Pure Mathematics
All seminars will take place in the International Building, Room IN028, on Tuesdays at 2 pm, unless stated otherwise. Tea will be served after the seminar at 3pm in Room 237 of the McCrea Building. All are welcome!
Autumn 2016
27th September Ben Barber (Bristol)
"Density methods for partition regularity"
Abstract: In this talk I will give density proofs of Ramsey theorems for which the corresponding density version is false. A system of linear equations with integer coefficients is partition regular if, whenever the natural numbers are finitely coloured, there is a monochromatic solution. The finite partition regular systems were completely characterised by Rado and are very well understood. Infinite partition regular systems are very poorly understood, and only a few families of examples are known. I'll describe a new family of infinite partition regular systems built using density methods. The construction is very flexible, and provides examples to settle numerous long standing conjectures in the area. As one example, there is an uncountable chain of subgroups of Q such that each element can be distinguished from its predecessors by its Ramsey properties.
4th October Fiona Skerman (Bristol)
"Modularity of Random Graphs"
Abstract: An important problem in network analysis is to identify highly connected components or `communities'. Most popular clustering algorithms work by approximately optimising modularity. Given a graph G, the modularity of a partition of the vertex set measures the extent to which edge density is higher within parts than between parts; and the maximum modularity q*(G) of G is the maximum of the modularity over all partitions of V(G) and takes a value in the interval [0,1) where larger values indicates a more clustered graph.
Knowledge of the maximum modularity of random graphs helps determine the significance of a division into communities/vertex partition of a real network. We investigate the maximum modularity of Erdos-Renyi random graphs and find there are three different phases of the likely maximum modularity. Concentration of the maximum modularity about its expectation and structural properties of an optimal partition are also established. This is joint work with Prof. Colin McDiarmid.
11th October Trevor Wooley (Bristol)
"Subconvexity in certain Diophantine problems via the circle method"
Abstract: The subconvexity barrier traditionally prevents one from applying the Hardy-Littlewood
(circle) method to Diophantine problems in which the number of variables is smaller than twice the inherent total degree. Thus, for a homogeneous polynomial in a number of variables bounded above by twice its degree, useful estimates for the associated exponential sum can be expected to be no better than the square-root of the associated reservoir of variables. In consequence, the error term in any application of the circle method to such a problem cannot be expected to be smaller than the anticipated main term, and one fails to deliver an asymptotic formula. There are perishingly few examples in which this subconvexity barrier has been circumvented, and even fewer having associated degree exceeding two. In this talk we review old and more recent progress, and exhibit a new class of examples of Diophantine problems associated with, though definitely not, of translation-invariant type.
18th October !!at 4pm in MC229!! David Stewart (Newcastle)
"Maximal subalgebras of modular Lie algebras"
Abstract: The question of classifying maximal subalgebras of Lie algebras goes all the way back to papers of Sophus Lie himself in the 1890s and has a long history from Dynkin onwards. We report on the latest developments in classifying the maximal subalgebras of Lie algebras of simple algebraic groups over algebraically closed fields of positive characteristic, a reasonable task thanks to the Premet-Strade classification of simple Lie modular algebras in characteristics at least 5. This is joint work with Sasha Premet.
25th October No Seminar!
1st November !! in ABLT1 !! Sejong Park (Southampton)
"Biset functors and double Burnside algebras"
Abstract: Biset functors for finite groups are functors compatible with operations induced by bisets - sets with commuting actions of two groups. Examples include Burnside ring functor, representation ring functors and cohomology functors. The double Burnside algebra of a finite group G is generated by (G, G)-bisets and acts on the evaluation of any biset functor F at G. I will review the theory of biset functors and what is known about the structure of the double Burnside algebras. Then I will explain an attempt (joint work with Goetz Pfeiffer) to clarify the structure of the double Burnside algebras in characteristic zero.
8th November !! in ABLT1 !! Nadia Mazza (Lancaster)
"Endotrivial modules for finite groups - a survey"
Abstract: Given a field k of positive characteristic p and a finite group G, a finitely generated kG-module M is endotrivial module if its endomorphism algebra End_k(M) of k-linear transformation of M splits as kG-module as the direct sum of the trivial module k plus some projective kG-module. Equivalently, M is "invertible" in the stable module category of G.
These modules are relevant in the study of the stable module category of G, and also in the study of the so-called "source algebras" of p-blocks. Furthermore, the set of stable iso classes of endotrivial modules forms a finitely generated abelian group T(G) and the ultimate objective is to determine T(G) for any finite group G and any prime p. In this talk, we'll survey the topic, giving examples, results and open questions.
15th November Enric Ventura (Barcelona)
"The degree of commutativity/nilpotency of an infinite group"
Abstract: (joint work with Y. Antolin and A. Martino) There is a classical result saying that, in a finite group, the probability that two elements commute is never between 5/8 and 1 (i.e., if it is bigger than 5/8 then the group is abelian). We make an adaptation of this notion for finitely generated infinite groups (w.r.t. a fixed finite set of generators) as the limit of such probabilities, when counted over successively growing balls in the group. This asymptotic notion is a lot more vague than in the finite setting, but we are still able to prove some interesting results concerning this new concept. The main one is the following Gromov-like result: "for any finitely generated residually finite group G of subexponential growth, the commuting degree of G is positive if and only if G is virtually abelian". In a similar way, we define the degree of r-nilpotency, and prove the analogous result as well.
29th November !! in ABLT1 !! Sarah Rees (Newcastle)
"Rewriting in Artin groups"
Abstract: The class of Artin groups is easy to define, via presentations,
but contains a variety of groups with apparently quite different properties.
For the class as a whole, many problems remain open, including the word problem;
this is in contrast to the situation for Coxeter groups, which arise as quotients of Artin groups.
I'll discuss what is known about rewrite systems for Artin groups, and evidence
for the possibility of a general approach to rewriting in these groups.
I'll refer in particular to my own recent work with Derek Holt, and with Eddy Godelle, as well as work of Dehornoy and Godelle, some of this very recent.
6th December Rachel Camina (Cambridge)
"Vanishing class sizes"
Abstract: For many years authors have considered the algebraic implications of arithmetic conditions on conjugacy class sizes for finite
groups. We look at recent results and consider the restricted case when just vanishing class sizes are considered.
Previous seminars