Intersemester 2016/17:

Title:

Abstract: In this talk I present a new take on polyhedral symmetries. I begin by describing that many viruses have icosahedrally symmetric surface structures. I briefly review recent work (with Reidun Twarock and Celine Boehm) to try and extend this symmetry principle also to the interior of viruses and carbon onions via suitable notions of affine extensions of non-crystallographic Coxeter groups. I have argued that in such reflection group settings (a vector space with an inner product) Clifford algebras are very natural objects to consider and in fact provide a very simple reflection formula. Applying this framework to root systems has led to the construction of the exceptional root system $E_8$ from the icosahedron and a proof that each 3D root system induces a corresponding 4D root system. In particular, the Trinity of irreducible 3D root systems $(A_3, B_3, H_3)$ gives rise to the Trinity of exceptional 4D root systems $(D_4, F_4, H_4)$. These exceptional root systems can thus be viewed as intrinsically three-dimensional phenomena. The countably infinite family $A_1\times I_2(n)$ gives rise to $I_2(n)\times I_2(n)$. Arnold had found a very cumbersome and indirect connection between $(A_3, B_3, H_3)$ and $(D_4, F_4, H_4)$ essentially via exponents in the Coxeter plane. This in fact extends to my full correspondence between 3D and 4D root systems, establishing an ADE correspondence related to the McKay correspondence. Furthermore, one can fully factorise the Coxeter element in the Clifford algebra with the exponents and complex structures of the eigenplanes arising purely from the geometry, without the need to complexify the real vector space.

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Abstract: A self-similar space is a compact metrizable space endowed with some form of self-similar structure. We will show that such spaces sometimes admit many homeomorphisms that preserve the self-similar structure, which we refer to as rearrangements of the space. The resulting groups of rearrangements are closely related to the Thompson groups F, T, and V. This is joint work with Bradley Forrest.

The schedule for Semester 1 of 2016-17 is:

Title:

Abstract: Let $\mathcal{P}$ be a partition of a finite set $X$. We say that a transformation $f:X\to X$ stabilises the partition $\mathcal{P}$ if for all $P\in \mathcal{P}$ there exists $Q\in \mathcal{P}$ such that $Pf\subseteq Q$. Let $T(X,\mathcal{P})$ denote the semigroup of all full transformations of $X$ that preserve the partition $\mathcal{P}$.

In 2005 Pei Huisheng found an upper bound for the minimum size of the generating sets of $T(X,\mathcal{P})$, when $\mathcal{P}$ is a partition in which all of its parts have the same size. In addition, Pei Huisheng conjectured that his bound was exact. In 2009, Araújo and Schneider used representation theory to solve Pei Huisheng's conjecture.

A more general task is to find the minimum size of the generating sets of $T(X,\mathcal{P})$, when $\mathcal{P}$ is an arbitrary partition. In this talk we presents the solution of this problem and discuss some of the proof techniques, which range from representation theory to combinatorial arguments.

This is joint work with João Araújo (Universade Aberta/CEMAT), James Mitchell (University of St Andrews), and Csaba Schneider (Universidade Federal de Minas Gerais).

Title:

Abstract: A measurable transformation $T$ of a probability space $(\Omega,\mathcal{B},m)$ is quasi invariant if it preserves the $\sigma$-ideal of measure $0$ sets. An old question of Halmos, which was answered in the affirmative by Ornstein and L. Arnold, is whether there exists such transformations which have recurrent dynamics but there exists no $\sigma$-finite, $m$-absolutely continuous $T$ invariant measures. Such systems are called type $III$. The type $III$ transformations can be further classified according to their Krieger types $III_\lambda, 0 \leq \lambda \leq 1$ where being type $III_1$ is equivalent to the Maharam extension being ergodic.

In this talk we will discuss these notions and more in the context of the dynamics of the shift with respect to products measures (not necessarily i.i.d.). If time permits we will discuss an application of these results to symmetric $\alpha$-stable processes, some extensions to the case of the shift of inhomogeneous Markov chain and the construction of a new class of Anosov diffeormorphisms of the torus.

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Abstract: 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.

13th Oct, 2016: DOUBLE BILL:

Title:

Abstract: Bernoulli convolutions are arguably the simplest fractal measures on the unit interval, parametrized by a factor t between 0 and 1. They have been studied for almost 80 years, without much success in the overlapping case. Only for countably many parameters it is exactly known whether the measure admits a density function. We introduce these measures from the viewpoints of probability, fractals, number systems, and dynamical systems. Then we present a new approach which represents the whole parametric family by a function of two parameters. The structure of that function is studied with computer assistance.

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Abstract: I will talk about some similarities between Fractal Geometry and Probability Theory, in particular the Markov Chains, Random Walks and Iterated Function System. I will then talk about some recent progress on projections and slices of random and deterministic fractal measures on the plane.

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Abstract: 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.

Title:

Abstract: 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.

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Abstract: 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.

Title:

Abstract: 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.

Talks in Summer 2016:

Title:

Abstract: Martin Gardner, The Best Friend Mathematics Ever Had, was best known for his 300 "Mathematical Games" columns in Scientific American, in which he introduced thousands of budding mathematicians to elegant problems and magical items which still lead to "Aha!" moments today. "Celebration of Mind" is an international initiative each October to continue what he did best, connecting Mathematics, Magic and Mystery. Gardner also asked simple questions that inspired serious research, and some of those questions remain unanswered today. We'll survey what he achieved and the legacy he leaves behind.

Title:

Abstract: Recently Nicolas Monod showed that the group of all piecewise projective homeomorphisms of the unit interval is nonamenable and yet does not contain a nonabelian free subgroup. This provided a new, very accessible example of a group admitting no finitely additive translation invariant probability measure yet not containing a nonabelian free subgroup. This talk will explore a finitely presented subgroup of Monod's group which is closely related to Thompson's group F and already exhibits these striking properties. This is joint work with Yash Lodha.

The schedule for Semester 1 of 2015/16 is:

Title:

Abstract: The plactic monoid (the monoid of Young tableaux) is closely connected with representations of the special linear Lie algebra and with the theory of symmetric functions. In particular, the representation-theoretic notions of Kashiwara operators and crystal bases can be applied to the plactic monoid in a purely combinatorial and monoid-theoretical way, with a very elegant interaction between the resulting crystal structure and the algebraic and combinatorial properties of the plactic monoid. Indeed, one can view the crystal structure as an alternative definition of the plactic monoid.

Krob & Thibon showed that the hypoplactic monoid (the monoid of quasi-ribbon tableaux, and a quotient of the plactic monoid) has a role for quasi-symmetric functions that is analoguous to the role of the plactic monoid for symmetric functions. However, there was no natural crystal structure known for the hypoplactic monoid.

This seminar will describe recent joint work with Malheiro, in which we detach the notion of Kashiwara operators from the underlying representation theory, and introduce a notion of "quasi-Kashiwara operators" that give rise to a "quasi-crystal" structure for the hypoplactic monoid. This quasi-crystal structure leads to new results and improved proofs for known results. It also sheds light on the relationship between the plactic monoid, the hypoplactic monoid, and the sylvester monoid (the monoid of binary search trees).

The exposition will be elementary. Representation theory and the theory of (quasi-)symmetric functions will only appear for motivation; no special knowledge of these areas will be assumed.

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Abstract: 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.

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Abstract: 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 \(\mathbb Z\), \(\mathbb N\) or \(\mathbb R\)-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 \(\mathbb Z\) and \(\mathbb Z^d\) 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 \(\mathbb Z\)-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).

Title:

Abstract: 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.

Title:

Abstract: 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).

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Abstract: 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.

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Abstract: 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).

Title:

Abstract: 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.

Title:

Abstract: The clique-width parameter provides a rough measure of the complexity of structure in (classes of) graphs. A well-known result of Courcelle, Makowsky and Rotics shows that many problems on graphs which are NP-hard in general can be solved in polynomial time in any class of graphs of bounded clique-width. Unlike the better-known treewidth graph parameter, clique-width respects the induced subgraph ordering, and in particular it can handle dense graphs. However, also unlike treewidth there is no known characterisation of the minimal classes of graphs which have unbounded clique-width.

In this talk, I will survey a number of results and techniques for studying the interface between bounded and unbounded clique-width. Of particular interest are insights from the combinatorial study of permutations (``permutation patterns''), which has brought to light several more minimal graph classes with unbounded clique-width, and also suggests that a restricted version of the parameter, called linear clique-width, often appears to characterise the interface.

Time-permitting, I will also discuss recent developments and open problems in the relationship between clique-width and well-quasi-ordering.

21st Jan, 2016:

Title:

Abstract: 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.

Title:

Abstract: 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.

The schedule for Semester 2 of 2014/15 is:

Title:

Abstract: 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.

Title:

Abstract: An Artin group is a group with a presentation of the form \[ \langle x_1,x_2,\cdots,x_n \mid \overbrace{x_ix_jx_i\cdots}^{m_{ij}}= \overbrace{x_jx_ix_j \cdots}^{m_{ij}}, i,j \in \{1,2,\cdots,n\}, i\neq j\rangle\] for \(m_{i,j} \in \mathbb{N} \cup \infty, m_{ij} \geq 2\), 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 \(m_{ij} \in \{2,\infty\}\)), large and extra-large type (with each \(m_{ij}\geq 3\) 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.

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Abstract: 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 \((D_1,D_2)\), and are computed from those colourings of the edges of \(D_1\) that yield \(D_2\) 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

Title:

Abstract: In asymptotic group theory, one associates to a group \(\Gamma\) a sequence of numbers \(a_n\) and studies the behaviour of \(a_n\) as \(n\) tends to infinity (for instance, \(a_n\) can be the number of subgroups of \(\Gamma\) of index \(n\), or the number of isomorphism classes of irreducible complex representations of \(\Gamma\) of degree \(n\)). One way to do this is to use the \(a_n\) as the coefficients of a zeta function \(\zeta_\Gamma(s):= \sum_{n=1}^\infty a_n n^{-s}\), 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.

Title:

Abstract: 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 (\(x \mapsto 1/x \mod 1\)). 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).

Title:

Abstract: 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.

Title:

Abstract: 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.

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Abstract: 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.

Title:

Abstract: 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.

Title:

Abstract: 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 \(\beta\)-shift will be our main examples.

NOTE: This will be in PHYSICS Theatre B

Title:

Abstract: In this talk I will give an overview of work with Eric Friedlander, Julia Pevtsova and Andrei Suslin on module of constant Jordan type. Given a nilpotent linear operator on a vector space, the Jordan type is the partition of the dimension that describes the Jordan canonical form of the operator.

Given two commuting nilpotent operators X and Y, we can ask about the configuration of the Jordan types of the operators aX+bY, for a and b elements of the base field. At its most basic level, the work is concerned with linear algebra. However, it has implications for group representation theory and algebraic geometry.

The schedule for Semester 1 of 2014/15 is:

Title:

Abstract: Ergodic theory studies the long term behaviour of dynamical systems. It has been successfully applied to a number of problems, such as finding patterns in prime numbers, understanding growth in groups, and understanding various geometric properties of measures in Euclidean space, which don't appear to involve any dynamics.

In this talk I will give a basic introduction to ergodic theory, followed by several examples of how one can inject dynamics into problems which don't involve dynamical systems. If time permits, we'll give a simple overview of how recent advances in ergodic theory are being used to generalise the Green-Tao theorem on arithmetic progressions in the prime numbers.

No knowledge of ergodic theory, geometry or number theory will be assumed!

Title:

Abstract: 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.

Title:

Abstract: 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.

Title:

Abstract: 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.

Title:

Abstract: Let \(L\) be quadratic extension of a \(p\)-adic number field \(K\). The ring of integers \(\mathcal{O}_L\) has a non-trivial involution induced by the Galois automorphism of \(L\), which induces an involution \(*\) on \(M_n(\mathcal{O}_L)\) in a manner that reminds us of the conjugate-transpose operation. The resulting unitary group \(U^*_n(\mathcal{O}_L) = \{ X \in M_n(\mathcal{O}_L): XX^* = I \}\).

The congruence subgroup property implies that any continuous finite-dimensional representation of \(U_n(\mathcal{O}_L)\) 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 \(U_n(\mathcal{O}_L)\). 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.

Title:

Abstract: 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

Title:

Abstract: 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.

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Abstract: 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.

The schedule for Semester 2 of 2013/14 is:

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Abstract: 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.

Title:

Abstract: 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.

Title:

Abstract: 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.

Title:

Abstract: 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.

Title:

Abstract: 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.

Title:

Abstract: 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.

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Abstract: A relational structure is homomorphism-homogeneous if any homomorphism between its finite substructures extends to an endomorphism of the structure in question. After providing a short survey of previous results, I will discuss the characterisation of all permutations on a finite set enjoying this property, obtained recently by Ă‰va JungĂˇbel and myself. To this end, I will review the more traditional view of a permutation as a set endowed with two linear orders (which eventually led to the theory of permutation patterns), and then switch to a different representation by a single linear order (considered as a directed graph with loops) whose non-loop edges are coloured in two colours, thereby 'splitting' the linear order into two posets.

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Abstract: 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.

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Abstract: A K3 surface is a kind of two-dimensional analogue of the elliptic curve or complex torus. In theoretical physics, one is interested in dualities between pairs of K3 surfaces and (vector bundles of) matrix algebras. An extensive theoretical framework has been developed to determine criteria for such dualites; however, there are not many explicit examples. We use Clifford algebras to construct an explicit example on a particular K3 surface. The corresponding duality is the inverse of a classical example due to Mukai. Based on joint work with Colin Ingalls.

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Abstract: The problem of reconstructing a measure in \(R^d\) from a (truncated) multi-sequence of its moments has important applications, and is in general very hard to solve. We concentrate on a natural case of a measure \(\mu\) with piecewise-polynomial density supported on a compact polyhedron P, and show that such problems can be solved exactly, due to existence of a natural integral transform of the measure (known as Fantappie transformation), which is a rational function \(F_\mu(u)\). The denominator of \(F_\mu(u)\) is the product of linear functions of the form \( 1-\langle u,v\rangle \), with \(v\) belonging to certain finite multiset V(P).

There are interesting applications of \(F_\mu(u)\) to compact (not necessarily convex) polyhedra. Let \(I(P)\) be the indicator function of \(P\). Then \(I(P)\) can be decomposed (up to a measure 0 subset) as a sum, with real coefficients, of \(I(D)\), where \(D\) runs through simplices with vertices in \(V(P)\). This can be viewed as a non-convex generalisation of triangulations of convex polytopes. Laplace transforms of cones related to such decompositions arise in the theory of hyperplane arrangements. Further refinements and applications will be discussed.

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Abstract: Generalised beta-transformations are the class of piecewise continuous interval maps given by taking the beta-transformation \(x\mapsto \beta x \mod1\), where \(\beta > 1\), and replacing some of the branches with branches of constant negative slope. We would like to describe the set of beta for which these maps can admit a Markov partition. We know that beta (which is the exponential of the entropy of the map) must be an algebraic number. Our main result is that the Galois conjugates of such beta have modulus less than 2. This extends an analysis of Solomyak for the case of beta-transformations, who obtained a sharp bound of the golden mean in that setting.

I will also describe a connection with some of the results of Thurston's fascinating final paper, where the Galois conjugates of entropies of post-critically finite unimodal maps are shown to describe a beautiful fractal. The talk will be suitable for non-specialists, and all technical terms in this abstract will be explained!

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Abstract: Interval Exchange Transformations are invertible piecewise order preserving isometries of the unit interval that generalize rotations. Masur and Veech independently showed that they are typically uniquely ergodic. There are know to be minimal and not uniquely ergodic interval exchanges. In interests of quantifying the size of this measure 0 (and meager) set the main results of this talk are:

a) The Hausdorff dimension of not-uniquely 4-IETs is 2 1/2 as a subset of the 3 dimensional simplex

b) The Hausdorff dimension of flat surfaces in H(2) whose vertical flow is not uniquely ergodic is 7 1/2 as a subset of an 8 dimensional space

c) For almost every flat surface in H(2) the set of directions where the flow is not uniquely ergodic has Hausdorff dimension 1/2.

These results all say that the Hausdorff codimension of these exceptional sets is 1/2. Masur-Smillie showed that the Hausdorff codimension was less than 1. It follows from work of Masur that the Hausdorff codimension is at least 1/2. This is joint work with J. Athreya.

Past colloquia can be found here.

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