The PM-seminar takes place on Wednesday at 2pm in McCrea 219.

Here is a campus plan.

January 10 !!Extra - Seminar at 2pm in McCrea 219!!

"Locally compact piecewise full groups"

Abstract: 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).

January 11 !!Mini-conference "Geometric Group Theory at Royal Holloway"!!

January 16

Abstract: We survey algorithms for computing in (generally infinite) groups defined by finite presentations, with an emphasis on the existence and availability of effective implementations.

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.

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.

January 23

Abstract: 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.

January 30

Abstract:
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.

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.

February 6

Abstract: 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.

February 13

Abstract:
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).

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).

February 20 No Seminar!

February 27

Abstract: 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.

March 6

Abstract: 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.

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.

March 13

March 20

Abstract

March 27

May 22