Autumn 2016

27th September | Ben Barber (Bristol) Density methods for partition regularity (abstract) |

4th October | Fiona Skerman (Bristol) Modularity of Random Graphs (abstract) |

11th October | Trevor Wooley (Bristol) Subconvexity in certain Diophantine problems via the circle method (abstract) |

18th October | David Stewart (Newcastle) Maximal subalgebras of modular Lie algebras (abstract) |

1st November | Sejong Park (Southampton) Biset functors and double Burnside algebras (abstract) |

8th November | Nadia Mazza (Lancaster) Endotrivial modules for finite groups — a survey (abstract) |

15th November | Enric Ventura (Barcelona) The degree of commutativity/nilpotency of an infinite group (abstract) |

29th November | Sarah Rees (Newcastle) Rewriting in Artin groups (abstract) |

6th December | Rachel Camina (Cambridge) Vanishing class sizes (abstract) |

Spring 2017

January 10 | Quentin Guignard (Paris) A zetafunctionological proof of Schanuel's theorem (abstract) |

January 17 | David Masser FRS (Basel) The unlikelihood of integrability in elementary terms |

January 24 | Julia Boettcher (LSE) Packing degenerate graphs (abstract) |

January 31 | Andrew McDowell (King's College) Target sets in degree proportional percolation (abstract) |

February 7 | Ilaria Castellano (Southampton) Cohomology for totally disconnected locally compact groups (abstract) |

February 21 | Sara Checcoli (Grenoble) A walk on the zeroes of polynomials (abstract) |

February 28 | Mike Harrison A computational algorithm for semi-stable models over p-adic fields (abstract) |

March 7 | Fabien Pazuki (Copenhagen) Heights and regulators of number fields and elliptic curves (abstract) |

March 13 | Yago Antolin (Madrid) Dehn fillings theorem and applications (abstract) |

March 14 | Laura Capuano (MPI Bonn) Unlikely intersections in Diophantine Geometry (abstract) |

March 21 | Holly Krieger (Cambridge) Height functions in arithmetic dynamics (abstract) |

Summer 2017

May 9 | Bill O'Donovan (Royal Holloway) Conjugation modules for symmetric groups (abstract) |

May 23 | Ivan Fesenko (Nottingham) The Mochizuki theory and its group-theoretical aspects (abstract) |

Ben Barber: Density methods for partition regularity

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.

Fiona Skerman: Modularity of Random Graphs

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.

Trevor Wooley: Subconvexity in certain Diophantine problems via the circle method

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.

David Stewart: Maximal subalgebras of modular Lie algebras

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.

Sejong Park: Biset functors and double Burnside algebras

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.

Nadia Mazza: Endotrivial modules for finite groups — a survey

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.

Enric Ventura: The degree of commutativity/nilpotency of an infinite group

(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.

Sarah Rees: Rewriting in Artin groups

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.

Rachel Camina: Vanishing class sizes

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.

Quentin Guignard: A zetafunctionological proof of Schanuel's theorem

In his 1950 thesis, John Tate gives an expression of the completed zeta function of a number field in terms of an idelic integral. I will recall the derivation of this expression, and I will explain how it can be extended to the completed height zeta function of rational points in projective spaces. I will then deduce from it a new proof of Schanuel's theorem of rational points of bounded height in projective space, as well as the meromorphic continuation and the functional equation of the corresponding height zeta function.

Julia Boettcher: Packing degenerate graphs

There has recently been much progress on (hyper)graph packing problems in Combinatorics, which became possible through the development of new techniques in the area. Highlights include the celebrated proof by Keevash of the existence of combinatorial designs whenever certain divisibility conditions are satisfied, or the proof of the so-called Tree Packing Conjecture for large bounded degree trees by Joos, Kim, Kühn and Osthus. Here, a family of graphs is said to pack into a given host graph if the members of the family can be embedded into the host-graph edge-disjointly.

Contributing to this body of research and improving on several other results, we can prove the following packing result for graphs of bounded degeneracy (but with potentially large maximum degree). A graph *H* has degeneracy at most *D* if there is an ordering of its vertices such that every vertex has at most *D* neighbours among the vertices preceding it in this order. For large n and arbitrarily small ε > 0 any family of graphs *G*_{1},...,*G*_{m} each of which has degeneracy bounded by a constant *D*, at most *n* vertices, and maximum degree o(*n*/log *n*), and whose number of edges sum to at most (1-ε)binom(*n*,2) packs into the complete graph *K*_{n} on *n* vertices. For obtaining this result we use a rather natural randomised packing strategy and show that this preserves certain pseudorandomness properties.

In the talk I shall give an overview of the mentioned progress in the area, explain our result and its background, and give an outline of the methods that we use. Joint work with Peter Allen, Jan Hladký, Diana Piguet.

Andrew McDowell: Target sets in degree proportional percolation

We consider a degree proportional percolation model in which an infection spreads through a graph by infecting any vertex which has a fixed proportion of its neighbours already infected. This talk will present the background and solution to the problem of identifying bounds on the size of a minimal initial target set which will spread the infection to the entire graph. Joint work with Richard Mycroft and Frederik Garbe.

Ilaria Castellano: Cohomology for totally disconnected locally compact groups

For a topological group *G* several cohomology theories have been introduced and studied in the past. In many cases the main motivation was to obtain an interpretation of the low-dimensional cohomology groups by analogy to discrete case. The aim of this talk is to present the rational discrete cohomology for totally disconnected locally compact groups and some results that can be obtained by investigation of the first-degree cohomology group.

Sara Checcoli: A walk on the zeroes of polynomials

This walk will bring us from a Babylonian tablet to Mordell'sconjecture, passing through Fermat and Galois, presenting several aspects of the old problem of solving polynomial equations.

Mike Harrison: A computational algorithm for semi-stable models over *p*-adic fields

In arithmetic investigations of a curve *C* of genus > 1 over a number field *K* and its Jacobian *J*(*C*), the regular minimal model of *C* over the *p*-adic completion *K*_{p} — a 'nice' scheme over the ring of integers of *K*_{p} giving a good *p*-integral structure for *C* — is important for computing *p*-local properties. However, it doesn't in general give the full Tate-modules of *J*(*C*) as local Galois modules, or the conductor in the wildly ramified case. The semi-stable reduction theorem says that there is a finite Galois extension *L*/*K*_{p} over which *C* attains semi-stable reductionand the semi-stable regular minimal model of *C* over *L* along with the action of *G*(*L*/*K*_{p}) on this does give the full Tate-modules and conductor of *J*(*C*) over *K*_{p}. Recent new proofs of the semi-stable reduction theorem based on finite covers of semi-stable curves have led me to a reasonably efficient computational algorithm for computing an *L*, which is close to smallest possible, and the semi-stable model over *L* in explicit form. I shall describe the components of the algorithm along with the theory of Liu and Lorenzini that lies behind it and give an example where the regular model and semi-stable model are quite different.

Fabien Pazuki: Heights and regulators of number fields and elliptic curves

The aim of the talk is to explain that there are only finitely many number fields with bounded regulator, except for the particular case of 'CM-fields'. We will then explain how to study a similar question in the context of elliptic curves defined over a fixed number field, with the regulator of the Mordell–Weil group as the central object. In this setting and under the ABC conjecture, there are only finitely many classes of elliptic curves defined over a fixed number field, with bounded positive rank and bounded regulator.

Yago Antolin: Dehn fillings theorem and applications

In this talk I will recall the version of the Dehn fillings theorem proved by Dahmani, Guirardel and Osin. I will present an extension of this theorem and use it to show that groups hyperbolic relative to residually finite groups satisfying the Farrell–Jones conjecture, also satisfy this conjecture. This is based on a joint work with R. Coulon and G. Gandini.

Laura Capuano: Unlikely intersections in Diophantine Geometry

What makes an intersection likely or unlikely? A simple dimension count shows that two varieties of dimension *r* and *s* are non 'likely' to intersect if *r* < codim *s*, unless there are some special geometrical relations among them. A series of conjectures due to Bombieri–Masser–Zannier, Zilber and Pink rely on this philosophy. I will give a general overview on these problems in the special cases of tori and of abelian varieties.

Holly Krieger: Height functions in arithmetic dynamics

I will discuss the role of height functions in complex and arithmetic dynamics, with a focus on applications to problems in unlikely intersections. This includes a dynamical analogue of the André-Oort conjecture proved by myself, Ghioca, Nguyen, and Ye, as well as recent work on dynamical bounded height theorems, analogous to those of Bombieri–Masser–Zannier. Time permitting, I will also discuss recent work on reduction of dynatomic curves and connections to the dynamical uniform boundedness conjecture.

Bill O'Donovan: Conjugation modules for symmetric groups

The study of the structure of permutation modules for symmetric groups over fields of positive characteristic is an active area of research in the modular representation theory of symmetric groups. In this talk, I will introduce a family of permutation modules for the symmetric group *S*_{pn}, the so-called conjugation modules, obtained by inducing the trivial module for the wreath product *C*_{p} wr *S*_{n} to *S*_{pn}. I will present results parametrising the vertices of the conjugation modules (and explain the modular representation theory background, e.g. the definition of a vertex of a module), and outline a connection to Foulkes' Conjecture, one of the most important open problems in the characteristic zero representation theory of the symmetric group (no knowledge of modular representation theory will be assumed).

Ivan Fesenko: The Mochizuki theory and its group-theoretical aspects

Traditional parts of algebraic number theory operate with abelian quotients of Galois group or their representations. The ground breaking IUT theory of Shinichi Mochizuki involves full Galois groups and arithmetic fundamental groups. His theory works with elliptic curves over number fields and implements deformation of multiplication of various associated rings. These deformations are not compatible with ring structure but Galois groups pass through such deformations. IUT then uses deep reconstructions algorithms, to restore from a Galois group as a pro-finite group the scheme whose fundamental group are the given pro-finite group. Measuring the result of the deformation produces bounds which eventually lead to solutions of several most celebrated problems in number theory. Properties of pro-finite groups such as elastic (for every open subgroup each of its nontrivial normal closed subgroups topologically finitely generated as a group is open) and slim (every open subgroup is centre-free) play central role.

Slides of very basic talks about IUT, with further references, are available.