RHUL Mathematics Seminar

Steven Galbraith: Supersingular isogeny graphs and applications
The graph of isogenies of supersingular elliptic curves has many applications in computational number theory and public key cryptography. I will present some of these applications and I will also discuss some open problems.

Waltraud Lederle: Almost automorphism groups of trees and topological full groups
We will introduce the concepts mentioned in the title, explain why they are interesting and show a connection between them. Thereby we will see examples of simple locally compact groups without lattices; finitely generated, simple amenable groups and the famous Higman-Thompson groups.

Mark Grant: An equivariant Eilenberg-Ganea Theorem
There are (at least) 3 competing notions of "dimension" for discrete groups: cohomological dimension, geometric dimension (the smallest dimension of a classifying space), and Lusternik-Schnirelmann category (of a classifying space). Theorems of Eilenberg-Ganea and Stallings and Swan from the 1950’s and 60's imply that these all coincide, except for the possible existence of a group with cat=cd=2 and gd=3.

I will discuss equivariant generalisations of these theorems to the setting of groups with operators. The statements involve Bredon cohomological dimension with respect to families of subgroups, which I'll define during the talk.

Ian Leary: CAT(0) groups need not be biautomatic
Ashot Minasyan and I construct groups that establish the result in the title, resolving a question that has been around for almost 30 years. I will start by explaining the phrases 'CAT(0)' and 'biautomatic'. After that I will talk about our groups and why they have the properties that we claim.

Arno Fehm: Deciding solvability of polynomial equations over local fields
The celebrated Hasse—Minkowski local-global principle relates the existence of zeros of quadratic forms over the field of rational numbers, and more generally number fields ('global'), to the existence of zeros over various completions ('local'), namely the real and complex numbers, and the p-adics, where analytic methods can be applied. A similar local-global principle holds in positive characteristic, where the 'local' fields are the fields of formal Laurent series over finite fields. However, while it is classical that for each local field K of characteristic zero there is an algorithm that determines whether a given system of polynomial equations has a common zero in K, and even more generally whether a given first-order sentence in the language of rings holds in K, for Laurent series fields over finite fields the existence of such algorithms is only partially understood. In this talk I will report on what is known about this, which will lead us from number theory and algebraic geometry to the model theory of valued fields.

Jessica Banks: Knots (and surfaces)
We will consider some questions in classical knot theory, and see how surfaces play a part. In particular, we will look at how knot theory compares with the physical world, and how hard it is to untangle a knot.

Xin Li: Semigroup C*-algebras
This talk is about semigroup C*-algebras, i.e., C*-algebras generated by left regular representations of semigroups. After a general introduction, we will focus on classes of one-relator monoids and Artin monoids of finite type. We present structural results and K-theory computations for the corresponding semigroup C*-algebras. This is joint work with Tron Omland and Jack Spielberg.

Evgeny Khukhro: Almost Engel compact groups
We say that a group G is almost Engel if for every g ∈ G there is a finite set E(g) such that for every x ∈ G all sufficiently long commutators [...[[x, g], g], . . . , g] belong to E(g), that is, for every x ∈ G there is a positive integer n(x, g) such that [...[[x, g], g], . . . , g] ∈ E(g) if g is repeated at least n(x, g) times. (Thus, Engel groups are precisely the almost Engel groups for which we can choose E(g) = {1} for all g ∈ G.)

We prove that if a compact (Hausdorff) group G is almost Engel, then G has a finite normal subgroup N such that G/N is locally nilpotent. If in addition there is a uniform bound |E(g)| ≤ m for the orders of the corresponding sets, then the subgroup N can be chosen of order bounded in terms of m. The proofs use the Wilson—Zelmanov theorem saying that Engel profinite groups are locally nilpotent.

This is joint work with Pavel Shumyatsky.

Krystal Guo: Transversals in covers of graphs
We study a polynomial with connections to correspondence colouring, a recent generalization of list-colouring, and the Unique Games Conjecture. Given a graph G and an assignment of elements of the symmetric group S_r to the edge of G, we define a cover graph: there are sets of r vertices corresponding each vertex of G, called fibres, and for each edge uv, we add a perfect matching between the fibres corresponding to u and v. A transversal subgraph of the cover is an induced subgraph which has exactly one vertex in each fibre. In this setting, we can associate correspondence colourings with transversal cocliques and unique label covers with transversal copies of G.

We define a polynomial which enumerates the transversal subgraphs of G with k edges. We show that this polynomial satisfies a contraction deletion formula and use this to study the evaluation of this polynomial at -r-1. This is joint work with Chris Godsil and Gordon Royle.

Vincent Jansen: Using mathematics to understand biology: on hotspots, homoclinics and horseshoes
I will start with a few words about the joint history of math and biology. I will touch on the work of Fibonacci and Volterra. What I then will talk about in detail is a neat biological problem: the paradoxical existence of recombination hotspots in the genome. I will start with a model for that system based on two alleles, and I will explain how we analysed the behaviour. This will lead me to the existence of a homoclinic cycle, and the analysis of the stability of the homoclinic cycle. As a finale I will then take that same model but now for more alleles, explain how the homoclinic cycles join up to form a homoclinic network in the vicinity of which we find chaotic dynamics (which is where the horseshoes come in). After that I will interpret these results in terms of biology

Laura Ciobanu: Conjugacy growth in groups, combinatorics and geometry
In this talk I will give an overview of what is known about conjugacy growth in infinite discrete groups and the formal series associated with it. In particular, I will highlight the connections between the rationality or lack of rationality of these series to analytic combinatorics and geometry.

Colin Reid: Locally compact piecewise full groups
I will be talking about joint work with Alejandra Garrido and David Robertson. Let X be the Cantor set (or any compact zero-dimensional Hausdorff space) and let G be a subgroup of the group Homeo(X) of homeomorphisms from the Cantor set to itself. We say h in Homeo(X) is 'piecewise in G' if there is an open cover O of X, such that for every Y in O, there is some g_Y in G such that hx = g_Yx for all x in Y. The piecewise elements of G themselves form a group, called the piecewise full group [[G]] of G (also known as the topological full group). Piecewise full groups have been extensively studied since the 1990s, but mostly in the context of countable groups. We consider instead those piecewise full groups that admit a nondiscrete locally compact group topology. In this context, we find that given a locally compact group acting on the Cantor set, then the group topology extends to the piecewise full group if and only if the action is locally decomposable, meaning that for every clopen partition of the space, there is an open subgroup that acts independently on each part of the partition. Moreover, if in addition G is compactly generated and acts minimally and expansively, then the derived group of [[G]] is open, compactly generated and abstractly simple. This construction gives rise to all local isomorphism types of compactly generated simple locally compact groups in which some compact open subgroup admits a nontrivial direct factorization; in this context, we note that there are uncountably many such groups up to isomorphism (by a construction of Smith), but it is unknown whether there are uncountably many up to local isomorphism. The piecewise full group construction also sheds light on a class of abstract commensurators of profinite groups, in the sense of Barnea–Ershov–Weigel: we show that a certain local condition suffices to ensure that the abstract commensurator has a simple monolith, and also obtain sufficient conditions for the abstract commensurator to be simple-by-(discrete abelian).

Cornelia Drutu: Superexpanders and warped cones
Superexpanders are expanders that do not embed coarsely in super reflexive Banach spaces. There are for the moment very few constructions of super expanders. The latest displays interesting connections with John Roe's concept of warped cone, a metric space as- sociated to the action of an infinite group on a compact space, which turned out to be a metric counter-example to the Baum–Connes conjecture. Random graphs are presumably sources of superexpanders even though for the moment this is not known. Some results are known, about expansion features of random graphs considered with respect to particular types of Banach spaces. In this talk I shall present the latest results on super expanders, based on work of several authors (Tim de Laat, Mikael de la Salle, Federico Vigolo, Damian Sawicki, John MacKay and myself)

Jose Burillo: An irrational-slope Thompson's group
Let τ = 0.618... be the small golden ratio, zero of the polynomial x² + x - 1. In 1995 Sean Cleary introduced the irrational-slope Thompson's group Fτ, which is the group of piecewise-linear maps of the interval [0, 1] with breaks in Z[τ] and slopes powers of τ. In this talk I will describe this group using tree-pair diagrams in Thompson's group style, and then I will show a finite presentation, a normal form, and prove that its commutator subgroup is a simple group. This group is the first example of a group of piecewise-linear maps of the interval whose abelianisation has torsion, and it is an open problem whether this group is a subgroup of Thompson's group F.

Lawrence Reeves: Commutators in groups of piecewise projective homeomorphisms
The Lodha–Moore groups are finitely presented counterexamples to the von Neumann–Day conjecture. They appear as subgroups of a group of piecewise projective maps constructed by Monod. We study their commutators and second commutators, showing some of them are simple. This is joint work with Jose Burillo and Yash Lodha

Derek Holt: Computing in finitely presented groups
The talk will focus mainly on the Word Problem, the Generalised Word Problem and the Conjugacy Problem.

For the Word Problem we will concentrate on the use of automatic structures for defining an effective normal form for the groups elements, with some applications to deciding finiteness, and to drawing pictures.

There are not many methods available for the Generalised Word Problem when the index of the subgroup H of G is infinite. The Stallings Folding method for subgroups of free groups can be generalised to quasiconvex subgroups of automatic groups provided that we can compute a so-called coset automatic structure. This is possible, for example, for quasiconvex subgroups of hyperbolic groups.

The Conjugacy Problem in certain types of groups (braid groups and polycyclic groups, for example) has received a lot of attention recently resulting from potential applications to cryptography. But in this talk we focus on the problem in hyperbolic groups, which can theoretically be solved in (almost) linear time, but for which effective implementations seem to be more difficult.

Fredrik Stromberg: Computational aspects of Spectral theory for subgroups of the modular group and Maass waveforms
I will give a brief introduction to the spectral theory of Fuchsian groups and in particular finite index subgroups of the modular group. There are many interesting and fundamental questions in this area (e.g. Selberg's eigenvalue conjecture) which are (currently) mainly approachable using computations and I will give an overview of some of the computational methods, main problems and recent developments.

Tamas Makai: Bootstrap percolation on inhomogenous random graphs
Bootstrap percolation is a deterministic process on a graph. It starts out from a set of infected vertices and in every step each vertex, which has at least r infected neighbours becomes infected. Once a vertex becomes infected it remains infected forever. The process stops once no additional vertices can become infected.

We analyse this process on the Chung-Lu random graph. In this model every vertex is assigned a weight and the vertices are connected proportional to the product of the vertex weigths. Each edge is inserted independently.

We determine the weight sequences where a small number of randomly infected vertices leads with high probability to the infection of a linear fraction of the vertices by the end of the process. We also establish the threshold on the number of initially infected vertices required for this linear outbreak to occur.

Ana Khukhro: Geometry and topology of finite quotients
A group's natural habitat is surely in the world of geometry — whether acting on metric or topological spaces, or being viewed as geometric objects themselves, groups are happiest when coupled with geometric notions. Geometric group theory is now a thriving topic in pure mathematics, with many connections to areas as varied as analysis, dynamics, and cryptography. One of the fundamental ideas of this subject is that a finitely generated group can be viewed as graph, called a Cayley graph, which can encode many of its algebraic properties geometrically. In this talk, we will introduce a variant on this geometric object for groups with many finite quotients, and see how its geometry can be used to create and study examples of interest in a coarse geometric setting.

Kitty Meeks: Reducing Reachability in Temporal Networks
The concept of reachability sets (i.e. which vertices in a network can be reached by travelling along edges from a given starting vertex) is central to many network-based processes, including the dissemination of information or the spread of disease through a network. Depending on the application, it might be desirable to increase or decrease the number of vertices that are reachable from any one starting vertex. In most applications, time plays a crucial role: each contact between individuals, represented by an edge, will only occur at certain time(s), when the corresponding edge is "active". The relative timing of edges is clearly crucial in determining the reachability set of any vertex in the network.

In this talk, I will address the problems of reducing the maximum reachability of any vertex in a given temporal network by two different means:

1. we can remove a limited number of time-edges (times at which a single edge is active) from the network, or

2. the number of timesteps at which each edge is active is fixed, but we can change the relative order in which different edges are active (perhaps subject to constraints on which edges must be active simultaneously, or restrictions on the timesteps available for each edge).

Mostly, we find that these problems are computationally intractable even when very strong restrictions are placed on the input, but we identify a small number of special cases which admit polynomial-time algorithms, as well as some general upper and lower bounds on what can be achieved.

Everything in this talk is based on joint work with Jessica Enright (University of Edinburgh); I will also mention some joint results with George B. Mertzios and Viktor Zamaraev (University of Durham) and Fiona Skerman (Uppsala University).

Wajid Mannan: An exotic presentation of Q₂₈
Whether or not the quaternion group of order 28 has a finite balanced presentation with non-standard homotopy has been an open problem for some time. I will provide such a presentation, recently discovered, and convey why it is of interest in low-dimensional topology.

Chris Smyth: Coxeter–Dynkin diagrams: old mysteries and new extensions
I first survey some of the wide variety of mathematical objects that turn out, mysteriously, to be classified by Coxeter-Dynkin diagrams.

Then I describe how, in joint work with James McKee, we extended these diagrams in a new but natural way. However, these extensions seem to generate further mysteries.

Alexandre Martin: The Tits alternative for two-dimensional Artin groups
Many groups of geometric interest present an interesting dichotomy at the level of their subgroups: their subgroups are either "very large" (they contain a non-abelian free subgroup) or "very small" (they are virtually abelian). In this talk, I will explain how one can use ideas from group actions in negative curvature to prove such a dichotomy. In particular, I will show how one can prove such a strengthening of the Tits Alternative for a large class of Artin groups. This is joint work with Piotr Przytycki.

Gareth Tracey: Generation properties in finite groups: Tools and applications
A well-developed branch of finite group theory compares different invariants in certain classes of finite groups. For example, one may take certain classes of permutation groups G, of degree n, and ask: How big can |G| be, in terms of n? How many generators will G need in terms of n? If one chooses generators of G at random, with replacement, then how long will it take before a generating set is found? In this talk, we will study these questions in various different classes of finite groups, from permutation and linear groups to simple groups and their subgroups. We will also outline some useful techniques, and some interesting applications to enumeration problems in graph theory.

Jan-Christoph Schlage-Puchta: The loneliness of the Wollmilchsau
Origami (aka square tiled surfaces) bring together various different branches of mathematics. They can be seen as a method to lift results on elliptic curves to higher genus curves, as manifolds with the best possible transition maps, or as a special class of dynamical systems. Their combinatorial description is linked to the product replacement algorithm and to finite index subgroups of F₂, the free group in two generators.

In 2008, Herrlich and Schmithüsen found an origami consisting of eight squares, which served as an example to so many interesting questions, that it became known as the eierlegende Wollmilchsau. Here we present joint work with Gabriela Weitze-Schmithüsen, showing that this origami is unique in the sense that every origami of singularity type (1,1,1,1) and full Veech group is an unramified covering of the Wollmilchsau.

Indira Chattereji: Property (T) and actions on L_p spaces
Discrete countable groups with property (T) are exactly those that do not admit any proper affine isometric action on a Hilbert space. I will recall basic facts about property (T) and discuss possible variations, involving actions on L_p spaces.

Jonathan Gross: Partial-Dual Genus Polynomials: Ribbons, Permutations, and Flags
We are concerned with a reconciliation of some alternative representations of embedded graphs and of the corresponding representations of partial duals. We define the partial-dual genus polynomial of an embedded graph G to be the generating function for the numbers of partial duals of G according to the genus of the embedding surfaces. We demonstrate that using the permutation representation (monodromy) by a bi-rotation system and an involution facilitates face-tracing and thereby simpifies the automated calculation of partial-dual genus polynomials.

Ellen Henke: Introduction to fusion systems
In the study of saturated fusion systems, previously independent developments in local finite group theory, in modular representation theory and in homotopy theory come together. I will give an introduction to the theory and mention some of my own results along the way. In particular, I will talk about a programme of Michael Aschbacher to revisit the classification of finite simple groups using fusion systems.

Imre Leader: The Graham–Pollak Problem for Hypergraphs
How many complete bipartite graphs do we need to decompose the complete graph on n vertices? It is easy to achieve this with n-1 complete bipartite graphs, and the Graham–Pollak Theorem states that this is the minimum. What happens for hypergraphs? For example, how many complete tripartite 3-graphs do we need to decompose the complete 3-graph on n vertices? We will report on recent progress on this question. Joint work with Luka Milicevic and Ta Sheng Tan.

Geetha Venkataraman: Enumeration of Groups in Varieties of A-groups
Let f(n) denote the number of isomorphism classes of groups of order n. Let S be a class of groups and let f_S(n) be the number of isomorphism classes of groups in S, of order n. Some interesting classes that have been studied are the class of soluble groups, varieties of groups (see [NeumannVarietiesOfGroups]), p-groups and A-groups (see [BlackburnNeumannVenkataram). Finite A-groups are those with abelian Sylow subgroups.

In 1993, L Pyber proved a result [PyberEnumeratingFiniteGroups], which bettered a conjecture, when he showed that f(n) ≤ n^{2/27μ(n)2 + O(μ(n)5/3)} where μ(n) denotes the maximum α, such that p^α divides n for any prime p.

While Pyber's upper bound has the correct leading term, it is certainly not the case for the error term. The key to this puzzle may lie in deeper investigation of A-groups and varieties of A-groups (see [VenkataramanEnumerationOfFiniteSolubleGroups] and [VenkataramanOnIrreducibilityOfInducedModules]).

We present results concerned with asymptotic bounds for f_S(n) when S is a variety of A-groups, namely, U = A_pA_q, and V = A_pA_q ⋁ A_qA_p and discuss open questions in this area.

The talk will be self-contained and should be accessible to anyone with a basic knowledge of group theory.