Royal Holloway - Pure Mathematics Seminar


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!! Colin Reid (Newcastle AUS)

"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 Derek Holt (Warwick)

"Computing in finitely presented groups"

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.


January 23 Fredrik Stromberg (Nottingham)

"Computational aspects of Spectral theory for subgroups of the modular group and Maass waveforms"
 
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 Tamas Makai (Queen Mary)

"Bootstrap percolation on inhomogenous random graphs"

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.


February 6 Ana Khukhro (Cambridge)

"Geometry and topology of finite quotients"

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 Kitty Meeks (Glasgow)

"Reducing Reachability in Temporal Networks"
 
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).
 

February 20 No Seminar!

February 27 Wajid Mannan (
Queen Mary)

"An exotic presentation of Q_28"

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 Chris Smyth (Edinburgh)

"Coxeter-Dynkin diagrams: old mysteries and new extensions"

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.


March 13 Alexandre Martin (Heriot-Watt)

March 20 Gareth Tracey (Bath)

"Generation properties in finite groups: Tools and applications"

Abstract



March 27 Jan-Christoph Schlage-Puchta (Rostock)



Summer term 2019:

May 22 Imre Leader (Cambridge)



Previous Pure Mathematics Seminars