Autumn 2015

29th September | Lars Louder (UCL) Wise's w-cycle conjecture and homological coherence for one-relator groups (abstract) |

6th October | Carolyn Chun (Brunel) A splitter theorem for internally 4-connected binary matroids (abstract) |

13th October | Steven Galbraith (University of Auckland, LMS Aitken Lecturer) Linear algebra with errors, coding theory, cryptography and Fourier analysis on finite groups (abstract) |

20th October | Sarah Hart (Birkbeck) Product-free sets in groups (abstract) |

27th October | Bernhard Koeck (Southampton) Doing algebraic K-theory algebraically (abstract) |

3rd November | Henry Bradford (Oxford) New uniform diameter bounds in pro- p groups (abstract) |

10th November | Nicholas Gill (South Wales) On Cherlin's conjecture for finite permutation groups (abstract) |

17th November | Sean Eberhard (Oxford) Permutations fixing a k-set (abstract) |

24th November | Christina Goldschmidt (Oxford) The Brownian continuum random tree as a unique fixed point (abstract) |

1st December | Andrew Tonks (Leicester) Möbius inversion and incidence algebras in decomposition spaces (abstract) |

8th December | Colva Roney-Dougal (St Andrews) Some relations relating to generation (abstract) |

Spring 2016

12th January | David Robertson (Newcastle) Conjugacy in Higman-Thompson groups (abstract) |

26th January | Nikolay Nikolov (Oxford) Words in profinite groups: beyond finite generation (abstract) |

2nd February | Philipp Habegger (Basel) Complex Multiplication or is e ^{π√163} an integer? (abstract) |

3rd February | Jose Burillo (Barcelona) Normal subgroups of the Lodha–Moore groups (abstract) |

9th February | Alex Wilkie (Manchester) Analytic Hardy fields |

16th February | Jelena Grbic (Southampton) Homotopy Rigidity of the Functor ΣΩ (abstract) |

23rd February | Zoltan Leka (Royal Holloway) Leibniz inequality in probability spaces (abstract) |

1st March | Jochen Koenigsmann (Oxford) Fields with the absolute Galois group of Q (abstract) |

15th March | Sandro Mattarei (Lincoln) Exponentials of derivations in prime characteristic (abstract) |

Lars Louder: Wise's *w*-cycle conjecture and homological coherence for one-relator groups

A two-complex *X* has non-positive immersions if, for every immersion *Y* → *X*, *Y* compact, either χ(*Y*) ≤ 0 or *Y* has trivial fundamental group. In this talk I'll show that presentation complexes for one-relator groups have non-positive immersions, and, as a corollary, that finitely generated subgroups of one-relator groups have finite second betti numbers. This is joint work with Henry Wilton.

Carolyn Chun: A splitter theorem for internally 4-connected binary matroids

Two powerful inductive tools for dealing with 3-connected matroids are Tutte's Wheels-and-Whirls Theorem and Seymour's Splitter Theorem. The first shows that it is always possible to remove one or two elements from a 3-connected matroid *M* to get another 3-connected matroid. The second shows that such removals can be done to maintain not only 3-connectivity but also a copy of a specified 3-connected minor of *M*. In this talk we present the analogues of Tutte's Wheels-and-Whirls Theorem and Seymour's Splitter Theorem adapted to the class of internally 4-connected binary matroids. We also present corollary results for internally 4-connected graphs.

Steven Galbraith: Linear algebra with errors, coding theory, cryptography and Fourier analysis on finite groups

Solving systems of linear equations *Ax* = *b* is easy, but how can we solve such a system when given a 'noisy' version of *b*? Over the reals one can use the least squares method, but the problem is harder when working over a finite field. Recently this subject has become very important in cryptography, due to the introduction of new cryptosystems with interesting properties.

The talk will survey work in this area. I will discuss connections with coding theory and cryptography. I will also explain how Fourier analysis in finite groups can be used to solve variants of this problem, and will briefly describe some other applications of Fourier analysis in cryptography. The talk will be accessible to a general mathematical audience.

Sarah Hart: Product-free sets in groups

A set *S* of positive integers is sum-free if for all *a*, *b* in *S*, *a* + *b* is not in *S*. We can ask questions like what is the maximum cardinality of a sum-free set of [1..*n*]; how many sum-free sets of [1..*n*] are there; can we categorize sum-free sets that are maximal not by cardinality but by inclusion, how small can such sets be, and so on. Naturally it didn't take long before people generalised first to cyclic groups, then abelian groups, and then to general groups, where we speak of 'product-free sets'. In this talk I'll give an overview of what's known in the general groups case, before describing some recent work with Chimere Anabanti.

Bernhard Koeck: Doing algebraic K-theory algebraically

In 2012 Grayson surprised the mathematical community with a description of higher algebraic K-groups in terms of generators and relations. After introducing that description we explain some fundamental theorems and constructions in algebraic K-theory in this new context. (This is joint work with Tom Harris and Lenny Taelman.)

Henry Bradford: New uniform diameter bounds in pro-*p* groups

We give new upper bounds for the diameters of finite groups arising as images of certain profinite groups, including p-adic analytic groups and the Nottingham group. Our bounds do not depend on a choice of generating set. Our method exploits the commutator structure of the profinite groups we study, in a manner reminiscent of the Solovay-Kitaev procedure from quantum computation.

Nicholas Gill: On Cherlin's conjecture for finite permutation groups

Motivated by questions arising in model theory, Cherlin has made an interesting conjecture concerning finite permutation groups. Work of Wiscons has reduced the conjecture to the following statement: The only almost simple *binary* permutation groups are *S*_{n} acting on *n* points. (The adjective 'binary' means, roughly, that the orbits of *k*-tuples can be deduced from the orbits of *2*-tuples, for any *k* > 2.)

We will report on current work aimed at proving Cherlin’s conjecture. In particular we will focus on the case where our permutation group is a group of Lie type (for instance PGL_{n}(*q*)); in this situation we can say rather a lot. This is joint work with Francis Hunt and Liam Harris, both of the University of South Wales.

Sean Eberhard: Permutations fixing a *k*-set

How many permutations on *n* letters have a fixed set of size *k*? This innocent-looking question turns out to be intimately connected to the well known 'multiplication table problem' from analytic number theory: how many distinct integers appear in the *n* by *n* multiplication table? The multiplication table problem was solved up to a constant factor by Kevin Ford in 2008. By borrowing techniques from Ford's paper we prove that the number of permutations on *n* letters having a fixed set of size *k* is, up to a constant factor, *n*!*k*^{-δ}(log *k*)^{-3/2}, where δ = 1 - (1+log log 2)/log 2. In this talk I will explain the connection between these two problems, as well as some of the ideas used in the proof. This is joint work with Kevin Ford and Ben Green.

Christina Goldschmidt: The Brownian continuum random tree as a unique fixed point

Aldous' Brownian continuum random tree (BCRT) is the scaling limit of a broad class of random trees, including all critical Galton-Watson trees with finite offspring variance. There are various different characterisations of the BCRT already in existence, and these different viewpoints each give access to different properties. In this talk, I will give an introduction to the BCRT (no prior knowledge necessary!) and I will show that it can also be characterised as the unique fixed point of a certain natural operation on continuum random trees. This is joint work with Marie Albenque (Ecole polytechnique, Paris).

Andrew Tonks: Möbius inversion and incidence algebras in decomposition spaces

The classical theory of incidence algebras and Möbius inversion for locally finite posets can be generalised in two ways. Firstly, one can replace posets by categories, as in the work of Leroux. Secondly, the numerical coefficients can be seen as the result of taking (homotopy) cardinality after working directly with basic combinatorial and algebraic objects, as in the work of Lawvere and Menni. In this talk, we introduce the 'ultimate' generalisation of these constructions, that we call decomposition spaces. Fundamental examples arise from weak category objects in infinity-groupoids. In particular, we obtain the Connes-Kreimer Hopf algebra from a decomposition space of combinatorial trees, and derived Hall algebras from Waldhausen's S. construction on a stable infinity category.This is joint work with Imma Gálvez Carrillo (Universitat Politècnica de Catalunya) Joachim Kock (Universitat Autònoma de Barcelona) [arxiv:1404.3202]

Colva Roney-Dougal: Some relations relating to generation

We investigate the structure of generating sets of finite groups, and other algebraic structures. In particular we define some new equivalence relations on the elements of a finite group that give insight into many generation properties of the group, and use this to investigate the generating graph of a finite group. This is joint work in progress with Peter Cameron.

David Robertson: Conjugacy in Higman-Thompson groups

In 1965, Richard Thompson introduced three groups *F* < *T* < *V* which provided the first examples of finitely presented, infinite simple groups. They remain intensely studied to this day. Nine years later, Higman discussed a family generalising *V* which we call *V*_{n,r}. Higman went further, giving a solution to the conjugacy problem in these groups.

In this talk, I'll introduce these groups and illustrate how we compute with their elements. Then I'll explain how Higman's solution works, including the details of a missing piece of his algorithm. If there's time, I'll demonstrate a live implementation of this algorithm and discuss future work on this subject.

Nikolay Nikolov: Words in profinite groups: beyond finite generation

Let *G* be a profinite group and *w* be a group word. There has been a lot of progress towards understanding the verbal subgroup *w*(*G*) for many interesting words *w* (for example commutators and powers) when *G* is topologically finitely generated. In particular in the examples above *w*(*G*) is closed in *G* and *G* is strongly complete (i.e. each subgroup of finite index is open in *G*). None of this remains true when *G* is not finitely generated. Nevertheless in some situations we can prove suitable analogues even when *G* is not finitely generated. In this talk I will discuss some natural questions and partial results in this more general setting.

Philipp Habegger: Complex Multiplication or is e^{π√163} an integer?

Roots of unity are algebraic values of the exponential function at algebraic arguments. They can be used to recover all normal finite field extensions of the rational numbers with abelian Galois group. Kronecker's Jugendtraum was to find analytic functions that account for such extensions over base fields other than the rationals. The theory of complex multiplication of elliptic curves provides a rich trove of examples of such functions with many surprising symmetries. It originated in the 19th century in work of Kronecker and Weber and underwent a remarkable development in the 20th century by Hilbert, Shimura, Deligne and many others.

In this talk I will provide a glimpse into some classical and elementary aspects of complex multiplication. Then I will discuss recent questions connected to problems in diophantine geometry, some of them are joint work with Jonathan Pila.

Jose Burillo: Normal subgroups of the Lodha–Moore groups

In 2012 Monod described a new example of a geometric group which gave a counterexample to the von Neumann conjecture, namely, it is not amenable and it does not contain free subgroups. More recently, Lodha and Moore found a subgroup of Monod's group which admits a reasonably simple finite presentation, with three generators and nine relators, and which is still a counterexample to von Neumann. This group is a generalisation of Thompson's group *F* and contains it as a subgroup. I will introduce this group, show its properties, and show some recent results about their commutator subgroups and their quotients, also generalising known results for *F*. This is joint work with Yash Lodha and Lawrence Reeves.

Jelena Grbic: Homotopy Rigidity of the Functor ΣΩ

I shall discuss the problem of the homotopy rigidity of the functor ΣΩ. Our solution to this problem depends heavily on new decompositions of looped co-*H*-spaces. I shall start by recalling some classical homotopy theoretical decomposition type results. Thereafter, I shall state new achievements and discuss how new functorial decompositions of looped co-*H*-spaces arise from an algebraic analysis of functorial coalgebra decompositions of tensor algebras. This is a joint work with Jie Wu.

Zoltan Leka: Leibniz inequality in probability spaces

Recently M. Rieffel showed that the (non-commutative) standard deviation is a strongly Leibniz seminorm. It seems to be a natural question whether seminorms determined by higher-order central moments have the same property or not. In this talk we shall investigate this question and present a few affirmative answers in ordinary probability spaces.

Jochen Koenigsmann: Fields with the absolute Galois group of **Q**

We will put forward a conjecture about fields whose absolute Galois group is isomorphic to that of the field **Q** of rational numbers. We will relate this conjecture to the Section Conjecture in Grothendieck's anabelian geometry. Indicating the first steps towards proving the conjecture we will see that (in a precise sense) most of the arithmetic of **Q** is encoded in its absolute Galois group.

Sandro Mattarei: Exponentials of derivations in prime characteristic

Historically, algebra derivations first arose in the context of Lie groups and algebras as infinitesimal automorphisms via a process of differentiation. The traditional way to go back to automorphisms from derivations is via the exponential series. Broadly speaking, under suitable conditions, 'the exponential of a derivation is an automorphism'.

The most delicate condition for this to make sense and be true is that one works over a field of characteristic zero, as in prime characteristic *p* most factorials at the denominators of the exponential series vanish. Nevertheless, variations of the traditional exponential, starting with a truncated exponential where all those terms of the exponential series are simply discarded, have been used for decades in the theory of modular Lie algebras, in the crucial technique of 'toral switching'.

I will start this talk with analysing the essential algebraic property of the truncated exponential which makes its application to derivations in prime characteristic successful, relating it to some weakened form of the familiar functional equation exp(*x*+*y*)=exp(*x*)exp(*y*). I will then discuss increasingly powerful variations of the traditional exponential in this context, such as Artin–Hasse exponentials, and certain generalized Laguerre polynomials.