2020 is a Special Year in Mathematics Communication, hosted by the Mathematical Education Research Group in CARMA.
Upcoming events include:
as well as regular seminars during the teaching semesters. Events and seminars will address the increasing importance of mathematics communication for and amongst a wide range of contexts and audiences, including across disciplines and industries, with the general public, and in education from kindergarten to PhD.
Further information and details of events will appear on the MathsComm web page.
Optimisation is a branch of applied mathematics that focuses on using mathematical techniques to optimise complex systems. Real-world optimisation problems are typically enormous in scale, with hundreds of thousands of inter-related variables and constraints, multiple conflicting objectives, and numerous candidate solutions that can easily exceed the total number of atoms in the solar system, overwhelming even the fastest supercomputers. Mathematical optimisation has numerous applications in business and industry, but there is a big mismatch between the optimisation problems studied in academia (which tend to be highly structured problems) and those encountered in practice (which are non-standard, highly unstructured problems). This lecture gives a non-technical overview of the presenter's recent experiences in building optimisation models and practical algorithms in the oil and gas, mining, and agriculture sectors. Some of this practical work has led to academic journal articles, showing that the gap between industry and academia can be overcome.
In number theory, special values coming from arithmetic generating functions always provide information about certain geometric invariants of the corresponding objects. For instance, the logarithmic derivative of the Riemann zeta function at s=0 is equal to the natural log of the length of the unit circle. Moreover, the celebrated Kronecker limit formula expresses the logarithmic derivative of the non-holomorphic Eisenstein series at s=0 in terms of the “periods” of “unit” elliptic curves.
In this talk, I will discuss the classical theory on “Kronecker terms”, and mention a similar phenomenon in the “mix characteristic” settings if time allows.
Profinite groups are the inverse limits of finite groups, or equivalently, the compact totally disconnected groups. First-order logic in the signature of groups can directly talk only about their algebraic structure. We address the question whether a profinite group $G$ can be determined by a single first-order sentence: is there a sentence $\phi$ such that $H \models \phi$ if and only if $H$ is topologically isomorphic to $G$, for each profinite group $H$?
Let $p\ge 3$ be a prime. We show that this property holds for the groups $SL_2(\mathbb Z_p)$ and $PSL_2(\mathbb Z_p)$ where $\mathbb Z_p$ is the ring of $p$-adic integers. If we restrict the reference class to the inverse limits of $p$-groups, we obtain many further examples, e.g.\ all groups with a bound on the dimension of the closed subgroups (such as the abelian group $\mathbb Z_p$).
This is joint work with Dan Segal and Katrin Tent.
A major change in the educational policy landscape in many countries has been the introduction of computing into the school curriculum, either as part of Mathematics or as a separate subject. This has often happened alongside the establishment of ‘Coding’ in out-of school clubs. In this talk, we will reflect on the situation in England where computing has been a compulsory subject since 2014 for all students from age 7 to 16 years. We will describe the research project, UCL ScratchMaths, designed to introduce students, aged 9-11 years, to both core computational and mathematical ideas. We will discuss the findings of the project, the challenges faced in its implementation and the exciting next steps in the computing/mathematics initiative from a more international perspective.
Please visit the lecture's Eventbrite page for more information and to register for this free event.
We prove a local-to-global result for fixed points of groups acting on $2$-dimensional affine buildings (possibly non-discrete, and not of type $\tilde{G}_{2}$). In the discrete case, our theorem establishes two conjectures by Marquis. (joint work with Koen Struyve and Anne Thomas)
A classical result of Siegel asserts that the (2,3,7)-triangle group attains the smallest covolume among lattices of $\mathrm{SL}_2(\mathbb{R})$. In general, given a semisimple Lie group $G$ over some local field $F$, one may ask which lattices in $G$ attain the smallest covolume. A complete answer to this question seems out of reach at the moment; nevertheless, many steps have been made in the last decades. Inspired by Siegel's result, Lubotzky determined that a lattice of minimal covolume in $\mathrm{SL}_2(F)$ with $F=\mathbb{F}_q((t))$ is given by the so-called characteristic $p$ modular group $\mathrm{SL}_2(\mathbb{F}_q[1/t])$. He noted that, in contrast with Siegel’s lattice, the quotient by $\mathrm{SL}_2(\mathbb{F}_q[1/t])$ was not compact, and asked what the typical situation should be: "for a semisimple Lie group over a local field, is a lattice of minimal covolume a cocompact or nonuniform lattice?".
In the talk, we will review some of the known results, and then discuss the case of $\mathrm{SL}_n(\mathbb{R})$ for $n > 2$. It turns out that, up to automorphism, the unique lattice of minimal covolume in $\mathrm{SL}_n(\mathbb{R})$ ($n > 2$) is $\mathrm{SL}_n(\mathbb{Z})$. In particular, it is not uniform, giving a partial answer to Lubotzky’s question in this case.
We define the almost automorphism group of a regular tree, also known as Neretin's group, and determine when two elements are conjugate. (joint work with Gil Goffer)
Whyte introduced translation-like actions of groups as a geometric generalization of subgroup containment. He then proved a geometric reformulation of the von Neumann conjecture by demonstrating that a finitely generated group is non amenable if and only it admits a translation-like action by a non-abelian free group. This provides motivation for the study of what groups can act translation-like on other groups. As a consequence of Gromov’s polynomial growth theorem, virtually nilpotent groups can act translation-like on other nilpotent groups. We demonstrate that if two nilpotent groups have the same growth, but non-isomorphic Carnot completions, then they can't act translation-like on each other. (joint work with David Cohen)
A special public event for Pi Day! Join us at NewSpace for MathsCraft activities to suit all ages from 8 years up, from origami to hyperbolic crocheting!
There will be two public talks:
Please drop by and celebrate pi.
In their celebrated 1993 paper, Brink and Howlett proved that all finitely generated Coxeter groups are automatic. In particular, they constructed a finite state automaton recognising the language of reduced words in a Coxeter group. This automaton is not minimal in general, and recently Christophe Hohlweg, Philippe Nadeau and Nathan Williams stated a conjectural criteria for the minimality. In this talk we will explain these concepts, and outline the proof of the conjecture of Hohlweg, Nadeau, and Williams. We will also describe an alternative algorithm to minimise any finite state automaton recognising the language of reduced words in a Coxeter group, which utilises the associated root system of the group.
This work is joint with James Parkinson.
Free groups, and free products of finite groups, are the easiest non-abelian infinite groups to think about. Yet the automorphism groups of such groups still present significant mysteries. We discuss a program of research concerning automorphisms of easily understood infinite groups.
Bridson, Burillo, Elder and Šunić asked if there exists a group with intermediate geodesic growth and if there is a characterisation of groups with polynomial geodesic growth. Towards these questions, they showed that there is no nilpotent group with intermediate geodesic growth, and they provided a sufficient condition for a virtually abelian group to have polynomial geodesic growth. In this talk, we take the next step in this study and show that the geodesic growth for a finitely generated virtually abelian group is either polynomial or exponential; and that the generating function of this geodesic growth series is holonomic, and rational in the polynomial growth case. To obtain this result, we will make use of the combinatorial properties of the class of linearly constrained language as studied by Massazza. In addition, we show that the language of geodesics of a virtually abelian group is blind multicounter.
Many well-known families of groups and semigroups have natural categorical analogues: e.g., full transformation categories, symmetric inverse categories, as well as categories of partitions, Brauer/Temperley-Lieb diagrams, braids and vines. This talk discusses presentations (by generators and relations) for such categories, utilising additional tensor/monoidal operations. The methods are quite general, and apply to a wide class of (strict) tensor categories with one-sided units.
Dye-Sensitized Solar Cells (DSSCs) have remained a viable source of renewable energy since their introduction in 1991 for their novel choice of materials. In particular, the substitution of a high-purity Silicon semiconductor for a nanoporous Titanium Dioxide greatly lowers production costs. Mathematical modelling for DSSCs must account for the electrochemical nature of DSSCs over the traditional models inherited from Shockley's work in the 1940s. Though the literature has developed a diffusion model for this purpose, there is sparse mathematical treatment in this area. The objective of this thesis is to provide mathematical insight with the goals of increasing our understanding of DSSCs and maximising their efficiency. In addition to providing new analytical solutions for linear diffusion models, we also apply Lie symmetry analysis to the nonlinear diffusion model and develop a new fractional diffusion equation based on subdiffusion equations derived from random-walk simulations.
The construction of compactly supported smooth orthonormal wavelets has been reformulated as feasibility problems. This feasibility approach to wavelet construction has been successful in reproducing Daubechies' wavelets and in building non-separable examples of wavelets on the plane. We discuss the extensions of these constructions to allow for the optimization of wavelets' cardinality and symmetry. We also present relevant optimization techniques that we have developed to solve wavelet feasibility problems. Finally, we tackle the under way application of the feasibility approach to construct compactly supported quaternionic orthonormal wavelets with prescribed regularity.
Hierarchically hyperbolic groups (HHGs) and spaces are recently-introduced generalisations of (Gromov-) hyperbolic groups and spaces. Other examples of HHGs include mapping class groups, right-angled Artin/Coxeter groups, and many groups acting properly and cocompactly on CAT(0) cube complexes. After a substantial introduction and motivation, I will present a combination theorem for hierarchically hyperbolic groups. As a corollary, any graph product of finitely many HHGs is itself a HHG. Joint work with B. Robbio.
The notion of a "hierarchically hyperbolic space/group" grows out of geometric similarities between CAT(0) cubical groups and mapping class groups. Hierarchical hyperbolicity is a "coarse nonpositive curvature" property that is more restrictive than acylindrical hyperbolicity but general enough to include many of the usual suspects in geometric group theory. The class of hierarchically hyperbolic groups is also closed under various procedures for constructing new groups from old, and the theory can be used, for example, to bound the asymptotic dimension and to study quasi-isometric rigidity for various groups. One disadvantage of the theory is that the definition - which is coarse-geometric and just an abstraction of properties of mapping class groups and cube complexes - is complicated. We therefore present a comparatively simple sufficient condition for a group to be hierarchically hyperbolic, in terms of an action on a hyperbolic simplicial complex. I will discuss some applications of this criterion to mapping class groups and (non-right-angled) Artin groups. This is joint work with Jason Behrstock, Alexandre Martin, and Alessandro Sisto.
In this talk I will introduce a new concept in graph theory known as generalized graph truncations. Although graph truncations have appeared throughout history, few papers have studied them and only from quite focused perspectives. Here I will give a general outline of how generalized truncations can be constructed as well as a characterisation of them. I will also outline some results I have had including eulerian truncations, planarity, edge-connectivity, and edge-colourings.
We explain the proof that Neretin groups have no nontrivial ergodic invariant random subgroups (IRS). Equivalently, any non-trivial ergodic p.m.p. action of Neretin’s group is essentially free. This property can be thought of as simplicity in the sense of measurable dynamics; while Neretin groups were known to be abstractly simple by a result of Kapoudjian. The heart of the proof is a “double commutator” lemma for IRSs of elliptic subgroups.
I will begin with the definition of topological full groups and explain various examples of them. The topological full group arising from a minimal homeomorphism on a Cantor set gave the first example of finitely generated simple groups that are amenable and infinite. The topological full groups of one-sided shifts of finite type are viewed as generalization of the Higman-Thompson groups. Based on these two fundamental examples, I will discuss recent development of the study around topological full groups.
We show that for almost all primitive integral cohomology classes in the fibered cone of a closed fibered hyperbolic 3-manifold, the monodromy normally generates the mapping class group of the fiber. The key idea of the proof is to use Fried’s theory of suspension flow and dynamic blow-up of Mosher. If the time permits, we also discuss the non-existence of the analog of Fried’s continuous extension of the normalized entropy over the fibered face in the case of asymptotic translation lengths on the curve complex. This talk is based on joint work with Eiko Kin, Hyunshik Shin and Chenxi Wu.
A sequence of expanders is a family of finite graphs that are sparse yet highly connected. Such families of graphs are fundamental object that found a wealth of applications throughout mathematics and computer science. This talk is centred around an "asymptotic" weakening of the notion of expansion. The original motivation for this asymptotic notion comes from the study of operator algebras associated with metric spaces. Further motivation comes from some recent works which established a connection between asymptotic expansion and strongly ergodic actions. I will give a non-technical introduction to this topic, highlighting the relations with usual expanders and group actions.
In 1967 Richard Thompson introduced the group $F$, hoping that it was non-amenable, since then it would disprove the von Neumann conjecture. Though the conjecture has subsequently been disproved, the question of the amenability of Thompson's group F has still not been rigorously settled. In this talk I will present the most comprehensive numerical attack on this problem that has yet been mounted. I will first give a history of the problem, including mention of the many incorrect "proofs" of amenability or non-amenability. Then I will give details of a new, efficient algorithm for obtaining terms of the co-growth sequence. Finally I will describe a number of numerical methods to analyse the co-growth sequences of a number of infinite, finitely-generated groups, and show how these methods provide compelling evidence (though of course not a proof) that Thompson's group F is not amenable. I will also describe an alternative route to a rigorous proof. (This is joint work with Andrew Elvey Price).
The theory of EG, the class of elementary amenable groups, has developed steadily since the class was introduced constructively by Day in 1957. At that time, it was unclear whether or not EG was equal to the class AG of all amenable groups. Highlights of this development certainly include Chou's article in 1980 which develops much of the basic structure theory of the class EG, and Grigorchuk's 1985 result showing that the first Grigorchuk group $\Gamma$ is amenable but not elementary amenable. In this talk we report on work where we demonstrate the existence of a family of finitely generated subgroups of Richard Thompson’s group $F$ which is strictly well-ordered by the embeddability relation in type $\varepsilon_{0}+1$. All except the maximum element of this family (which is $F$ itself) are elementary amenable groups. In this way, for each $\alpha<\varepsilon_{0}$, we obtain a finitely generated elementary amenable subgroup of F whose EA-class is $\alpha+2$. The talk will be pitched for an algebraically inclined audience, but little background knowledge will be assumed. Joint work with Matthew Brin and Justin Moore.
A non-compact, compactly generated, locally compact group whose proper quotients are all compact is called just-non-compact. Discrete just-non-compact groups are John Wilson’s famous just infinite groups. In this talk, I’ll describe an ongoing project to use permutation groups to better understand the class of just-non-compact groups that are totally disconnected. An important step for this project has recently been completed: there is now a structure theorem for non-compact tdlc groups G that have a compact open subgroup that is maximal. Using this structure theorem, together with Cheryl Praeger and Csaba Schneider’s recent work on homogeneous cartesian decompositions, one can deduce a neat test for whether the monolith of such a group G is a one-ended group in the class S of nondiscrete, topologically simple, compactly generated, tdlc groups. This class S plays a fundamental role in the structure theory of compactly generated tdlc groups, and few types of groups in S are known.
We show that many 2-dimensional Artin groups are residually finite. This includes Artin groups on three generators with labels at least 3, where either at least one label is even, or at most one label is equal 3. The result relies on decomposition of these Artin groups as graphs of finite rank free groups.
I will describe a new proof, joint with Adam Piggott (UQ), that groups presented by finite convergent length-reducing rewriting systems where each rule has left-hand side of length 3 are exactly the plain groups (free products of finite and infinite cyclic groups). Our proof relies on a new result about properties of embedded circuits in geodetic graphs, which may be of independent interest in graph theory.
A finite graph that can be obtained from a given graph by contracting edges and removing vertices and edges is said to be a minor of this graph. Minors have played an important role in graph theory, ever since the well-known result of Kuratowski that characterised planar graphs as those that do not admit the complete graph on 5 vertices nor the complete bipartite graph on (3,3) vertices as minors. In this talk, we will explore how this concept interacts with some notions from geometric group theory, and describe a new characterisation of virtually free groups in terms of minors of their Cayley graphs.
A graph is vertex-transitive if its group of automorphism acts transitively on its vertices. A very important concept in the study of these graphs is that of local action, that is, the permutation group induced by a vertex-stabiliser on the corresponding neighbourhood. I will explain some of its importance and discuss some attempts to generalise it to the case of directed graphs.
The concept of a synchronising permutation group was introduced nearly 15 years ago as a possible way of approaching The \v{C}ern\'y Conjecture. Such groups must be primitive. In an attempt to understand synchronising groups, a whole hierarchy of properties for a permutation group has been developed, namely, 2-transitive groups, $\mathbb{Q}$I-groups, spreading, separating, synchronsing, almost synchronising and primitive. Many surprising connections with other areas of mathematics such as finite geometry, graph theory, and design theory have arisen in the study of these properties. In this survey talk I will give an overview of the hierarchy and discuss what is known about which groups lie where.
We give a survey of recent results exploring connections between the Higman-Thompson groups and their automorphism groups and the group of autmorphisms of the shift dynamical system. Our survey takes us from dynamical systems to group theory via groups of homeomorphisms with a segue through combinatorics, in particular, de Bruijn graphs.
Answer: Only when it's an ample group in the sense of Krieger (in particular, discrete, countable and locally finite) and has a Bratteli diagram satisfying certain conditions.
Complaint: Wait, isn't Neretin's group a non-discrete, locally compact, topological full group?
Retort: It is, but you need to use the correct topology!
A fleshed-out version of the above conversation will be given in the talk. Based on joint work with Colin Reid.
There has been a long interest in embedding and non-embedding results for groups in the Thompson family. One way to get at results of this form is to classify maximal subgroups. In this talk, we will define certain labelings of binary trees and use them to produce a large family of new maximal subgroups of Thompson's group V. We also relate them to a conjecture about Thompson's group T.
This is joint, ongoing work with Jim Belk, Collin Bleak, and Martyn Quick at the University of Saint Andrews.
We consider irrational slope versions of T and V. We give infinite presentations for these groups and show how they can be represented by tree-pair diagrams. We also show that they have index-2 normal subgroups that are simple.
This is joint work with Brita Nucinkis and Pep Burillo.
In the 1950's and 60's, the field of general relativity was revolutionised by the introduction of advanced mathematical analysis, in particular through the work of Choquet-Bruhat and Penrose. This revolution put (relativistic) astrophysics and cosmology on a firm mathematical foundation and culminated in definitive theoretical evidence for the formation of "singularities" in stellar collapse and the beginning of (the current phase of?) the universe. I will present an introduction to these advances and some of the mathematics behind them. The talk is aimed at "the lay mathematician".
Topological full groups of minimal subshifts are an important source of exotic examples in geometric group theory, as well as being powerful invariants of symbolic dynamical systems. In 2011, Grigorchuk and Medynets proved that TFGs are LEF, that is, every finite subset of the multiplication table occurs in the multiplication table of some finite group. In this talk we explore some ways in which asymptotic properties of the finite groups which occur reflect asymptotic properties of the associated subshift. Joint work with Daniele Dona.
It is well-known that the Galois group of an (infinite) algebraic field extension is a profinite group. When the extension is transcendental, the automorphism group is no longer compact, but has a totally disconnected locally compact structure (TDLC for short). The study of TDLC groups was initiated by van Dantzig in 1936 and then restarted by Willis in 1994. In this talk some of Willis' concepts, such as tidy subgroups, the scale function, flat subgroups and directions are introduced and applied to examples of automorphism groups of transcendental field extensions. It remains unknown whether there exist conditions that a TDLC group must satisfy to be a Galois group. A suggestion of such a condition is made.
I will show how to construct field extensions with Galois groups isomorphic to general linear groups (with entries in various rings and fields) from the torsion of elliptic curves and Drinfeld modules. No prior knowledge of these structures is assumed.