Autumn 2017

September 27 | Alex Fink (Queen Mary) Tutte characters for combinatorial bialgebras (abstract) |

October 4 | Anitha Thillaisundaram (Lincoln) On branch groups (abstract) |

October 11 | Sibylle Schroll (Leicester) New varieties of algebras (abstract) |

October 18 | Rachel Newton (Reading) Counting failures of a local-global principle (abstract) |

October 25 | Karin Bauer (Graz) Dimers with boundary, associated algebras and module categories (abstract) |

November 1 | Aurelien Galateau (Besancon) Explicit versions of the Manin–Mumford conjecture (abstract) |

November 8 | Gareth Jones (Manchester) Pfaffian functions and elliptic functions (abstract) |

November 15 | Alison Parker (Leeds) Some central idempotents for the Brauer algebra (abstract) |

November 22 | Tim Browning (Bristol) Diophantine equations: use and misuse (abstract) |

November 29 | Simon Smith (Lincoln) Infinite primitive permutation groups (abstract) |

December 6 | Xiaolei Wu (Bonn) On the finiteness of the classifying space for the family of virtually cyclic subgroups (abstract) |

December 6 | Matteo Vannacci (Duesseldorf) Two proofs of a remarkable identity and some zeta functions associated to groups (abstract) |

Spring 2018

January 24 | Sam Chow (York) A Galois counting problem (abstract) |

January 31 | Christopher Frei (Manchester) Bounds for l-torsion in class groups (abstract) |

February 7 | Dan Segal (Oxford) Groups of finite upper rank (abstract) |

February 14 | Sofia Lindqvist (Oxford) Rado's criterion over kth powers (abstract) |

February 20 | Christian Elsholtz (Graz) Iterated divisor functions (abstract) |

February 21 | Amanda Cameron (Leipzig) An Ehrhart theory generalisation of the Tutte polynomial (abstract) |

March 21 | Eugene Shargorodsky (Kings College) On the Amick–Fraenkel conjecture (abstract) |

Summer 2018

April 26 | Victor Beresnevich (York) Diophantine approximation and sums of reciprocals of fractional parts (abstract) |

April 30 | Philip Dittmann (Oxford) The totally p-adically integral elements of a number field (abstract) |

May 2 | Matthew Tointon (Cambridge) Nilpotent approximate groups (abstract) |

May 9 | Brent Everitt (York) (Co)Homology of arrangements (abstract) |

May 16 | Martin Liebeck (Imperial College) Simple groups, random generation, and algorithms for finitely presented groups (abstract) |

Alex Fink: Tutte characters for combinatorial bialgebras

In the forty years since Rota introduced it, the perspectiveon combinatorial objects through Hopf algebras has continued to growin productivity. For one, Krajewski, Moffatt, and Tanasa found thatmany famous Tutte-like graph polynomials arise in a uniform fashionfrom the Hopf algebras formulated from various classes of topologicalgraph or their associated matroid-like structures. With Clément Dupont and Luca Moci, we tried to understand *arithmetic matroids* andthe convolution formula of Backman and Lenz for their Tutte polynomialin this fashion. We found an obstruction in the lack of a(convincingly canonical) antipode, but managed to extend the theory tothe bialgebra case. I'll explain.

Anitha Thillaisundaram: On branch groups

Stemming from the Burnside problem, branch groups have delivered lots of exotic examples over the past 30 years. Among them are easily describable finitely generated torsion groups, as well as the first example of a finitely generated group with intermediate word growth. We will investigate a generalisation of the Grigorchuk-Gupta-Sidki branch groups and talk about their maximal subgroups and about their profinite completion. Additionally, we demonstrate a link to a conjecture of Passman on group rings.

Sibylle Schroll: New varieties of algebras

In this talk I will report on joint work with Ed Green and Lutz Hille introducing new varieties whose points are in bijections with algebras. Each variety has a distinguished point corresponding to a monomial algebras and all algebras in a variety have a properties that are governed by those of the monomial algebra.

Rachel Newton: Counting failures of a local-global principle

The search for rational solutions to polynomial equations is ongoing for more than 4000 years. Modern approaches try to piece together 'local' information to decide whether a polynomial equation has a 'global' (i.e. rational) solution. I will describe this approach and its limitations, with the aim of quantifying how often the local-global method fails within families of polynomial equations arising from the norm map between fields, as seen in Galois theory. This is joint work with Tim Browning.

Karin Bauer: Dimers with boundary, associated algebras and module categories

Dimer models with boundary were introduced in joint work with King and Marsh as a natural generalisation of dimers. We use these to derive certain infinite dimensional algebras and consider idempotent subalgebras w.r.t. the boundary.The dimer models can be embedded in a surface with boundary. In the disk case, themaximal CM modules over the boundary algebra are a Frobenius category which categorifies the cluster structure of the Grassmannian.

Aurelien Galateau: Explicit versions of the Manin–Mumford conjecture

The Manin–Mumford conjecture describes the distribution of torsion points in subvarieties of abelian varieties. It was proven by Raynaud thirty years ago, and some explicit versions were later given by Coleman, Buium or Hrushovski. I will discribe these classical results as well as a recent joint work with César Martinez, in which we give uniform bounds for the distribution of torsion points with essentially sharp dependence on the geometry of the subvariety.

Gareth Jones: Pfaffian functions and elliptic functions

I will discuss work with Harry Schmidt in which we give a definition of Weierstrass elliptic functions in terms of pfaffian functions, refining a result due to Macintyre. I'll also mention an application in which we give an effective version of a result of Corvaja, Masser and Zannier on a sharpening of the Manin–Mumford conjecture for non-split extensions of elliptic curves by the additive group.

Alison Parker: Some central idempotents for the Brauer algebra

The Brauer algebra is an important algebra in representation theory, partly because it includes the representation theory of the symmetric group as a special case.I will introduce this algebra and give some background as well as explain why it is so important. I will then describe a method for constructing central idempotents in the Brauer algebra that is more efficient and computational tractable than previous methods. This is joint work with Oliver King and Paul Martin.

Tim Browning: Diophantine equations: use and misuse

Integer solutions to polynomial equations have been studied since the dawn of time. In this talk I will discuss some of the surprising contexts that these equations arise, such as in quantum computing, before describing some recent work specific to cubic equations.

Simon Smith: Infinite primitive permutation groups

In this talk I will summarise recent developments in the structure theory of infinite permutation groups. An important class of permutation groups are those that are *primitive*. These groups are indecomposable in some sense, and are often thought of as being the 'atoms' of permutation group theory.

I will present a recent result which describes the structure of all subdegree-finite primitive permutation groups. This class of groups includes, for example, all automorphism groups of locally finite primitive graphs.

Xiaolei Wu: On the finiteness of the classifying space for the family of virtually cyclic subgroups

In this talk, I will first give a gentle introduction to the classifying space for the family of subgroups with many examples. Note that when the family consists of only the trivial subgroup, the classifying space is just the Eilenberg-Maclane space which is a classical object to study. Then I will concentrate on the case when the family is the set of virtually cyclic subgroups. In particular, I will discuss an interesting conjecture due to Juan-Pineda and Leary. The conjecture says agroup admits a finite model for the classifying space for the family of virtually cyclic subgroups if and only if it is virtually cyclic. I will talk about some recent progress on this conjecture and some ideas behind the proof. This is a joint work with Timm von Puttkamer.

Matteo Vannacci: Two proofs of a remarkable identity and some zeta functions associated to groups

As a way to encode asymptotic aspects of a group, one can define several "zeta functions" associated to counting problems on groups. I will give an overview of the theory of these zeta functions and, through basic yet intriguing examples, I will present a way of computing two of these zeta functions.

Sam Chow: A Galois counting problem

We count monic quartic polynomials with prescribed Galois group, by box height. Among other things, we obtain the order of magnitude for *D*_{4} quartics, and show that non-*S*_{4} quartics are dominated by reducibles. Weapons include determinant method estimates, the invariant theory of binary forms, the geometry of numbers, and diophantine approximation. Joint with Rainer Dietmann.

Christopher Frei: Bounds for *l*-torsion in class groups

The arithmetic of a number field is determined in large parts by its class group. While it is a classical result of algebraic number theory that this group is always finite and abelian, precise information on its structure remains quite elusive. To emphasise how little we know, it is still unknown whether the class group is trivial for infinitely many number fields. In this talk, we introduce class groups and survey classical and recent results (conditional and unconditional) bounding the cardinality of their *l*-torsion subgroups, for a natural number *l*. In the remaining time, we discuss recent joint work with Martin Widmer on average bounds for this *l*-torsion in certain families of number fields.

Dan Segal: Groups of finite upper rank

The upper rank of a group is the supremum of the Prufer ranks of its finite quotient groups. The upper *p*-rank, for a prime *p*, is defined analogously, using the Sylow *p*-subgroups of finite quotients. I’ll discuss to what extent the upper rank of a finitely generated group is controlled by its upper *p*-ranks.

Sofia Lindqvist: Rado's criterion over *k*th powers

An equation is said to be partition regular if for any finite colouring of the integers there is a monochromatic solution to the equation. In the case of linear homogeneous equations Rado showed that an equation is partition regular iff the coefficients satisfy something known as Rado's criterion. We extend this result to sums of *k*th powers, provided the number of variables is sufficiently large in terms of *k*. This is joint work with Sam Chow and Sean Prendiville.

Christian Elsholtz: Iterated divisor functions

The divisor function d(*n*) is large, when *n* is a product of many small primes. For example d(2.3.5.7)=16. The iterated divisor function d(d(*n*)) is quite large, when d(*n*) is such a product of small primes. But what is the *maximal* order of magnitude of d(d(*n*))? This question was raised by Ramanujan in 1915, and was later studied by Erdos, Katai and Ivic. In joint work with Y. Buttkewitz, K Ford, J.C. Schlage-Puchta we determined the maximal order. With M. Technau and N. Technau we generalized this result to a class of related multiplicative functions, which includes the case *r*_{2}(*n*), counting the number of representations as sums of two squares, also investigated by Ramanujan.

Amanda Cameron: An Ehrhart theory generalisation of the Tutte polynomial

The Tutte polynomial is one of the most important and well-known graph polynomials, and also features prominently in matroid theory. It is however not directly applicable to polymatroids, these being a natural generalisation of matroids. For instance, deletion-contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. This is based on joint work with Alex Fink.

Eugene Shargorodsky: On the Amick–Fraenkel conjecture

In 1987, C.J. Amick and L.E. Fraenkel published a paper on the behaviour of the Stokes wave of extreme form near its crest, where they obtained a complete asymptotic expansion for the angle between the wave profile and the horizontal. Their derivation of the expansion relied on an assumption of linear independence over the rationals of solutions of a certain transcendental equation, which is still an open question. The talk is intended to be a non-technical survey of some results related to the Amick-Fraenkel conjecture and ranging from fluid dynamics to number theory.

Victor Beresnevich: Diophantine approximation and sums of reciprocals of fractional parts

I will discuss several results concerting sums of reciprocals of fractional parts of arithmetical progressions as well as their relationship to basic results in Diophantine approximation and some techniques for obtaining upper bounds, in particular, the use of the so-called Three Distance Theorem. I will also explain the role of these sums in problems of uniform distribution and metric number theory and conclude the talk with some yet unanswered questions.

Philip Dittmann: The totally *p*-adically integral elements of a number field

A theorem of Siegel asserts that every element of a number field which is totally positive, i.e. positive under each real embedding, can be expressed as a sum of four squares. This generalises the well-known fact that every positive rational is a sum of four squares.I will present an analogous representability result about elements which are *p*-adic integers under any embedding into **Q**_{p}, as well as some of its consequences.This is joint work with Sylvy Anscombe and Arno Fehm.

Matthew Tointon: Nilpotent approximate groups

In recent years there has been a large volume of work studying so-called 'approximate groups', not least because of their varied and powerful applications to fields as wide-ranging as number theory, differential geometry and theoretical computer science. Roughly, an 'approximate subgroup' of a group is a symmetric set that is 'approximately closed' under the group operation. In this talk I will introduce approximate groups, in particular stating a celebrated result of Breuillard, Green and Tao saying that every approximate group has a 'large nilpotent piece' in a certain precise sense. In the main part of the talk I will explain how to obtain further structure of this nilpotent piece. If I have time at the end I will sketch an application, joint with Tessera, to growth in groups.

Brent Everitt: (Co)Homology of arrangements

An arrangement is a finite collection of linear hyperplanes in some vector space. In this talk we survey a number of different answers to the question, 'what is the cohomology of an arrangement?'

Martin Liebeck: Simple groups, random generation, and algorithms for finitely presented groups

If one picks two elements at random in a finite non-abelian simple group *G* (such as an alternating group), then these two elements will generate *G* with probability tending to 1 as the order of *G* tends to infinity. I will discuss this result and variations, and show how they connect with some basic questions concerning the existence of algorithms that determine finite images of finitely presented groups. No specialist knowledge will be assumed.