Autumn 2012

2nd October | Lukasz Grabowski (Imperial College) Spectral properties of group rings of semidirect products (abstract) |

9th October | Owen Cotton-Barratt (University of Oxford) Scale invariance in geometry and topology (abstract) |

16th October | Jörg Brüdern (Universität Göttingen) Anisotropic diagonal forms (abstract) |

23rd October | Lilian Matthiesen (University of Bristol) Rational points on conic bundle surfaces via additive combinatorics (abstract) |

30th October | David Singerman (University of Southampton) The geometry of Galois' last theorem (abstract) |

6th November | Steven Noble (Brunel University) The Merino–Welsh conjecture: an inequality for Tutte polynomials (abstract) |

7th November | Sergei Chmutov (Ohio State University) Beraha numbers and graph polynomials (abstract) |

13th November | Gareth Jones (University of Southampton) Bipartite graph embeddings, Riemann surfaces and Galois groups (abstract) |

20th November | Oleg Pikhurko (University of Warwick) On possible Turán densities (abstract) |

27th November | Andrei Yafaev (UCL) Some applications of model theory to diophantine geometry (abstract) |

4th December | Dan Loughran (Bristol) Rational points of bounded height and the Weil restriction (abstract) |

13th December | Dorothy Buck (Imperial) The Topology DNA (abstract) |

Spring 2013

8th January | Iain Moffat (Royal Holloway) Duality and equivalence of graphs in surfaces (abstract) |

15th January | Igor Wigman (King's College London) The distribution of defect and other nonlinear functionals of random spherical harmonics (abstract) |

22nd January | Carolyn Chun (Brunel University) Fundamental questions in matroids (abstract) |

29th January | Chris Gill (University of Oxford) Multiplicities and vertices in tensor products of Young modules (abstract) |

5th February | Mark Walters (QMUL) Optimal resistor networks (abstract) |

12th February | Stephen Huggett (University of Plymouth) Newton, the geometer (abstract) |

19th February | Brita Nucinkis (RHUL) On Richard Thompson's groups and their generalisations (abstract) |

26th February | Bill Jackson (QMUL) Chromatic polynomials (abstract) |

5th March | Ashot Minasyan (University of Southampton) Hyperbolically embedded subgroups in groups acting on trees (abstract) |

12th March | Martin Loebl (Charles University, Prague) On Rota's bases conjecture (abstract) |

19th March | Francesco Matucci (Paris) The conjugacy problem in extensions of Thompson's group (abstract) |

21st March | Conchita Martinez-Perez (Zaragoza) Isomorphisms between Brin–Higman–Thompson groups (abstract) |

Summer 2013

26th June | Steven Galbraith (University of Auckland) A tutorial on recent work of Joux on the discrete logarithm problem |

Lukasz Grabowski: Spectral properties of group rings of semidirect products

Consider a discrete group *G* and its regular representation on the Hilbert space l_{2}(*G*). We want to understand spectral properties (i.e. kernels, eigenspaces, etc.) of elements of the group ring of *G*. In the recent years it has been observed that if *G* is a semidirect product, we can often get a very good description of such properties. This has led to various counterexamples to the Atiyah conjecture, examples of interesting spectral measures (in particular to manifolds with trivial Novikov–Schubin invariants), and to relating random walks on semidirect products with random Schroedinger operators. I will start by giving some motivations for the study of spectral invariants; then I'll explain why are the semidirect products amenable to computations, and I'll indicate the ways to obtain the examples mentioned above.

Owen Cotton-Barratt: Scale invariance in geometry and topology

The traditional scope of geometry and topology has been idealised objects. Despite no perfect examples of such ideal objects existing in the external world, we often feel our intuitions in these subjects do relate to it. We will explore this discrepancy, and develop some theory to explain it. We take some inspiration from coarse geometry (which cannot see finite structures) to generalise the ideas behind persistent homology, a recent tool for detecting scale-dependent structures in finite metric spaces.

Jörg Brüdern: Anisotropic diagonal forms

This talk addresses the classical question whether a diagonal form admits non-trivial zeros. We shall discuss this problem in the obviously related cases where the underlying field is finite, or *p*-adic, or the rational numbers. In all these cases, the familiar conjecture of Artin has been confirmed long ago, and our primary concern is to go well beyond this. It turns out that a complete classification of the anisotropic forms is possible if the number of variables is only half as large as is needed for Artin's conjectures.

Lilian Matthiesen: Rational points on conic bundle surfaces via additive combinatorics

Methods of Green and Tao can be used to prove the Hasse principle and weak approximation for some special intersections of quadrics defined over the rational numbers. This implies that the Brauer–Manin obstruction controls weak approximation on conic bundles with an arbitrary number of degenerate fibres, all defined over **Q**. This is joint work with Tim Browning and Alexei Skorobogatov.

David Singerman: The geometry of Galois' last theorem

We are referring to the Theorem in Galois' last letter, written the night before his fatal duel. PSL(2,*p*), (*p* prime) acts transitively on the *p*+1 points of the projective line. In this letter Galois showed that if *p* is not equal to 2, 3, 5, 7, 11, then PSL(2,*p*) does not act transitively on fewer points. The cases of most interest are *p* = 7 and 11. Now PSL(2,7) is the automorphism group of the Fano plane, the projective plane with 7 points and also the automorphism group of the Klein quartic, a famous Riemann surface of genus 3 with its well known map of type {3,7}. PSL(2,11) is the automorphism group of a biplane with 11 points. Is there a corresponding Riemann surface? Yes! There is a Riemann surface of genus 70 and a corresponding map of type {5,11}. This surface is composed of Buckyballs and is known as the Buckyball curve. This curve is a direct cousin of the Klein quartic.

Steven Noble: The Merino–Welsh conjecture: an inequality for Tutte polynomials

While investigating convexity properties of the Tutte polynomial, Criel Merino and Dominic Welsh conjectured that in any 2-connected loopless, bridgeless graph, the larger of the number of acyclic orientations and the number of totally cyclic orientations is at least the number of spanning trees of the graph. Each of these invariants is an evaluation of the Tutte polynomial. We will discuss the background to this conjecture and explain why various 'obvious' approaches do not work. We show that a stronger version of it holds for series-parallel networks. This is joint work with many co-authors including, for the most recent result, Gordon Royle.

Sergei Chmutov: Beraha numbers and graph polynomials

The *n*-th Beraha number is defined as *B*_{n} = 2+2cos(2π/*n*). According to W. Tutte the Beraha numbers are tightly related to chromatic polynomials of graphs. It is known that a non-integer Beraha number can never be a root of the chromatic polynomial of any graph. Nevertheless, conjecturally the roots of the chromatic polynomial tend to accumulate around the Beraha numbers. In the talk I first briefly review the Beraha numbers and then I turn to the Tutte polynomial which specializes to the chromatic polynomial. After that I plan to discuss applications of the Tutte polynomial in knot theory motivated by topology.

Gareth Jones: Bipartite graph embeddings, Riemann surfaces and Galois groups

I shall explain how surface embeddings of bipartite graphs (called dessins d'enfants by Grothendieck) give a link between compact Riemann surfaces and algebraic number fields, and how they provide a faithful representation of the absolute Galois group Γ = Gal(**Q**/**Q**), an important profinite group. I shall also explain how results of Hall on finite solvable groups and of Huppert and Wielandt on products of cyclic groups allow the classification of the most symmetric dessins, namely the regular embeddings of complete bipartite graphs *K*_{n,n}, and a description of how Γ acts on them.

Oleg Pikhurko: On possible Turán densities

Let *k* ≥ 3 and *F* be a family of *k*-graphs, i.e. *k*-uniform set systems. The Turán function ex(*n*,*F*) is the maximum number of edges in a *F*-free *k*-graph on *n* vertices. The Turán density π(*F*) is the limit of ex(*n*,*F*) / binom(*n*,*k*) as *n* tends to infinity. We disprove the conjecture of Chung and Graham that π(*F*) is rational for every finite family *F*. The conjecture was independently disproved by Baber and Talbot. Also, we show that the set of possible Turán densities has cardinality of the continuum.

Andrei Yafaev: Some applications of model theory to diophantine geometry

Very recently Jonathan Pila came up with a new and very promising approach to the Andre-Oort conjecture involving the ideas from Model Theory (more specifically, the theory of o-minimality). This approach had already been sucessfully applied to many special cases. In this talk I will explain the approach, focussing on the simplest case of the Andre–Oort conjecture, namely that of products of two modular curves.

Dan Loughran: Rational points of bounded height and the Weil restriction

If one is interested in studying diophantine equations over number fields, there is a clever trick due to Weil where one may move the problem from the number field setting to the usual field of rational numbers by performing a 'restriction of scalars'. In this talk, we consider the problem of how the height of a solution (a measure of the complexity of a solution) changes under this process, and in particular how the number of solutions of bounded height changes.

Dorothy Buck: The Topology DNA

The central axis of the famous DNA double helix is typically topologically constrained or circular, and can become knotted or linked during important cellular reactions. The shape of this axis can influence which proteins interact with the underlying DNA.

I will give an overview of some of the methods from 3-manifold topology that are used to model both these DNA molecules and a variety of DNA-protein reactions. We'll conclude with a few examples showing how the answers from these models aid biologists.

Iain Moffat: Duality and equivalence of graphs in surfaces

This talk revolves around two fundamental constructions in graph theory: duals and medial graphs. There are a host of well-known relations between duals and medial graphs of graphs drawn in the plane. By considering these relations we will be led to the working principle that duality and equality of plane graphs are equivalent concepts. It is then natural to ask what happens when we change our notion of equality. In this talk we will see how isomorphism of abstract graphs corresponds to an extension of duality called twisted duality, and how twisted duality extends the fundamental relations between duals and medial graphs from graphs in the plane to graphs in other surfaces. We will then go on to see how this group action leads to a deeper understanding of the properties of, and relationships among, various graph polynomials, including the chromatic polynomial, the Penrose polynomial, and topological Tutte polynomials.

Igor Wigman: The distribution of defect and other nonlinear functionals of random spherical harmonics

This work is joint with Domenico Marinucci. Partially motivated by questions in mathematical physics and cosmology, we study the distribution of "defect" (or 'signed measure') of high degree random spherical harmonics. We were able to determine the asymptotic shape of the defect variance precisely, and moreover prove a version of Central Limit Theorem for its distribution; our techniques yield similar results for other functionals, provided they satisfy some generic condition. Our proofs combine asymptotic analysis of the Legendre polynomials, together with a recently developed inequality of Nourdin–Peccati, based on the Malliavin Calculus. In this talk I plan to introduce the subject and discuss, at least briefly, the proofs of the main results.

Carolyn Chun: Fundamental questions in matroids

A matroid is a mathematical structure that generalizes the notion of linear independence in a matrix. In this talk, I will discuss recent progress on the most compelling open question in matroid theory, Rota's conjecture.

Chris Gill: Multiplicities and vertices in tensor products of Young modules

One of the main outstanding questions in the representation theory of the symmetric groups, and the Schur algebras, is to determine the decomposition numbers. Recently Cohen, Hemmer, and Nakano have shown that determining the multiplicities of the direct summands occurring in a tensor product of Young modules is equivalent to determining these decomposition numbers. In this talk I will describe recent work giving certain restrictions on the vertices of direct summands in such a tensor product, and some reductions theorems for these multiplicities.

Mark Walters: Optimal resistor networks

Suppose we have a finite graph. We can view this as a resistor network where each edge has unit resistance. We can then calculate the resistance between any two vertices and ask questions like 'which graph with *n* vertices and *m* edges minimises the average resistance between pairs of vertices?' We are not able to fully answer this question but we can show that the obvious answer is not correct. In this talk we will discuss the obvious answer, why it is wrong, and why the correct solution seems hard to find.

This problem was motivated by some questions about the design of statistical experiments (and has some surprising applications in chemistry) but this talk will not assume any statistical knowledge.

Stephen Huggett: Newton, the geometer

We describe some of Newton's most profound geometrical discoveries, arguing that by thinking of him as a geometer we gain a deep insight into his peculiar genius. We pay particular attention to Newton's work on the organic construction, which deserves to be better known, being a classical geometrical construction of the Cremona transformation (1862). Newton was aware of its importance in geometry, using it to generate algebraic curves, including those with singularities. This is joint work with Nicole Bloye.

Brita Nucinkis: On Richard Thompson's groups and their generalisations

Richard Thompson's groups *F*, *T* and *V* are groups of homeomorphisms of the unit interval, the circle and the Cantor-set respectively. In this talk I will describe how one can view these groups and their generalisations as groups of automorphisms of so called Cantor-algebras, and how this viewpoint can be used to derive cohomological finiteness properties.

Bill Jackson: Chromatic polynomials

I will describe some of my favourite results and open problems for the chromatic polynomial and its relatives: flow polynomials, characteristic polynomials, Tutte polynomials and Potts model partition functions. I will also outline some techniques for working with these polynomials.

Ashot Minasyan: Hyperbolically embedded subgroups in groups acting on trees

The concept of a hyperbolically embedded subgroup was introduced in a recent paper of Dahmani, Guirardel and Osin, where it was used to solve a number of open problems about the mapping class groups of closed surfaces and the outer automorphism groups of free groups.

Martin Loebl: On Rota's bases conjecture

The long-standing Rota's bases conjecture asserts that the columns of arbitrary *n* non-singular *n*×*n* matrices can be partitioned into *n* sets such that each set forms a non-singular matrix, and each set is a transversal, i.e., it has exactly one column of each of the original matrices. The long-standing Alon Tarsi conjecture asserts that for n even, the sum of sign(*L*), *L* *n*×*n* latin square, is non-zero, where sign(*L*) is the product of the 2*n* permutations given by the rows and the columns of *L*. Note that the analogous sum is zero for *n* odd.

In the 90's it was proven that the Alon Tarsi Conjecture implies the Rota's bases conjecture for *n* even. I will speak about our recent result asserting that for each *n* (odd or even), Rota's bases conjecture is implied by the assertion that the sum of sign(*L*'), *L'* *n*×*n* latin square with 1st row and 1st column equal to the identity permutation, is non-zero. Note that for *n* even, this condition is equivalent to the Alon Tarsi conjecture. The starting point in our reasoning is a non-commutative generalization of a formula of Shmuel Onn. We also establish a link between the Rota's bases conjecture and the Ryser Brualdi Stein conjecture. This is joint work with Ron Aharoni.

Francesco Matucci: The conjugacy problem in extensions of Thompson's group

In a recent paper, Bogopolski, Martino and Ventura develop a criterion to study the conjugacy problem in extension of groups. This is achieved by reducing to the study of two other decision problems: the twisted conjugacy problem and the orbit decidability problem. We describe a simplified point of view of the conjugacy in Thompson's group *F* which allows us to attack these decision problems. We produce examples of extensions of *F* where the conjugacy problem is solvable and others where it is unsolvable. We also prove related results on twisted conjugacy class and discuss possible generalizations. This is joint work with Jose' Burillo and Enric Ventura.

Conchita Martinez-Perez: Isomorphisms between Brin–Higman–Thompson groups

This is a joint work with Warren Dicks. We review arguments in the literature that together with a new result yield the complete classification up to isomorphism of the Brin–Higman–Thompson groups.