Autumn 2011

27th September | Uri Onn (Ben Gurion University of the Negev) Representation zeta functions (abstract) |

4th October | Sinead Lyle (UEA) Homomorphisms between Specht modules (abstract) |

11th October | Julia Brandes (Bristol) A generalised version of the multi-dimensional Waring's problem (abstract) |

18th October | Gary Greaves (RHUL) Algebraic integers and combinatorial objects (abstract) |

25th October | Glyn Harman (RHUL) Primes whose sums of digits are prime (abstract) |

1th November | Chris Dowden Extremal planar graphs (abstract) |

8th November | Martin Kassabov (Southampton) Groups of oscillating intermediate growth (joint work with I. Pak) (abstract) |

15th November | Robert Brignall (Open University) Grid Classes (abstract) |

29th November | Inna (Korchagina) Capdeboscq (Warwick) Cocompact Lattices in some Kac-Moody groups (abstract) |

6th December | Michael Harvey (RHUL) Representations of quadratic forms (abstract) |

Spring 2012

10th January | Jan-Christoph Schlage-Puchta (Ghent) Random actions of p-groups on the p-adic tree and subgroup growth of pro-p-group (abstract) |

17th January | Matthew Towers (Kent) Koszul duality and small quantum sl _{3} (abstract) |

24th January | Nikolay Nikolov (Imperial College) Words with few values in finite simple groups (abstract) |

31th January | David Conlon (Oxford) Graph regularity and removal lemmas (abstract) |

7th February | Amarpreet Rattan (Birkbeck) Star factorisations in the symmetric group (abstract) |

14th February | Eira Scourfield (RHUL) Divisors of a polynomial over the integers (abstract) |

21th February | Lutz Warnke (Oxford) Explosive percolation is continuous (abstract) |

28th February | Tim Jones (Bristol) Points, lines, and arithmetic (abstract) |

6th March | Peter Pappas (Oxford) Combinatorial aspects of the Unit Conjecture for group rings (abstract) |

20th March | Tim Browning (Bristol) Sums of three squareful numbers (abstract) |

Uri Onn: This is an introductory talk to the subject of representation growth and representation zeta functions of groups. I will also present some recent results, including joint work with N. Avni, B. Klopsch and C. Voll.

Sinead Lyle: The Specht modules are important modules for the symmetric group algebra and the Hecke algebras of type A. A lot of their properties can be encoded combinatorially and partitions, Young diagrams and tableaux are used extensively to study them. In this talk, we show how we may use combinatorics to construct homomorphisms between Specht modules indexed by certain pairs of partitions. This is joint work with Andrew Mathas.

Julia Brandes: A generalisation of Waring's problem, considered first by Arkhipov and Karatsuba, is the question of representing not an integer, but a given polynomial, as a sum of powers of linear polynomials. We investigate a related problem and prove a Hasse principle for the number of identical representations of a set of given forms by homogeneous polynomials of general shape. The result leads to sizeable improvements for estimates of the number of linear spaces on the intersection of hypersurfaces.

Gary Greaves: One gains a better understanding of certain sets of algebraic integers by associating them to combinatorial objects. In a series of recent papers, McKee and Smyth have used various kinds of graphs and matrices to study certain algebraic integers. I will present results supporting a conjecture of Lehmer that have been obtained by classifying combinatorial objects having a certain property.

Glyn Harman: Thanks to a recent extraordinary result given by Drmota, Mauduit and Rivat, we know that every sufficiently large prime is the sum of the digits of another prime. In this talk, which is aimed at a general mathematically literate audience, we explore the question of counting the number of primes whose sum of digits is prime, and show that an asymptotic formula is only possible if we assume a hypothesis somewhat stronger than the Riemann hypothesis. Nevertheless, we are able to obtain a formula for the sum of reciprocals of these primes. In addition we can show that if we look at the “decimal” expansion (in any base) of a “typical” real number, we find infinitely many primes whose sum of digits is prime. Technical details will be kept to a minimum!

Chris Dowden: One of the best known results in graph theory is Turan's Theorem, which concerns the maximum number of edges that a graph on *n* vertices can have without containing a “complete” subgraph of size *r* (i.e. *r* vertices with an edge between every pair). The Erdős-Stone Theorem then extends this to the case when other subgraphs are forbidden instead. Recently, I have started to look at the analogous question for planar graphs, which are graphs that can be drawn in the 2-dimensional plane without any edges crossing. In other words, how many edges can a planar graph on *n* vertices have without containing a specified forbidden subgraph? In this talk, I intend to present results for the case when the forbidden subgraph is a complete graph, which is quite easy to deal with, and also for the case when the forbidden subgraph is a small cycle, which is more difficult.

Martin Kassabov: We construct an uncountable family of finitely generated groups of intermediate growth, with growth functions of new type. These functions can have large oscillations between lower and upper bounds, both of which come from a wide class of functions. In particular, we can have growth oscillating between exp(*n*^{a}) (for 0.8 < *a* < 1) and any prescribed subexponential function, growing as rapidly as desired. Our construction is built on top of any of the Grigorchuk groups of intermediate growth, and is a variation on the limit of permutational wreath product.

Robert Brignall: The study of permutation classes (sets of permutations closed downwards under the containment partial order) has recently begun to move from the ad-hoc consideration of specific examples to a more general structure theory. Central to this theory are “grid classes” — classes consisting of the permutations which when viewed graphically can be divided into cells using a collection of horizontal and vertical lines, with each cell satisfying some additional conditions. For example, we may specify that the points of a certain cell must form an increasing sequence.

Highlights in the study of grid classes include startling results in the asymptotic growth rates of so-called “small permutation classes”, and general techniques to apply Higman's theorem or construct infinite antichains to resolve the question of well-quasi-order for a large number of permutation classes. Meanwhile, other problems such as finding the minimal set of forbidden patterns (the “basis”) for a given grid class remain essentially wide open.

In this talk, I will present some of the most pertinent results and open problems in the study of grid classes. Of particular interest are monotone grid classes (where every non-empty cell of the grid must form an increasing or a decreasing sequence), and results relating to their structure, enumeration and well-quasi-order.

Inna (Korchagina) Capdeboscq: After introducing Kac-Moody groups and recalling the necessary prerequisites about lattices, I will discuss existence and constructions of cocompact lattices in some topological Kac-Moody groups.

Michael Harvey: In this talk, we shall discuss the problem of representing an integral positive-definite quadratic form of rank *m* by another of rank *n*. When *m* = 1, this reduces to the classical case of representing a positive integer. Siegel obtained an exact formula for the number of averaged representations. Here we shall give an asymptotic formula for the number of representations, provided *n* is large enough in term of *m*.

Jan-Christoph Schlage-Puchta: Let *G* be a *p*-group, *P*_{n} the *p*-Sylow subgroup of the symmetric group $S_{p^n}$. Homomorphisms of *G* into *P*_{n} can be viewed as actions of *G* on the *p*-adic tree of height *n*. We show how the behaviour of random actions is linked to the subgroup growth of *G*, and describe some phenomena for random actions. As application we show that the subgroup growth of large pro-*p*-groups is much more irregular than the subgroup growth of large discrete groups.

Matthew Towers: Abstract: Certain graded algebras have a remarkable relationship to their Ext algebras called Koszul duality. The small quantum groups, deformed versions of restricted enveloping algebras of semisimple Lie algebras, have long been conjectured to have this property. I will try to explain exactly what Koszul duality means and to illustrate it with the case of small quantum sl_{3}.

Nikolay Nikolov: Abstract: I will survey some recent results about word values in finite groups and the open problems that remain. I will also present a new result with M. Kassabov constructing words which take values only the 3-cycles and the identity in a given alternating group.

David Conlon: Szemerédi's regularity lemma states that every large graph may be partitioned into a small number of parts so that the bipartite graph between almost all pairs of parts is random-like. One of the most important applications of this theorem is the graph removal lemma, which roughly says that every graph with few copies of a fixed graph *H* can be made *H*-free by removing few edges. In this talk, we will discuss recent progress on bounds for these theorems and for several important variants. This is joint work with Jacob Fox.

Amarpreet Rattan: I will be talking about certain factorisations in the symmetric group known as star factorisations. I will put these factorisations into the context of general transitive factorisations, as well as giving an overview of the results to date. I will end with some open questions.

Eira Scourfield: The motivation for the topic of this talk is a problem, still unsolved, raised in a 1952 paper by Paul Erdos. Let *f* be a product of irreducible polynomials with integer coefficients. Various authors have investigated problems related to determining an asymptotic formula for the number of divisors *d* < *y* = *y*(*x*) of *f*(*n*) summed over *n* < *x*, where *d* may be restricted in some way and *f* itself may be irreducible. In particular we discuss the cases when *d* is smooth or unrestricted or *q*-free or exact, where *d* is exact if *d* divides *f*(*n*) but *dp* is not a divisor of *f*(*n*) for any prime divisor *p* of *d*, and *d* is smooth if its largest prime divisor is restricted in size.

Lutz Warnke: It is widely believed that certain simple modifications of the random graph process lead to discontinuous phase transitions. In particular, starting with the empty graph on *n* vertices, suppose that at each step two pairs of vertices are chosen uniformly at random, but only one pair is joined, namely one minimizing the product of the sizes of the components to be joined. Making explicit an earlier belief of Achlioptas and others, in 2009, Achlioptas, D'Souza and Spencer conjectured that there exists a δ > 0 (in fact, δ ≥ 1/2) such that with high probability the order of the largest component ‘jumps’ from o(*n*) to at least δ*n* in o(*n*) steps of the process, a phenomenon known as ‘explosive percolation’. We give a simple proof that this is not the case. Our result applies to all ‘Achlioptas processes’, and more generally to any process where a fixed number of independent random vertices are chosen at each step, and (at least) one edge between these vertices is added to the current graph, according to any (online) rule. (Joint work with Oliver Riordan.)

Tim Jones: What is the maximum number of incidences possible between a set of points and a set of lines in a plane? The extent to which answers are known depends on the plane over which the field is defined. Over **R** the question was settled by Szemeredi and Trotter in the 1980s but it is still open in other settings, such as finite fields. I'll explain recent progress in the case of a finite field of prime order, obtained via the close links between the incidences question and problems in combinatorial number theory.

Peter Pappas: The Unit Conjecture for group algebras asserts that if *K* is a field and *G* a torsion-free group, then all units in the group algebra *KG* are ‘trivial’; that is, non-zero scalar multiples of group elements. Unlike the related Zero-Divisor Conjecture, very little progress on this problem has been made. I shall give a brief description of the problem, offer alternate viewpoints which may be of interest to combinatorists, and present recent advances obtained recently in collaboration with David A. Craven.

Tim Browning: A number is called *r*-full if whenever a prime *p* divides it, then so does *p*^{r}. We discuss the frequency of solutions to the equation *x*+*y*=*z* in *r*-full positive integers *x*,*y*,*z*, relating it to work of Campana about integral points on orbifolds. We present a conjecture in the special case *r*=2, together with some partial results. This is joint work with Karl Van Valckenborgh.