an active area of research with a wide range of applications. I will begin my talk with a broad overview of this area. Then, I will narrow my focus onto symmetry-forced rigidity of plane frameworks to discuss a recent result. I will discuss the algebraic-geometric ideas involved in the proof, and how these same ideas can be used to address certain problems in matrix completion.

Smooth ergodic theory aims to analyse the long-term statistics of chaotic dynamical systems. There are several analytic and probabilistic tools that are used to answer such questions. Each of these approaches has its advantages and its shortcomings, depending on the system under consideration. In this talk, I will focus on transfer operator techniques and spectral methods. In the first half of this talk, I will explain ideas behind this approach through simple, yet important examples. In the second half of the talk, I will discuss a recent joint work with C. Liverani, whose long-term goal is to provide a good spectral picture for pricewise hyperbolic systems with singularities (e.g. billiard maps) in any dimension.

A pragmatic view of tropical geometry regards it as a combinatorialization of complex or algebraic geometric objects such that a surprisingly large number of algebraic invariants may be recovered by the corresponding combinatorial ones. I will survey this connection in the context of the moduli spaces of curves. In recent years the relationship between the algebraic and tropical moduli spaces of curves has been clarified by gaining a concrete understanding of how tropical curves capture the infinitesimal geometry of deformations of algebraic curves. My goal is to give a broad strokes description of the evolution of this story, suitable for a general audience. Time permitting, I will focus on some recent results (joint with Hannah Markwig and Andreas Gross) on the intersection theory on tropical moduli spaces of curves of higher genus.

One of many definitions gives the rank of an $m\times n$ matrix $M$ as the smallest natural number such that M can be factorized as $AB$, where $A$ and $B$ are $m\times r$ and $r\times n$ matrices respectively. In many applications, one is interested in factorizations of a particular form. For example, factorizations with nonnegative entries define the nonnegative rank which is a notion that is used in data mining applications, statistics, complexity theory etc. Nonnegative rank has geometric characterizations using nested polytopes. I will give an overview how these nested polytopes are related to characterizations of the set of matrices of given nonnegative rank and uniqueness of nonnegative matrix factorizations.

Evolutionary game theory describes the development of social behaviour not as careful, rational decision making but as a mixture of imitation and messy trial and error. I will discuss the evolutionary perspective in the context of iterated games: what the folk theorems mean for evolutionary dynamics and how choice of strategy space can have important consequences for the fitness landscape associated with an evolutionary process. I will then discuss a real-world application of evolutionary game theory to help understand the spread of misinformation online. I’ll summarize predictions for patterns of fake news sharing and engagement that arise from studying the evolutionary dynamics of an iterated “fake news game”. Finally i’ll present some recent data gathered to test those predictions.

Equipped with the latest technology and dreams of glory, international observing parties trekked around the globe in the second half of the 19th century in quest of new astronomical results. As developments in photography fueled a race to capture an image of the solar corona, scientists also hoped for insight about both the size of the universe and the chemical composition of the Sun. North American eclipse paths in 1860 and 1869 especially played into the scientific agenda articulated by mid-century mathematical practitioners in the United States. Results from 1869 heralded as huge success on American shores were met with suspicion in Europe. Americans viewed joint expeditions in 1870 and 1871 as opportunities to vindicate their work.

I will give an overview about results and open questions on equations defining algebraic varieties and present results for equations defining toric varieties.

Systems such as vertebrates, origami sheets, umbrellas and robots are flexible, meaning that they can deform at their joints despite the rigidity of their constitutive elements. Since the time of JC Maxwell, rigidity theory has described such phenomena by identifying the spatial embeddings of graphs that preserve the lengths of their edges. I will focus on the rigidity of metamaterials, which repeat periodically in their bulk but also include boundaries. Building on the work of Kane and Lubensky (2013), I will show that the bulk structure generates an abstract space of modes and that periodic bulk modes lie at the boundary between modes on disjoint surfaces. These modes then generate a topological invariant that generates and places boundary modes. Mechanical criticality, generalized from Calladine for multiply periodic structures, is a crucial control parameter.

Diagonal groups form an important class of permutation groups, arising as one of four basic classes in the description of primitive permutation groups given by the O'Nan--Scott theorem. In joint work with Cheryl Praeger and Csaba Schneider, we have produced a combinatorial description of objects (which we call "diagonal semilattices") whose automorphism groups (in dimension at least 3) are diagonal groups. The groups emerge naturally from combinatorial assumptions. The proof is by induction on the dimension; starting the induction in 3 dimensions is the hardest part of the argument, and involves characterising a certain class of Latin cubes as sets of partitions.

Each quiver appearing in a seed of a skew-symmetric cluster algebra determines a corresponding group, which we call a cluster group, which is defined via a presentation. Grant and Marsh showed that, for quivers appearing in skew-symmetric cluster algebras of finite type, the associated cluster groups are isomorphic to finite reflection groups and thus are finite Coxeter groups. There are many well-established results for Coxeter presentations and it is natural to ask whether the cluster group presentations possess comparable properties.I will define a cluster group associated to a cluster quiver and explain how the theory of cluster algebras forms the basis of research into cluster groups. As for Coxeter groups, we can consider parabolic subgroups of cluster groups. I will outline a proof which shows that, in the type A case, there exists an isomorphism between the lattice of subsets of the defining generators of the cluster group and the lattice of its parabolic subgroups.

One of the fundamental problems in dynamics is to understand the attractor of a system, i.e. the set where most orbits spent most of the time. As soon as the existence of an attractor is determined, one would like to know if it persists in a family of systems and in which way i.e. its stability. Attractors of one dimensional systems are well understood, and their stability as well. I will discuss attractors of two dimensional systems, starting with the special case of Henon maps. In this setting very little is understood. Already to determine the existence of an attractor is a very difficult problem. I will survey the known results and discuss the new developments in the understanding of attractors, coexistence of attractors and their stability for two dimensional dynamical systems.

Tropical geometry is geometry over the tropical semiring, wheremultiplication is replaced by addition and addition is replaced byminimum. This "tropicalization" procedure turns algebraic varieties(solutions to polynomial equations) into polyhedral complexes, whichare combinatorial objects. A surprisingly large amount of informationabout the variety is still present in tropical "combinatorialshadow". In this talk I will introduce tropical varieties, andindicate some of their applications, both inside and outside algebraicgeometry.

Tensor decomposition has many applications. However, it is often a hard problem. In this talk we discuss a family of tensors, called orthogonally decomposable tensors, which retain many of the nice properties of matrices that general tensors don't. A symmetric tensor is orthogonally decomposable if it can be written as a linear combination of tensor powers of $n$ orthonormal vectors. We will see that the decomposition of such tensors can be found efficiently, their eigenvectors can be computed efficiently, and the set of orthogonally decomposable tensors of low rank is closed and can be described by a set of quadratic equations. Analogously, we study nonsymmetric orthogonally decomposable tensors, and show that the same results hold. Finally, we discuss a generalization called orthogonal tensor networks, which allows to find an efficient decomposition of a larger set of tensors.

In this talk I will describe two spaces of groups. The first is the Grigorchukspace of marked groups, and the second is the Polish space of enumerated groups.Both spaces provide a useful framework for the study of countable, discretegroups. After a gentle introduction, I shall describe some recent results in thefield, such as the work of Minasyan, Osin and Witzel on quasi-isometricdiversity, and the work of Elayavalli and Goldbring on the generic version ofthe von Neumann-Day problem. I shall also describe how some of my past and alsosome recent work (in part with coauthors) fits into the picture.

Chip-firing is a single player game played on the vertices of a graph. The possible outcomes of the game reflect graph invariants such as gonality and complexity. It was recently discovered that there is a deep connection between chip-firing and algebraic geometry, and that the game may be used to encode geometric properties of algebraic curves. This approach has already led to new results in algebraic geometry and solutions to various open problems. In my talk, I will explain the rules of the game, and explore the interactions between winning strategies, the geometry of graphs and curves, and Kircchof's matrix tree theorem.

TBD

In 2001 Peres, Simon and Solomyak considered one-parameter families of self-similar Iterated Function Systems (IFSs) on the line satisfying the so-called transversality condition. They proved that if the similarity dimension is less than 1, then for a typical parameter (both in category and measure sense) the existence of overlaps between cylinders implies that the appropriate dimensional Hausdorff measure of the attractor is zero. We extend this result both for self-similar and self-conformal IFSs on the line. Moreover, combining with recent results of Fraser-Henderson-Olson-Robinson and Angelevska-Kaenmaki-Troscheit we obtain that the Assouad dimension of such systems is 1.(joint work with Balazs Barany, Mihal Rams, and Karoly Simon)

Computing the complexity of Markov bases is an extremely challenging problem; no formula is known in general and there are very few classes of toric ideals for which the Markov complexity has been computed. A monomial curve $C$ in $\mathbb{A}^3$ has Markov complexity $m(C)$ two or three. Two if the monomial curve is complete intersection and three otherwise. Our main result shows that there is no $d\in \mathbb{N}$ such that $m(C) \leq d$ for all monomial curves $C$ in $\mathbb{A}^4$. The same result is true even if we restrict to complete intersections. We extend this result to all monomial curves in $\mathbb{A}^n, n \geq 4$.

Signalling pathways can be modelled as a biochemical reaction network. When the kinetics follow mass-action kinetics, the resulting mathematical model is a polynomial dynamical system. I will overview approaches to analyse these models with steady-state data using computational algebraic geometry and statistics. Then I will present how to analyse such models with time-course data using differential algebra and geometry for model identifiability. Finally, I will present how topological data analysis can provide additional information to distinguish these models and experimental data from wild-type and mutant molecules. These case studies showcase how computational geometry, topology and dynamics can provide new insights in the biological systems, specifically how changes at the molecular scale (e.g. MEK WT and mutants) result in phenotypic changes (e.g. fruit fly mutations).

A countable relational structure (e.g. a graph or partial order) is said to be homogeneous if every isomorphism between finite (induced) substructures extends to an automorphism of the structure. I will give an overview of various classification and structure theorems for homogeneous structures (for example graphs, digraphs, permutations, finite homogeneous structures). I will then focus on Cherlin's programme to classify metrically homogeneous graphs, that is, graphs which become homogeneous when enriched by binary relation symbols interpreted by graph distance (this can be seen as a strengthening of the well-known property of distance transitivity for graphs). In particular, I will describe joint work with Amato and Cherlin classifying metrically homogeneous graphs of diameter 3.

In 2012 Hochman and Shmerkin proved that, given Borel probability measures on $[0,1]$ invariant under multiplication by 2 and 3 respectively, the Hausdorff dimension of the orthogonal projection of the product of these measures is equal to the maximum possible value in every direction except the horizontal and vertical directions. Their result holds beyond multiplication by 2,3 to natural numbers m,n which are multiplicatively independent. We discuss a generalisation of this theorem to include random cascade measures on subsets of $[0,1]$ invariant under multiplication by multiplicatively independent $m$,$n$. We will define random cascade measures in a heuristic way, as a natural randomisation of invariant measures on symbolic space. The theorem of Hochman and Shmerkin fully resolved a conjecture of Furstenberg originating in the late 1960s concerning sumsets of these invariant sets. We apply our main result to present a random version of this conjecture which holds for products of percolations on $\times m, \times n$-invariant sets.

Semi-Latin squares generalise Latin squares, and are used in thedesign of comparative experiments. My talk, including joint work withR.A. Bailey, will focus on uniform semi-Latin squares, which generalise completesets of mutually orthogonal Latin squares and have excellent statistical optimalityproperties. I will discuss the construction and properties of uniform semi-Latin squares, including connections with group theory, computation, and other types of combinatorial designs. This talk should be accessible to a general mathematical audience.

The study of the relationship between the size of a set and the existence of arithmetic progressions contained in it has been a major problem for a long time. For example, Szemerédi’s theorem provides an answer in the discrete context: sets of natural numbers of positive density contain arithmetic progressions of all finite lengths. In the continuous context, one can study the same kind of problem: How large can a set of real numbers that avoids arithmetic progressions be? It is well known that sets of positive Lebesgue measure contain a homothetic copy of every finite set (in particular, arithmetic progressions of every finite length), so it is of interest to study the question using other notions of size. I will present some results in the continuous context providing answers to the previous question for various concepts of size, and also a result of the same type (joint with J. Fraser and P. Shmerkin) for approximate arithmetic progressions.

Between any pair of points lying in a plane over any field we can define a distance quantity. If we have a finite number of points, we then have a finite number if distances. The question I would like to talk about is: how many distances must occur between any set of points? This question was originally asked by Erdös and, up to a logarithmic factor, resolved over the reals by Guth and Katz in 2010; I will discuss how the question can be approached over general fields. This is work joint with B. Murphy and M. Rudnev.

Automatic groups arose in the 80s and 90s and were hotly studied. If agroup admits an automatic structure, many nice consequences follow such asa quadratic time algorithm to solve the word problem. Recently a new kindof automatic group was defined: Cayley (graph) automatic groups preservethe nice algorithmic properties, but the definition allows for a muchlarger class of groups (for example, non finitely presented).New work of Berdinsky and Trakuldit (Mahidol, Thailand) proposes a way toquantify how close a Cayley automatic group is from being genuinelyautomatic; I will describe their work and some new contributions andquestions, joint with Berdinsky and Taback (Bowdoin, USA)

Studying generating sets for groups has led to many interesting and surprising results. For instance, every finite simple group can be generated by just two elements. In fact, Guralnick and Kantor, in 2000, proved that in a finite simple group every nontrivial element is contained in a generating pair, a property known as $3/2$-generation. This answers a 1962 question of Steinberg. This talk will be a survey of several lines of current research that are motivated by this result. In particular, I will discuss recent progress towards a complete classification of the finite $3/2$-generated groups, work with Tim Burness on the stronger notion of uniform domination (which has a close connection with bases for permutation groups), and work with Casey Donoven on analogous results for infinite groups related to Thompson's group $V$.

In a famous list of problems in combinatorial geometry from 1966, Leo Moser asked for the largest $s(n)$ such that every $3$-dimensional convex polyhedron with $n$ vertices has a $2$-dimensional shadow with at least $s(n)$ vertices. I will describe the main steps towards the answer, which is that $s(n)$ is of order $\log(n)/\log\log(n)$, found recently in collaboration with Jeffrey Lagarias and Yusheng Luo, and which follows from 1989 work of Chazelle, Edelsbrunner and Guibas. I will also report on current work with Alfredo Hubard concerning higher-dimensional generalizations of this problem.

In this talk, I will attempt to present the algorithms we have developed to find an infinite presentation for Thompson's group $F$ which reflects the geometric structure of the unit interval.

In the 1980s Dan Shechtman discovered quasicrystals, by producing metal alloys whose electron diffraction patterns exhibited a great deal of structural order but also 5-fold rotational symmetry, precluding the periodically repeating arrangement of a regular crystal. Before this discovery, mathematically idealised infinite, aperiodic patterns had already been considered. For example, in the 60s the first aperiodic tile set was found by Berger in his resolution of Wang’s Domino Problem, and in the 70s Penrose discovered his famous aperiodic tilings. In this talk I will introduce the field of Aperiodic Order: the study of infinite tilings or point sets of Euclidean space which have a rich structure of approximate symmetries but few, if any, global symmetries. The classical study of periodic patterns is conducted through consideration of their space groups of global symmetries. Since aperiodic patterns lack global symmetry, new mathematical tools need to be introduced. I will explain how one defines associated moduli spaces of patterns, whose topological invariants turn out to be natural and important invariants of the original patterns. I will explain some key aspects of recent joint work with John Hunton, which incorporates their rotational and translational structure into this topological analysis.

Over the past decades various advancements have appeared in the multiscale analysis of random graphs (clusters, loops, patterns, ...) probably due to their prevalence on big data. Motivated by random billiards, parallel to these results on random graphs many ‘dynamicists’/geometers established analogous results for random surfaces. In this talk we will present basics of this field and history, and present an overview of our work on the spectral theory of random surfaces.Joint work with Joe Thomas (Manchester), Etienne Le Masson (Cergy, Paris), Clifford Gilmore (Cork), Mostafa Sabri (Cairo)

We consider maps with invariant measuresand look at the time it takes for random pointsto enter a given (positive measure) set.For return times a theorem of Kac states that onaverage the return times are the reciprocal ofthe size of the return set. The actual returntimes however have a distribution which typicallyallows for arbitrarily long returns.We want to consider a sequence of shrinking setsthat contract to a single point or, possibly, tosome more complicated zero measure set.With good control over the shrinking target setsone can prove meaningful limiting resultswhich tend to be either a straight Poissondistribution or a compound Poisson distribution.

We will discuss the family of Braided Higman-Thompson groups and their finiteness properties. Along the way, we will come across certain subcomplexes of the curve complex and arc complex of a surface as well as introduce the notion of a disk complex.This is an ongoing joint work with Xiaolei Wu.

It is well-known that the number of periodic orbits of a geodesic flow over a compact negatively curved manifold (or, more generally, of an Anosov flow) grows exponentially fast. We are interested in the growth rate. If we pass to a finite regular cover then the growth rate remains the same but if the cover is infinite then, with an appropriate definition, the growth rate may decrease. It then becomes interesting to characterise when we still have equality. For geodesic flows, a combination of work of Roblin (2005) and Dougall and myself (2016) establishes that equality holds if and only if the covering group is amenable. A key feature underpinning this result if the time-reversal symmetry enjoyed by geodesic flows and the result fails for general Anosov flows, when this symmetry is absent. We will discuss these topics and a recent result that provides a natural generalisation to the Anosov setting. This is joint work with Rhiannon Dougall.

We consider some classes of piecewise expanding maps infinite dimensional spaces with absolutely continuous (with respect toLebesgue measure) invariant probability measures and derive anentropy formula for such measures. Using this entropy formula, wepresent sufficient conditions for the continuity of that entropy incertain parametrized families of maps. We apply our results to a familyof higher dimensional tent maps. Joint work with Antonio Pumariño.

We present joint work with D. Beltran and J. Hickman on sharp local smoothing estimates for general Fourier integral operators. Local smoothing bounds imply major estimates in harmonic analysis, including Bochner-Riesz estimates, oscillatory integral estimates and bounds for the size of Besicovitch sets and related problems involving Kakeya maximal functions. Our work gives a sharp resolution to the most general form of the local smoothing problem formulated by the speaker in the early 1990s. We rely on decoupling estimates of Bourgain and Demeter.

Representation theory aims to study an algebraic object, often a ring, by understanding how it acts on any given abelian group. In this talk we will be interested in studying a finite-dimensional algebra $A$ from the viewpoint of representation theory. An abelian group together with an action of $A$ defines an $A$-module and so we consider the collection of all (finite-dimensional) $A$-modules and the maps between them. This is the category $\mathrm{mod}(A)$. A classical problem, considered by Morita in 1958, is how to determine when we have an equivalence between the category $\mathrm{mod}(A)$ and the category $\mathrm{mod}(B)$ for two algebras $A$ and $B$. It turns out that such an equivalence can be detected by the presence of certain modules $P$ called progenerators. We will take this classical theorem as the starting point of our talk and go on to review how this idea has developed in representation theory up until the present day. We will end by presenting some joint work with Karin Baur, in which we classify the cosilting modules over cluster-tilted algebras of type $\tilde{\mathbb{A}}$.

Inverse monoids are monoids equipped with an involutive inverse operation that satisfies the identities $mm^{-1}m=m$ and $mm^{-1}nn^{-1}=nn^{-1}mm^{-1}$. They can be considered as the abstractions of partial symmetries, and are one of the many generalizations of groups. The Cayley graph of an inverse monoid need not be strongly connected, nevertheless, the strongly connected components, called Schützenberger graphs, form metric spaces. We prove that if a finitely presented inverse monoid has tree-like Schützenberger graphs, then the Schützenberger graphs have regular geodesics, and the inverse monoid has a solvable word problem.

Inverse monoids are monoids equipped with an involutive inverse operation that satisfies the identities $mm^{-1}m=m$ and $mm^{-1}nn^{-1}=nn^{-1}mm^{-1}$. They can be considered as the abstractions of partial symmetries, and are one of the many generalizations of groups. The Cayley graph of an inverse monoid need not be strongly connected, nevertheless, the strongly connected components, called Schützenberger graphs, form metric spaces. We prove that if a finitely presented inverse monoid has tree-like Schützenberger graphs, then the Schützenberger graphs have regular geodesics, and the inverse monoid has a solvable word problem.

In topological group theory, the study of locally compact groups naturally splits into looking at those that are connected and those that are totally disconnected. The connected case is broadly understood, following the solution to Hilbert's Fifth Problem. The totally disconnected case ("tdlc groups") was considered intractable until breakthrough work by George Willis in the 1990s. In 2002, work by Rögnvladur Möller showed that Willis' theory could be recovered via the theory of infinite permutation groups acting on locally-finite graphs.In this talk I will discuss some of the interplay between tdlc theory and the theory of infinite permutation groups. I will try to highlight places where the application of permutation groups has lead to breakthroughs in our understanding of tdlc groups. I will also present some high-profile open problems on tdlc groups that I think could potentially be solved using permutation groups.

I will give an overview of several results around the Ham Sandwich theorem. Conjectured by Steinhaus as the "leg of pork theorem" and proved by Banach, it states that for any three measures in 3 dimensional space (e.g. a slice of ham and the two pieces of bread), there exists a plane that cuts in half each of the three measures. Since then there has been a myriad of generalizations and applications to geometric analysis, incidence geometry and computational geometry. In this talk I will survey some of these results.

Studying algebras via their finitary properties is a classic approach stretching back to Noether and Artin in the early part of the last century. Here by a *finitary property* for a class of algebras $\mathcal{A}$ we mean a property, defined for algebras in $\mathcal{A}$, that is guaranteed to be satisfied by any finite member of $\mathcal{A}$. Of course, the idea is that it will also be satisfied by some infinite algebras in $\mathcal{A}$, and the fact that it is will ensure that such algebras behave in some way like the finite ones. This talk will focus on finitary properties for monoids. Many of these arise naturally from the representation of a monoid $S$ via mappings of sets or, equivalently and more concretely, by *$S$-acts*. A right $S$-act is a set $A$ together with a map $A\times S\rightarrow A$ where $(a,s)\mapsto as$, such that for all $a\in A$ and $s,t\in S$ we have $a1=a$ and $(as)t=a(st)$. For example, a monoid is *right noetherian* if every right congruence is finitely generated, and this is equivalent to every monogenic right $S$-act being finitely presented. A finitary property of particular interest to me is that of coherency. We say that a monoid $S$ is *right coherent* if every finitely generated $S$-subact of every finitely presented right $S$-act is finitely presented. *Left coherency* is defined dually and $S$ is *coherent* if it is both right and left coherent. These notions are analogous tothose for a ring $R$ (where, of course, $S$-acts are replaced by $R$-modules). Coherency arises naturally from several directions, as this talk will explain, and is closely related to the notion of acts being algebraically closed. We examine the connection with other finitary properties, such as that of being right noetherian. Of course one wants to know which monoids *are* (right) coherent, and how coherency behaves with respect to some standard constructions. These question turns out to be hard to answer, in spite of us having a condition for coherency analogous to that of Chase for rings. I will present a selection of the results in this area and also a number of open problems. The talk will be aimed at a non-specialist audience.The results in this talk are taken from many sources, the most recent being joint with Yang Dandan, Miklós Hartmann, Thomas Quinn-Gregson, Nik Ruškuc and Rida-e Zenab.

The speaker is one of a team of seven mathematicians and mathematics educators, representing different universities across the United States, who have been at work to design, author, classroom test, revise, evaluate, and disseminate classroom modules called Primary Source Projects (PSPs, which are meant to teach standard topics from across the early years of the undergraduate mathematics curriculum through primary historical source materials. This endeavor, called by the acronym TRIUMPHS, intends for PSPs to replace traditional textbook presentations of mathematical content by focusing student attention on the interpretation of historical source texts combined with brief contextual material and carefully crafted exercises meant to encourage sense-making by students. PSPs are also designed to incorporate principles of active learning, wherein the bulk of classroom time is given over to student work on project tasks and exercises, both alone and in discussion with small groups of classmates, or involving the entire class, rather than to traditional lectures by the instructor.The TRIUMPHS team, supported with funding from the US National Science Foundation, have created some 48 such modules together with a few external authors. These are now freely available from the TRIUMPHS website. Some PSPs can take as little as 30 minutes to implement, while others are designed to take up to four weeks (with median implementation time of about one week) of classroom time. There are modules written to support standard coursework from precalculus and calculus, linear algebra, differential equations, algebra, theory of numbers, geometry, analysis, statistics, and a few other subjects as well.This talk will discuss the TRIUMPHS endeavor generally but will show examples of PSPs at work through a pair of projects authored by the speaker, one of which is an introduction to the study of trigonometry, the other of which teaches the matrix determinant.

Many combinatorial problems call for an object a little bit moreinformative than a matroid to capture all the relevant information. Inthe topology of torus arrangements, e.g., this extra information is ofarithmetic flavour. I'll introduce the object Luca Moci and I definedto answer this call -- matroids (of modules) over rings, the ring inthis case being Z -- and then show how their structure relates tobetter-known matroid generalisations, in particular the valuatedmatroids of Dress and Wenzel.

In this talk, we first introduce an application of (undirected) two edge-colored graphs to group algebras of non-Noetherian groups, where a group is Noetherian if every subgroup is finitely generated. Almost all infinite groups are non-Noetherian except for polycyclic by finite groups. We have used these graphs to study primitivity of group algebras of non-Noetherian groups, where generally a ring is right primitive if it has a maximal right ideal which contains no non-trivial ideals. As every simple ring is primitive, we can regard primitivity as a generalization of simplicity. As for group algebras, any group algebra of a nontrivial group is not simple, but some group algebras of non-abelian infinite groups are primitive. In general our method using two edge-colored graphs seems to be effective to find a non-scalar element appeared in products of elements of a group algebra if its group has non-abelian free subgroups. But there exist some non-Noetherian groups with no non-abelian free subgroups; for example Thompson’s group $F$ and a free Burnside group of large exponent. Finally in order to be able to investigate group algebras of these groups, we use a ‘directed’ two edge-colored graph and improve our method.

Let $G$ be a permutation group of degree $n$ on the domain $\Omega$,and $k$ a positive integer with$k\le n$. We say that $G$ has the *$k$-existential property*, or*$k$-et*, if there exists a $k$-subset $A$ whose orbit under $G$contains transversals for all $k$-partitions $\mathcal{P}$ of$\Omega$.This property is a substantial weakening of the $k$-universaltransversal property, or $k$-ut,investigated by my coauthors, which required this condition to holdfor all $k$-subsets $A$ of the domain $\Omega$.Both of these conditions relate to the regularity of transformationsemigroups. In the earlier work, the regularity of $\langle G,t\rangle$ for anymap $t$ of rank $k$ (with $k < n/2$) was shown to be equivalent to the$k$-ut property. The question investigated in our new result is ifthere is a $k$-subset $A$ of the domain such that $\langle G, t\rangle$ isregular for all maps $t$ with image $A$. This turns out to be muchmore delicate: the $k$-et property(with $A$ as witnessing set) is a necessary, but not sufficient condition.In this talk we give a nearly complete characterizations of both$k$-et and its corresponding regularity condition in the case that$4\le k\le n/2$.This is joint work with João Araújo (Universidade Nova) and PeterJ. Cameron (St Andrews).

I'll begin by discussing Selberg's eigenvalue conjecture. This is an analog of the Riemann hypothesis for a special family of Riemann surfaces that feature heavily in number theory, for example in Wiles' proof of the Taniyama-Shimura conjecture. I'll explain how in the last 10 years, number theorists have had to turn to Anosov dynamics to obtain the approximations to Selberg's conjecture that became relevant to emerging 'thin groups' questions about Apollonian circle packings and continued fractions. Then if I have time, I'll explain how I have proved an extension of Selberg's 3/16 theorem to higher genus moduli spaces, and point out some interesting ingredients of the proof.Part of the talk is based on joint work with Oh and Winter and with Bourgain and Kontorovich.

The associahedron is a polytope whose vertices correspond to binary trees with n nodes and whose edges correspond to rotations between these trees. In this talk, I will review a classical construction of this polytope and present generalizations to related families of polytopes such as permutreehedra and quotientopes. Based on joint works with Viviane Pons (Univ. Orsay) and Francisco Santos (Univ. Cantabria).

In Diophantine Approximation we are often interested in the Lebesgue and Hausdorff measures of certain $\limsup$ sets. In 2006, motivated by such considerations, Beresnevich and Velani proved a remarkable result --- the Mass Transference Principle --- which allows for the transference of Lebesgue measure theoretic statements to Hausdorff measure theoretic statements for $\limsup$ sets arising from sequences of balls in $\mathbb{R}^k$. Subsequently, they extended this Mass Transference Principle to the more general situation in which the $\limsup$ sets arise from sequences of neighbourhoods of "approximating" planes. In this talk, I aim to discuss two recent strengthenings and generalisations of this latter result. Firstly, in a joint work with Victor Beresnevich (York), we have removed some potentially restrictive conditions from the statement given by Beresnevich and Velani. The improvement we obtain yields a number of interesting applications in Diophantine Approximation. Secondly, in a joint work with Simon Baker (Warwick), we have extended these results to a more general class of sets which include smooth manifolds and certain fractal sets.

I will discuss questions and problems concerning the rigidity of bar-joint frameworks as well as rigidity of other kinds of combinatorial structures. One fundamental question is whether, given a graph (or other appropriate combinatorial description), a "generic" framework with that underlying graph is rigid. In some circumstances, however, one would like to understand the rigidity of frameworks which have some symmetry and so are not fully generic. I will present some of the recent results in this direction as well as some of the difficulties and methods for proving such results.

A wide variety of optimization problems at the heart of machine learning, economics, and operations research can be cast as constrained submodular maximization problems. Unfortunately, in many emerging applications, the resulting submodular optimization problems are too large to be solved (or even stored) on a single machine. In this talk, I will describe a new framework for distributed submodular maximization. The framework is easily stated in the popular MapReduce model of computation, and provides a general condition under which centralised algorithms can be transformed into distributed algorithms with almost no loss in the approximation guarantee. This talk is based on joint work with Rafael da Pont Barbosa, Alina Ene, and Huy Nguyen.

A compact $K\subset\mathbb{R}^{2}$ is called self-affine if thereexist affine contractions $\varphi_{1},...,\varphi_{m}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$such that $K=\cup_{i=1}^{m}\varphi_{i}(K)$. In the 1980s, KennethFalconer introduced a value $s$, called the affinity dimension, whichis the ''expected'' value for the dimension of $K$. I will discussa recent project with Mike Hochman, building on our joint work withBalázs Bárány, in which we establish the equality $\dim K=s$ undermild assumptions.

An action of a group on a compact metric space is rational if there is an assignment of binary addresses to points in the space for which the elements of the group act as asynchronous, finite-state automata. In this talk I will discuss the theory of rational actions and sketch a proof that any hyperbolic group acts rationally on its Gromov boundary. This gives an embedding of any hyperbolic group into the group of rational homeomorphisms of the Cantor set defined by Grigorchuk, Nekrashevych, and Suschanskii. This is joint work with Collin Bleak and Francesco Matucci.

I will discuss Erdős–Rothschild type problems in discrete structures, in which we seek to maximise the number of non-Ramsey colourings. That is, the number of colourings of a discrete structure such that every colour class does not contain some forbidden substructure. For example, maximise the number of $r$-colourings without monochromatic* $k$-cliques among all $n$-vertex graphs;* Schur triples among all subsets of a given abelian group;* Schur triples among all subsets of $\{ 1,\ldots,n\}$.I will give an overview of the area, the number- and graph-theoretic tools which go into the proofs, and discuss some joint work with Oleg Pikhurko and Zelealem Yilma, and with Hong Liu and Maryam Sharifzadeh.

Izhakian and Margolis noted that the bicyclic monoid can be faithfully represented by $2 \times 2$ upper triangular tropical matrices, and furthermore that the semigroup of all such matrices satisfies Adjan's minimal length identity $ABBA AB ABBA = ABBA BA ABBA$ for the bicyclic monoid. In light of these results they posed the natural question of whether these two semigroups satisfy exactly the same semigroup identities. In joint work with Laure Daviaud and Mark Kambites, we provided a necessary and sufficient condition for an identity to hold in the semigroup $UT_n$ of $n \times n$ upper triangular tropical matrices, and used this result in the case $n=2$ to give a positive answer to the above. In further joint work with Ngoc Tran, we provide geometric methods and algorithms to verify, generate and enumerate $UT_n$ identities of a fixed length over a two letter alphabet. This leads to some new observations in the case of the bicyclic monoid.

A classical Poincaré inequality states that for a smooth function on a ball in Euclidean space, the L^p norm of the deviation of the function from its average is controlled by the L^p norm of the gradient of the function. In recent years analogous inequalities have been studied on general metric spaces, where they may or may not hold depending on the particular space. In recent work with Hume and Tessera, we have introduced the "Poincaré profile" of a space which measures, on large scale, to what extent such inequalities hold. This idea generalises the "Separation Profile" of Benjamini, Schramm and Timar. In this talk I'll survey some of the history and motivations of these ideas, and some results and applications of our work, for example to show non-embedding results for groups.

Algebraic perspectives and techniques have become increasingly prevalent in combinatorial design theory. The designs of interest can be viewed as square matrices whose rows (or columns) taken pairwise obey a fixed constraint depending on the matrix size and coefficient ring. Hadamard matrices and their generalisations are important examples; the constraint in these cases is orthogonality. The literature on Hadamard matrices is immense, covering numerous applications in areas such as signal processing, cryptography, and quantum computing. The algebraic approach has been especially successful in solving existence and classification problems for `cocyclic' designs, which are defined via certain regular subgroups of their automorphism groups. The pairwise row/column constraint for cocyclic designs translates to a simpler balance condition (e.g., a cocyclic matrix is Hadamard if and only if all non-initial row sums are zero). We present a survey of algebraic design theory, emphasizing some key results and open problems. In particular, we mention recent work extending the notion of cocyclic design when necessary restrictions on matrix size are modified (e.g., what is the analogue of cocyclic Hadamard matrix when the size is even but not divisible by 4?). This arose from considerations of the maximal determinant problem originally posed by Hadamard.

There is a widely studied connection between subgroups of the infinite symmetric group and automorphisms of first-order structures; in particular, homogeneous structures (as characterised in the celebrated theorem of Fraisse) provide examples of interesting infinite permutation groups. This connection can be generalised in a surprising fashion, as there exist submonoids of the infinite symmetric group that are not necessarily groups; these are called permutation monoids. Much like the group case, there is a correspondence between permutation monoids and bimorphism monoids (monoids of bijective endomorphisms) of first-order structures. In this talk, I will explore this connection, introduce the idea of an MB-homogeneous structure and characterise these by demonstrating a Fraisse-like theorem. I will then go on to examine MB-homogeneous graphs in more detail, leading to some surprising results. The material in this talk almost wholly consists of work done during my PhD, and is joint with David Evans (Imperial) and Bob Gray (UEA).

In this talk, I will go over some of the recent developments on nonlinear dispersive PDEs, such as the nonlinear Schroedinger equations (NLS) and nonlinear wave equations (NLW), with rough and random data and/or forcing. In particular, taking the stochastic NLS and the stochastic nonlinear heat equations as examples, I will describe the difference between the dispersive and parabolic problems from the viewpoint of critical regularities, etc. If time permits, I will also discuss some recent development for the stochastic NLW.

Suppose that we have a framework consisting of finitely many fixed-length bars connected at universal joints. Such frameworks (and variants) arise in many guises, with applications to the study of sensor networks, the matrix completion problem in statistics, robotics and protein folding. The fundamental question in rigidity theory is to determine if a framework is rigid or flexible. The standard approach in combinatorial rigidity theory is to differentiate the quadratic equations constraining the distances between joints, and work with these linear equations to determine if the framework is infinitesimally rigid or flexible. In this talk I will discuss recent progress using algebraic matroids that gives further insight into the infinitesimal theory and also provides methods for identifying special bar lengths for which a generically rigid framework is flexible. We use circuit polynomials to identify stresses, or dependence relations among the linearized distance equations and to find bar lengths which give rise to motions. This is joint work with Zvi Rosen, Louis Theran, and Cynthia Vinzant.

Stemming from the Burnside problem, branch groups have delivered lots of exotic examples over the past 30 years. Among them are easily describable finitely generated torsion groups, as well as the first example of a finitely generated group with intermediate word growth. We will investigate a generalisation of the Grigorchuk-Gupta-Sidki branch groups and talk about their maximal subgroups and about their profinite completion. Additionally, we demonstrate a link to a conjecture of Passman on group rings.

It is a classical problem to try to count the number of closed curves on (hyperbolic) surfaces with bounded length. Due to people such as Delsart, Huber, and Margulis it is known that the asymptotic growth of the number of curves is exponential in the length. On the other hand, if one only looks at simple curves the growth is polynomial. Mirzakhani proved that the number of simple curves on a hyperbolic surface of genus $g$ of length at most $L$ is asymptotic to $L^{6g-6}$. Recently, she extended her result to also hold for curves with bounded self intersection, showing that the same polynomial growth holds. In this talk I will discuss her results and some recent generalizations.

A subset $S$ of a group $G$ is product-free if for all $x$ and $y$ in $S$, the product $xy$ is not in $S$. This definition generalises the notion of sum-free sets of integers, where these sets were first studied. In this talk I'll: give an overview of what's known about sum-free and product-free sets in groups; introduce the related concept of filled groups; describe some joint work in this area with Chimere Anabanti and Grahame Erskine.

A quasisymmetric map is one that changes angles in a controlled way. As such they are generalizations of conformal maps and appear naturally in many areas, including Complex Analysis and Geometric group theory. A quasisphere is a metric sphere that is quasisymmetrically equivalent to the standard $2$-sphere. An important open question is to give a characterization of quasispheres. This is closely related to Cannon's conjecture. This conjecture may be formulated as stipulating that a group that ``behaves topologically'' as a Kleinian group ``is geometrically'' such a group. Equivalently, it stipulates that the ``boundary at infinity'' of such groups is a quasisphere. A Thurston map is a map that behaves ``topologically'' as a rational map, i.e., a branched covering of the $2$-sphere that is postcritically finite. A question that is analog to Cannon's conjecture is whether a Thurston map ``is'' a rational map. This is answered by Thurston's classification of rational maps. For Thurston maps that are expanding in a suitable sense, we may define ``visual metrics''. The map then is (topologically conjugate) to a rational map if and only if the sphere equipped with such a metric is a quasisphere. This talk is based on joint work with Mario Bonk.

The mapping class group of an orientable surface (of finite type) is the group or orientation preserving diffeormorphisms of the the surface modulo isotopy. There are three types of mapping classes (Thurston classification): finite order, reducible and pseudo-Anosov genearlising the classification for modular group $SL(2,\mathbb{Z})$. From multiple perspectives, pseudo-Anosov maps are the most interesting type. This talk will survey the theory of pseudo-Anosov maps with small entropy. It will subsequently focus on deriving bounds in terms of genus for a particular notion of entropy: "translation distance in the curve complex". The main result is joint work with Chia-yen Tsai.

Combinatorial structures have been considered under various orders, including substructure order and homomorphism order. In this talk, I will introduce and discuss the homomorphic image order, corresponding to the existence of a surjective homomorphism between two structures. I will focus on partial well-order and antichains, exploring how the homomorphic image order behaves in the context of graphs and graph-like structures. In particular, I will discuss a near-complete characterization of partially well-ordered avoidance classes with one obstruction. This is joint work with Nik Ruskuc.

Residual finiteness growth for a group provides a way of measuring how effectively the finite quotients measure the group. In this talk, we will look at a family of groups acting on a rooted tree and consider specifically the finite $p$-quotients. We will show how the action on the tree provides a method for constructing groups of arbitrarily large $p$-residual finiteness growth.

The notion of an amenable group dates back to von Neumann in 1929, and appears in many different guises; such as in the Banach-Tarski paradox, and in the spectral geometry of manifolds. We discuss a more dynamical setting where amenability appears. Namely, we are interested in the growth of periodic orbits of the geodesic flow arising from negative curvature. No prior knowledge of these objects is assumed! This is joint work with R. Sharp.

Two fundamental finiteness properties in group theory are those of being finitely generated and of being finitely presented. These two properties were generalised to higher dimensions by C. T. C. Wall in 1965. A group G is said to be of type Fn if it has an Eilenberg-MacLane complex K(G,1) with finite n-skeleton. (Here K(G,1) is a certain nice topological space with fundamental group G.) It may be shown that a group is finitely generated if and only if it is of type F1 and is finitely presented if and only if it is of type F2. Related to this is a certain homological finiteness property called FPn. This property is defined for a monoid S in terms of the existence of certain resolutions of free left ZS-modules. The property FPn was introduced for groups by Bieri in (1976). In monoid and semigroup theory the property FPn arises naturally in the study of string rewriting systems. The connection between complete rewriting systems and homological finiteness properties is given by a result of Anick (1986) which shows that a monoid that admits such a presentation must be of type FPn for all n. The properties Fn and FPn are closely related. In particular, for finitely presented groups they are equivalent. Because of the connection with rewriting systems, the finiteness property FPn for monoids has received a great deal of attention in the literature. It is sometimes easier to establish the topological finiteness properties Fn for a group than the homological finiteness properties FPn, especially if there is a suitable geometry/topological space on which the group acts in a nice way. Currently no theory of Fn exists for monoids. In this talk I will describe some recent joint work with Benjamin Steinberg (City College of New York) which was motivated by the question of developing a useful notion of Fn for monoids. This led us to develop a theory of monoids acting on CW complexes. I will explain the ideas we have developed and some of their applications.

Roughly speaking, a set is called a "Salem set" if it carries a measure whose Fourier transform decays polynomially with degree -s/2 where s is the Hausdorff dimension of the set (this is the fastest possible decay). Salem sets are often found via random processes, such as the random distortion of a Cantor set under Brownian motion. An old question of Kahane going back to the 1960s was whether or not the graph of classical Brownian motion is almost surely a Salem set. I will discuss this question in some detail: first I will show that the answer is 'no', and secondly I will show how to compute the optimal almost sure decay rate of the Fourier transform of measures supported on the graph. I will keep technical detail to a minimum and will not assume (much) a priori knowledge of Fourier analysis, or probability theory. This talk will include joint work with Tuomas Orponen (University of Helsinki) and Tuomas Sahlsten (University of Manchester).

Majorana theory is an axiomatic framework in which to study objects related to the Monster group and its 196,884 dimensional representation, the Griess algebra. The theory was first developed in 2009 and was inspired by results by mathematicians such as M. Miyamoto and S. Sakuma who studied the Griess algebra using vertex operator algbras, objects used in the proof of Monstrous moonshine. The objects at the centre of the theory are known as Majorana algebras and can be studied either in their own right, or as Majorana representations of certain groups. I will be discussing my work, which builds on that of A. Seress, developing an algorithm to construct Majorana representations.

(Joint work with A. Le Boudec) The tree almost automorphism groups are non-discrete locally compact completions of the Higman-Thompson groups. The tree almost automorphism groups are independently interesting locally compact groups, and furthermore every group that almost acts on a sufficiently regular rooted tree embeds into one of these groups. We begin by introducing the almost automorphism groups and describing their relationship to the Higman-Thompson groups. We then consider the subgroups such that every element is contained in a compact subgroup; such groups are the topological analogue of torsion subgroups and are called periodic. We show every periodic subgroup is indeed locally elliptic - i.e. every finite set is contained in a compact subgroup. As applications, we recover a result for Thompson's group V as well as a new observation about the Röver group. We finally consider the commensurated subgroups of almost automorphism groups; these subgroups generalize normal subgroups. We show every commensurated closed subgroup of an almost automorphism group is either finite, compact and open, or equal to the entire group. As an application, we obtain new information on the possible lattice envelopes of Thompson's group T.

Given a problem from number theory a useful technique is to rephrase it in terms of a property of a dynamical system. One can then use the statistical/topological properties of the dynamical system to gain more insight and hopefully solve the original problem. This talk will be an exposition of this technique and will include many examples.

The classical Analyst's Traveling Salesman Theorem of Peter Jones gives a condition for when a subset of Euclidean space can be contained in a curve of finite length (or in other words, when a "traveling salesman" can visit potentially infinitely many cities in space in a finite time). The length of this curve is given by a sum of quantities called beta-numbers that measure how non-flat the set is at each scale and location. Conversely, given such a curve, the sum of its beta-numbers is controlled by the total length of the curve, giving us quantitative information about how non-flat the curve is. This result and its subsequent variants have had applications to various subjects like harmonic analysis, complex analysis, and harmonic measure. In this talk, we will introduce a version of this theorem that holds for higher dimensional objects other than curves. This is joint work with Raanan Schul.

The Valued Constraint Satisfaction Problem (VCSP) is a well-known combinatorial problem. An instance of VCSP is given by a finite set of variables, a finite domain of labels for the variables, and a sum of functions, each function depending on a subset of the variables. Each function can take finite rational values specifying costs of assignments of labels to its variables or the infinite value, which indicates an infeasible assignment. The goal is to find an assignment of labels to the variables that minimizes the sum. The case when all functions take only values 0 and infinity corresponds to the standard CSP. We study (assuming that $Peq NP$) how the computational complexity of VCSP depends on the set of functions allowed in the instances, the so-called constraint language. Helped greatly by algebra, massive progress has been made in the last three years on this complexity classification question, and our work gives, in a way, the final answer to it, modulo the complexity of CSPs. This is joint work with Vladimir Kolmogorov and Michal Rolinek (both from IST Austria).

Starting with a substitution tiling, such as the Penrose tiling, we demonstrate a method for constructing infinitely many new substitution tilings. Each of these new tilings is derived from a graph iterated function system and the tiles typically have fractal boundary. As an application of fractal tilings, we construct an odd spectral triple on a C*-algebra associated with an aperiodic substitution tiling. Even though spectral triples on substitution tilings have been extremely well studied in the last 25 years, our construction produces the first truly noncommutative spectral triple associated with a tiling. My work on fractal substitution tilings is joint with Natalie Frank and Sam Webster, and my work on spectral triples is joint with Michael Mampusti.

Graphons are uncountable limits of sequences of finite graphs. Their invention in 2006 by Lovasz and Szegedy revolutionized both the finite and the infinite graph theory by bringing an unforeseen connection. Graphons, also known as combinatorial limits can be seen as certain ultraproducts, which makes them amenable to study using the methods of logic. We shall give a very general talk about this concept and at the end present some joint results with Tomasic.

The aim is to initiate a ``manifold'' theory for metric Diophantine approximation on the limit sets of Kleinian groups. We investigate the notions of singular and extremal limit points within the geometrically finite Kleinian group framework. Also, we consider the natural analogue of Davenport's problem regarding badly approximable limit points in a given subset of the limit set. Beyond extremality, we discuss potential Khintchine-type statements for subsets of the limit set. These can be interpreted as the conjectural ``manifold'' strengthening of Sullivan's logarithmic law for geodesics.

A (bar-joint) framework (G,p) is a graph G, along with a placement p of its vertices into R^d. A framework is said to be universally rigid if any other (G,q) in *any dimension* $D\geq d$ that has the same edge lengths as (G,p) is related to (G,p) by a rigid body motion. I'll describe an algebraic characterisation of which graphs G have generic universally rigid frameworks (G,p) and a close connection to a widely used semidefinite programming algorithm for the graph realisation or "distance geometry" problem. Joint work with Bob Connelly and Shlomo Gortler.

In the talk we will survey a novel domain of computational group theory: computing with infinite linear groups. We will provide an introduction to the area, and will discuss available methods and algorithms. Special consideration will be given to the most recent developments in computing with arithmetic groups and its applications. This talk is aimed at a general mathematical audience.

The number of irreducible polynomials over a finite field was first counted by Gauss. We will explain a connection between counting the number of irreducible polynomials over F_q with certain properties, and the number of rational points on some related algebraic curves. This idea can be used to count the number of irreducible polynomials with certain coefficients being 0. The appearance of supersingular curves explains the interesting periodic behaviour in the formulae, and new formulae are also obtained.

In this talk, we investigate the long standing problem of exact dimensionality of self-affine measures. We show that every self-affine measure on the plane is exact dimensional regardless of the choice of the defining iterated function system and it satisfy the Ledrappier-Young formula.

In this talk we will start by recalling the basic 1-dimensional derivatives: the usual one, the affine, and the projective. We will show how these tools allow establishing important results on the algebraic structure of groups of diffeomorphisms of 1-dimensional manifolds. For instance, a variation of the projective derivative leads to the following theorem of the speaker: every finitely-generated Kazhdan group of (smooth enough) circle diffeomorphisms is finite. Several open problems will be addressed.

This work (which is joint with Peter Neumann and Simon Smith) began as a study of the structure of infinite permutation groups G in which point stabilisers are finite and all infinite normal subgroups are transitive. This led to a generalisation in which point stabilisers are merely assumed to satisfy min-N, the minimal condition on normal subgroups. The groups G are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal non-trivial normal subgroups, or they have a regular normal subgroup M which is a divisible abelian p-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on the socle of M.

We present estimates for the Fourier transforms of Gibbs measures associated to iterated function systems generated by linear fractional transformations. As an application we obtain that the Patterson-Sullivan measure and other Gibbs measures on certain Fuchsian groups have a power decay for the Fourier transform and in particular showing that they are Rajchman measures. This yields that the limit sets for these Fuchsian groups have positive Fourier dimension and have a prevalence of numbers with strong equidistribution features. The talk is based on a joint work with Thomas Jordan (Bristol) and Tomas Persson (Lund).

Quasicrystals are ordered but aperiodic discrete point patterns. They were found in diffraction patterns of physical materials in the 80's, but models for quasicrystals, apieriodic tilings, had been investigated as mathematical objects a lot earlier. We give a very short introduction to the topic, describe the cut and project method for producing aperiodic tilings, and make an observation connecting regularity of the cut and project set to Diophantine approximation. We then explain implications of this observation, in both number theory and tiling theory. The talk is based on several recent works, joint with Alan Haynes, Antoine Julien, Lorenzo Sadun and James Walton.

A primitive root is a generator of the (cyclic) multiplicative group of a finite field. Consecutive elements in a finite field are formed by adding 1. Is it possible to guarantee the existence of two (or more) consecutive primitive roots? We consider this and other existence questions that can be resolved theoretically, perhaps with the aid of a "feasible" amount of computation. Some of the material described is joint work with Tomás Oliviera e Silva (Aveiro) and Tim Trudgian (Canberra).

A classical result of Gelfand shows that the exponential growth rate of the powers of a matrix is determined by its spectrum. This idea admits many inequivalent generalisations to sets of matrices, such as the joint spectral radius, lower spectral radius, Lyapunov exponent, and matrix pressure. I will describe some of the difficulties of working with these quantities and give some positive and negative results on their continuity and computability properties. Towards the end of the talk I will apply these results to show that the affinity dimension of a self-affine fractal is a computable function of the linear parts of the affinities.

One of the main purposes of dynamical systems is to understand the behavior of the space of orbits of continuous group and semigroup actions on compact metric spaces. For their simplicity, the most studied and well understood classes of such dynamical systems are Z, N or R-actions, which correspond to the dynamics of homeomorphisms, continuous endomorphisms or continuous flows, respectively. A notion of topological complexity for such dynamical systems has been proposed in the seventies and was very well studied by Goodwin, Bowen, Walters and Parry, among others. In particular, these dynamical systems satisfy a variational principle: the topological complexity of the dynamics is the supremum of the measure theoretical complexity among the space of invariant probability measures. Such strong relations between the topological and ergodic features of a dynamical system is still unavailable for general group actions. On the one hand, the theory is not unified since several notions of topological complexity have been proposed, and many of them depend on properties of the group action as commutativity or amenability. On the other hand, many group actions admit no common invariant measures and this notion should be replaced by a more flexible concept. In this talk I will first recall the concepts of topological pressure and the variational principles for Z and Zd actions. Then I will propose a notion of topological entropy and pressure for finitely generated semigroup actions and illustrate how this mimics some of the features of the notion proposed in the seventies for Z-actions, namely its regularity and bounds on the exponential growth of periodic orbits in the particular case of finitely generated semigroups of expanding maps. Focusing on the later setting for simplicity, we will also discuss some results on the ergodic properties of the semigroup dynamics and zeta functions. These results are part of joint works with F. Rodrigues (UFRGS, Brazil) and M. Carvalho (U. Porto, Portugal).

We describe some experiments and results on random polynomials with integer coefficients. Among the questions considered are the likelihood of the value of such a polynomial being prime, the distribution of the number of roots modulo primes (and not), and many others.

The dynamical properties of certain shift spaces are presented. We introduce two new classes of shifts, namely boundedly supermultiplicative (BSM) shifts and balanced shifts. It turns out that any almost specified shift is both BSM and balanced, and any balanced shift is BSM. However, there are examples of shifts which are BSM but not balanced. We also study the measure theoretic properties of balanced shifts and we show that a shift space admits a Gibbs state if and only if it is balanced. The S-gap shift and the β-shift will be our main examples.

Sinai billiards form a natural class of dynamical systems with chaotic properties. In this nontechnical talk, I will only consider 2-d billiards. Our understanding of the ergodic properties of the billiard map (from collision to collision) is fairly complete, and exponentially mixing was proved by L.-S. Young almost twenty years ago. Describing Sinai billiard flows (the continuous time dynamics) is more difficult. I will present joint recent results with Demers and Liverani.

Thompson's group V is probably the best known example of a finitely presented simple group. The presentation originally given by Thompson in his notes appears to have remained the best in terms of fewest generators and relations for decades. In this talk, I will give a number of different collections of generators, including a new smaller presentation, for V and comment on generating sets for its relatives nV. These will illustrate how Thompson's group V can be viewed as an infinite analogue of the finite alternating and symmetric groups. This is ongoing (and nearly finished!) joint work with Collin Bleak.

Generic analytic curves are dense in the plane. For particular paramatrised families of analytic curves, this need not be true (e.g. graphs of complex polynomials), or something stronger could be true (e.g. under the zeta-function, the image of every vertical line in the critical strip is dense). Not many classes of explicit dense curves were known. We show that exponential of exponential of almost every line in the complex plane is dense in the plane, along with some related results.

Let G be a transitive permutation group. If G is finite, then a classical theorem of Jordan implies the existence of fixed-point-free elements, which we call derangements. This result has some interesting and unexpected applications, and it leads to several natural problems on the abundance and order of derangements in G that have been the focus of recent research. In this talk, I will discuss some of these related problems, and I will report on recent joint work with Hung Tong-Viet on primitive permutation groups with extremal derangement properties.

Minkowski's question mark function is a Holder continuous bijection from the unit interval to itself which maps quadratic irrationals to rationals and can be defined using the continued fraction expansion. It is a singular function and so has 0 derivative almost everywhere (despite being strictly increasing). We will show that it crops up in dynamical systems through the topological conjugacy between the Farey map and the doubling map and as an invariant measure for the Gauss map (x↦1/xmod1). A natural question to ask about singular function is how their Fourier coefficients behave and in fact Salem asked whether the Fourier coefficients for the Minkowski question mark function decay as n tends to infinity. We will show that by viewing the Minkowski question mark function as an invariant measure for the Gauss map this question can be settled. If time permits we'll discuss some consequences of the Fourier transform of a singular measure decaying polynomially. This is joint work with Tuomas Sahlsten (Jerusalem).

In asymptotic group theory, one associates to a group Γ a sequence of numbers an and studies the behaviour of an as n tends to infinity (for instance, an can be the number of subgroups of Γ of index n, or the number of isomorphism classes of irreducible complex representations of Γ of degree n). One way to do this is to use the an as the coefficients of a zeta function ζΓ(s):=∑n=1∞ann−s, where s is a complex parameter. I will discuss subgroup and representation zeta functions of finitely generated nilpotent groups. This involves ideas from model theory and p-adic integration.

We introduce and study a new combinatorial object called a web world. A web world consists of a set of diagrams that we call web diagrams. The motivation for introducing these comes from particle physics, where web diagrams arise as particular types of Feynman diagrams describing scattering amplitudes in non-Abelian gauge (Yang-Mills) theories. The web world of a web diagram is the set of all web diagrams that result from permuting the order in which endpoints of edges appear on a peg. To each web world we associate two matrices called the web-colouring matrix and web-mixing matrix, respectively. The entries of these matrices are indexed by ordered pairs of web diagrams (D1,D2), and are computed from those colourings of the edges of D1 that yield D2 under a certain transformation determined by each colouring. One of the main goals is the calculation of the web-mixing and web-colouring matrices. In this talk I will give an overview of the results we have obtained so far. These include a decomposition theorems for disjoint web worlds, results pertaining to the diagonal entries of the matrices and how they relate to order preserving maps on posets, and a combinatorial proof of idempotency of the web-mixing matrices

An Artin group is a group with a presentation of the form ⟨x1,x2,⋯,xn∣xixjxi⋯⏞mij=xjxixj⋯⏞mij,i,j∈{1,2,⋯,n},i≠j⟩ for mi,j∈N∪∞,mij≥2, which can be described naturally by a Coxeter matrix or graph. This family of groups contains a wide range of groups, including braid groups, free groups, free abelian groups and much else, and its members exhibit a wide range of behaviour. Many problems remain open for the family as a whole, including the word problem, but are solved for particular subfamilies. The groups of finite type (mapping onto finite Coxeter groups), right-angled type (with each mij∈{2,∞}), large and extra-large type (with each mij≥3 or 4), FC type (every complete subgraph of the Coxeter graph corresponds to a finite type subgroup) have been particularly studied. After introducing Artin groups and surveying what is known, I will describe recent work with Derek Holt and (sometimes) Laura Ciobanu, Eddy Godelle, dealing with a big collection of Artin groups, containing all the large groups, which we call `sufficiently large'. For those Artin groups Holt and I have elementary descriptions of the sets of geodesic and shortlex geodesic words, and can reduce any input word to either form. So we can solve the word problem, and prove the groups shortlex automatic. And, following Appel and Schupp we can solve the conjugacy problem in extra-large groups in cubic time. For many of the large Artin groups, including all extra-large groups, Holt, Ciobanu and I can deduce the rapid decay property and verify the Baum-Connes conjecture. And although our methods are quite different from those of Godelle and Dehornoy for spherical-type groups, we can pool our resources and derive a weak form of hyperbolicity for many, many Artin groups. I'll explain some background for the problems we attach, and outline their solution.

In this talk I will first give an overview of the subject of open dynamical systems and present some key results in the case that the system preserves a finite measure. Then I will introduce the class of systems we recently studied, some of which preserve an infinite measure, and present the results. I will outline some results from infinite ergodic theory which might allow our results to extended to other systems. This is joint work with Georgie Knight.

I will describe some more-or-less natural models of random 3-dimensional manifolds, and describe recent advances (which draw from very diverse parts of mathematics) in understanding what a random such manifold looks like.

It is well known that the symmetric group on a countably infinite set is simple modulo the subgroup of finitary permutations; a similar result holds for countable-dimensional general linear groups. I will describe a sequence of general results, starting off with work of Lascar in 1992, which show that the automorphism groups of certain countable structures are simple, or are simple modulo a normal subgroup of 'bounded' automorphisms. I will discuss some recent applications of these results to structures constructed using Hrushovski amalgamation classes (first saying what these are and why they are interesting). This is joint work with Zaniar Ghadernezhad and Katrin Tent.

Acyclic orientations of graphs are related to the efficient use of radio spectrum on the one hand, and mathematical properties such as graph colouring on the other. In this talk we explore the solution space of acyclic orientations, focusing on the number associated with a given graph. We reveal what is known about the distribution of these numbers, focusing on extreme values. The graphs which provide the minimum value are shown to be extremal for other graph parameters, such as number of cliques and of forests, also. Of particular interest are the maximum values and the graph structures which achieve these. We have many conjectures and pose several open problems to the audience. This talk will highlight computational considerations as well as algebraic properties, illustrate how the two are complementary. I look forward to hearing about your research in St. Andrews and exploring various possible open problems of joint interest. Joint work with Peter Cameron, St. Andrew's University, and Robert Schumacher, Ph.D student, City University London

Let L be quadratic extension of a p-adic number field K. The ring of integers OL has a non-trivial involution induced by the Galois automorphism of L, which induces an involution ∗ on Mn(OL) in a manner that reminds us of the conjugate-transpose operation. The resulting unitary group Un∗(OL)={X∈Mn(OL):XX∗=I}. The congruence subgroup property implies that any continuous finite-dimensional representation of Un(OL) factors through a congruence subgroup. This reduces the study of these representations to that of describing the irreducible representations of unitary groups over finite local rings. Recently we have calculated the orders of unitary groups of finite local rings in both ramified and unramified cases, and constructed irreducible characters that arise as constituents of the Weil representation of Un(OL). These results rely on tools from Clifford theory and hermitian geometry that we will explore in this talk. This is based on joint work with Fernando Szechtman, Rachael Quinlan, and James Cruikshank.

The length of a subgroup chain in a group is bounded by the logarithm of the group order. This fails for semigroups, but it is perhaps surprising that there is a lower bound for the length of a subsemigroup chain in the full transformation semigroup which is a constant multiple of the semigroup order. In this talk I will discuss the latter, and some related, results. This is joint work with P. J. Cameron, M. Gadouleau and Y. Peresse.

The paper [J. Balogh, B. Bollobas, D. Weinreich, A jump to the Bell number for hereditary graph properties, J. Combin. Theory Ser. B 95 (2005) 29-48] identifies a jump in the speed of hereditary graph properties to the Bell number Bn and provides a partial characterisation of the family of minimal classes whose speed is at least Bn. In this talk, we give a complete characterisation of this family. Since this family is infinite, the decidability of the problem of determining if the speed of a hereditary property is above or below the Bell number is questionable. We answer this question positively by showing that there exists an algorithm which, given a finite set F of graphs, decides whether the speed of the class of graphs containing no induced subgraphs from the set F is above or below the Bell number. For properties defined by infinitely many minimal forbidden induced subgraphs, the speed is known to be above the Bell number. Joint work with Aistis Atminas, Andrew Collins and Jan Foniok.

James Clerk Maxwell (1831-1879) was, by any measure, a natural philosopher of the first rank who made wide-ranging contributions to science. He also, however, wrote poetry. In this talk examples of Maxwell's poetry will be discussed in the context of a biographical sketch. It will be argued that not only was Maxwell a very good poet, but that his poetry enriches our view of his life and its intellectual context.

I will motivate a couple of problems in discrete harmonic analysis by discussing their Euclidean counterparts. Specifically, I will focus on Stein's spherical maximal function, Magyar's discrete version and the ideas behind Magyar--Stein--Wainger's theorem (proving $L^p$ boundedness). We will pay particular attention to the synthesis of ideas from harmonic analysis and analytic number theory. I will then discuss higher degree versions and lacunary versions and conclude with applications to ergodic theory and combinatorics.

Mathematical billiards are popular models from mathematical physics of moving particles undergoing elastic collisions. In 1979, Pianigiani and Yorke posed the problem of characterizing escape rates and limiting distributions for a chaotic billiard table with small holes. In this talk, I will introduce the basic set-up and motivating questions in the study of open dynamical systems. I will then explain how a recently developed framework using functional analysis can be applied to billiard tables with a variety of holes and having either finite or infinite horizon. Recent results using this approach include the existence of physical limiting conditionally invariant measures, which are the analogue of physical measures for open systems and a variational principle connecting the escape rate to the entropy on the survivor set.

In this talk I will introduce the "rational hierarchy of semigroups". In this hierarchy, semigroups are compared based on the difficulty of their word problem. I will give some properties of semigroups in this hierarchy, and more importantly, I will give a survey of open questions and research directions I am interested in.

The development of algorithms for arbitrary precision computation in the symbolic computation system Maple has led to some interesting discoveries, as well as some interesting re-discoveries. In this talk, I will survey three areas of research and development that I have been involved in over the past 24 years.

The study of abstract combinatorial structures, like graphs, and their associated computational problems has been central to the development of the theory of algorithmic complexity. One reason for this is that such structures provide the right level of abstraction for both formulating and solving a large variety of problems appearing in practice. However, algorithms efficiently solving such problems often implicitly, and subtly, violate this abstraction when choosing of an arbitrary element from a set of elements, e.g., as in the selection of a pivot during the Gaussian elimination algorithm for matrix rank. In practice structures are represented in programs by a particular encoding, say, as a binary string, which contains information external to the abstraction. This information can be used to efficiently implement an arbitrary algorithmic choice, e.g., by selecting the element with the lexicographically first encoding. It is a major open question in descriptive complexity whether such violations of abstraction are necessary when efficiently solving graph problems. In this talk I will demonstrate that a host of fundamental combinatorial and geometric optimization problems can be efficiently solved on structures without violating their abstraction. In particular, I shall describe how to efficiently decide the size of a maximum matching in a graph without making arbitrary algorithmic choices, settling an open problem first posed by Blass, Gurevich, and Shelah. Along the way to this result, I will show, surprisingly, that the same can be done for the Ellipsoid Method for linear programming. This is joint work with Anuj Dawar and Bjarki Holm, which appeared in LICS 2013.

We go through a few examples of fractal sets and the definition of Hausdorff dimension, and list some classical results in dimension theory of fractal sets. Keeping the examples in mind, we take a general look at ways for finding the Hausdorff dimension. We finish with some recent results in the field, connecting them to this general framework.

We will briefly survey some of the issues that arise when we try to think of the space of all compact group automorphisms modulo various natural notions of dynamical equivalence. In particular, we will describe some recent work exhibiting continua of equivalence classes of automorphisms.