## Previous Pure Mathematics Colloquia - 2007 to 2008

Previous Pure Mathematics Colloquia from: 2013/14, 2012/13, 2011/12, 2010/11, 2009/10, 2008/09, 2007/08

### Thursday 29th of May

Alan Cain
(University of St Andrews)
Automaton semigroups

Automaton groups -- groups of automorphisms of labelled rooted trees generated by actions of finite automata -- were introduced in the early 1980s as interesting examples having exotic' properties, the most notable perhaps being the Gupta--Sidki group, an elegant example of a finitely generated infinite periodic group. Since then, a substantial theory has developed. Several authors have mentioned in passing that the concept naturally generalizes to semigroups, but no systematic study was undertaken.

Recently Victor Maltcev and I began this study. This talk will introduce the notion of automaton semigroups, give examples to illustrate the many semigroups that can arise in this way, describe some of their elegant properties and techniques for working with them, and discuss interesting open problems. Particular attention will be paid to semigroups arising from viewing Cayley graphs of finite semigroups as automata.

### Thursday 1st of May

Bob Gray
(University of St Andrews)
Graphs and digraphs with many symmetries, and a wonderfully elegant argument of Hikoe Enomoto (joint work with Dugald Macpherson and Cheryl E. Praeger)

A graph is set-homogeneous if for any two finite induced subgraphs A and B, if A and B are isomorphic then there is an automorphism of the graph that sends A to B (setwise). This is a natural weakening of the notion of a homogeneous graph, which is a graph where any isomorphism between finite induced subgraphs extends to an automorphism. Any homogeneous graph is set-homogeneous, and Ronse (J. London Math. Soc. (1978)) showed that for finite graphs the converse is also true. He did this by classifying the finite set-homogeneous graphs and then observing that each of them also happens to be homogeneous. Subsequently, Enomoto (J. Comb. Theory Ser. B (1981)) reproved Ronse's result by giving a devastatingly simple, and extremely short, direct proof of the fact that every finite set-homogeneous graph is homogeneous.

In this talk I will present Enomoto's argument and go on to explain how it inspired a recent collaboration, which began with the question of whether a similar approach might be applied to other kinds of relational structure. Specifically I will talk about set-homogeneous directed graphs, outlining a classification in the finite case, and giving some partial results in the countably infinite case, including a classification of those that are set-homogeneous but not 2-homogeneous.

### Thursday 24th of April

Alexander Konovalov
(University of St Andrews)
The Prime Graph of the Normalized Unit Group of the Integral Group Rings of Sporadic Simple Groups (joint work with Victor Bovdi, Eric Jespers, Steve Linton)

Let V(ZG) be the normalized unit group of the integral group ring ZG of a finite group G. The long-standing conjecture of H. Zassenhaus (ZC) says that every torsion unit from V(ZG) is conjugate within the rational group algebra QG to an element of G.

W. Kimmerle proposed to relate (ZC) with some properties of graphs associated with groups. The Gruenberg-Kegel graph (or the prime graph) of the group G is the graph with vertices labelled by the prime divisors of the order of G with an edge from p to q if and only if there is an element of order pq in the group G. Then Kimmerle's conjecture (KC) asks whether G and V(ZG) have the same prime graph.

We started the program of verifying (KC) for sporadic simple groups, using the Luthar-Passi method with recent extensions by M. Hertweck as a main tool. Now we are able to report that (KC) holds for the following twelve sporadic simple groups:

- Mathieu groups M_11, M_12, M_22, M_23, M_24; - Janko groups J_1, J_2, J_3; - Higman-Sims, McLaughlin, Rudvalis and Suzuki groups.

In my talk I will also introduce the notion of (p,q)-irreducible characters which in some cases may substantially reduce enumeration, and will describe the state of the problem for the O'Nan group, 3rd and 2nd Conway groups, and the 4th Janko group.

### Thursday 17th of April

Volodymyr Mazorchuk
(University of Glasgow)
Kiselman's semigroups

Kiselman's semigroups form a family of finite semigroups defined via a presentation, rather similar to the one of the symmetric group. The first semigroup of this family was studied by Christer Kiselman and it has origins in convexity theory. We will show that these semigroups have rather nice combinatorial properties, and make a special emphasise on the dimension of the minimal faithful representation. Finally, we will formulate some open problems. (This is a joint work with G. Kudryavtseva.)

### Tuesday 11th of March

George Havas
(University of Queensland)
Behind and beyond a theorem on groups related to trivalent graphs

In 2006 we completed the proof of a five-part conjecture which was made in 1977 about a family of groups related to trivalent graphs. This family covers all 2-generator, 2-relator groups where one relator specifies that a generator is an involution and the other relator has three syllables. Our proof relies upon detailed but general computations in the groups under question. The proof is theoretical, but based upon explicit proofs produced by machine for individual cases. I explain how we derived the general proofs from specific cases. The conjecture essentially addressed only the finite groups in the family. We extend the results to infinite groups, effectively determining when members of this family of finitely presented groups are simply isomorphic to a specific quotient.

### Thursday 6th of March

Des Fitzgerald
(University of Tasmania)
Diagrams and presentations for divided-difference semigroups

The symmetric group acts on multivariate polynomials by permutation of coordinates. Action of a transposition (i j), followed by differencing, gives a polynomial vanishing on a hyperplane x_i = x_j , and then division by (x_i - x_j) yields another polynomial, by the remainder theorem. The resulting operators are nilpotent generators of a divided-difference semigroup.

I shall give a presentation and normal form for two of these semigroups, and a representation by diagrams reminiscent of braids.

### Thursday 21st of February

Sanju Velani
(University of York)
A Mass Transference Principle in Diophantine Aprroximation

There are two fundamental results in the classical theory of metric Diophantine approximation: Khintchine's theorem and Jarnik's theorem. The former relates the size of the set of well approximable numbers, expressed in terms of Lebesgue measure, to the behavior of a certain volume sum. The latter is a Hausdorff measure version of the former. We start by discussing these theorems and show that the general Hausdorff theorem is infact a consequence of the Lebesgue theorem. The key is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for limsup subsets of R^k to Hausdorff measure theoretic statements.

Time permittimg, a Hausdorff measure version of the Duffin-Schaeffer conjecture in metric number theory is introduced and discussed. The general conjecture is established modulo the original conjecture.

### Thursday 7th of February

Debbie Lockett
(QMUL)
Connected-homomorphism-homogeneous graphs

A relational structure is homogeneous if any isomorphism between finite substructures can be extended to an automorphism of the whole structure. A nice way of saying this is that any local symmetry is global, and because of this rich symmetry homogeneous structures have proved to be very interesting since their introduction by Fraisse in about 1950. My research involves investigating a weakening of the notion of homogeneity where "isomorphism" is replaced by "homomorphism" in the definition, as first suggested by Cameron and Nesetril in 2004. In this talk I will give a friendly introduction to homogeneous structures, before looking in particular at a particular case of a generalisation of this generalisation of homogeneity for particular structures: connected-homomorphism-homogeneity of finite graphs.

### Friday 30th of November

Nikolay Nikolov
(Imperial)
Expansion and product decompositions of finite groups: variations on a theme of Gowers (joint work with Laszlo Babai and Laszlo Pyber)

Let $G$ be a finite group of order $n$ and let $m$ be the minimum dimension of its irreducible representations over the reals; and let $X,Y,Z$ be subsets of $G$. In a recent paper, Tim Gowers proved that if $|X||Y||Z|\ge n^3/m$ then there exist $x\in X$, $y\in Y$, and $z\in Z$ such that $xy=z$.

We observe that Gowers' result can be restated as saying that under the same condition, $XYZ=G$. We prove the following generalization: if $X_1,\dots,X_t$ are subsets of $G$ $(t\ge 3)$ such that

\beq \prod_{i=1}^t |X_i| \ge \frac{n^t}{m^{t-2}} \eeq

then\ $\prod_{i=1}^t X_i = G.$

A number of applications to the area of expansion of Cayley graphs and `bounded generation'' follow, resulting in considerably simpler proofs as well as improved bounds.

### Thursday 8th of November

Jeremie Guilhot
(Aberdeen)
On the computation of Kazhdan-Lusztig polynomials in affine Weyl groups

In this talk, i will present the notion of affine Weyl groups and Kazhdan-Lusztig cells. Then, using a geometric presentation of an affine Weyl group, I will establish that the Kazhdan-Lusztig polynomials are invariant under (long enough) translations. This, plus some computations with GAP, will allow us to determine the decomposition of $\tilde{G}_{2}$ into left cells, for certain choice of parameters.

### Thursday 25th of October

Sergey Kitaev

Sequences and morphisms

The Arshon sequence was given in 1937 in connection with the problem of constructing a square-free sequence on a given alphabet. The Dragon curve (the paperfolding sequence) was discovered by physicist John Heighway and was described by Martin Gardner in 1978. The Peano curve was studied by the Italian mathematician Giuseppe Peano in 1890 as an example of a continuous space filling curve. The Peano infinite word is a discrete analog of the Peano curve.

Are there any similarities between the Arshon sequence, the Dragon curve, and the Peano infinite word? In this talk, which does not require any special background, I will answer this question in the affirmative.

### Thursday 18th of October

Max Neunhoeffer
(St Andrews)
Iwahori-Hecke algebras and James' conjecture

In this talk I explain what James' conjecture about Iwahori-Hecke algebras is. To this end I define the algebras, introduce some concepts of their representation theory over different fields and finally present a formulation of James' conjecture.

### Thursday 11th of October

Mike Evans
(Washington and Lee University, USA)
Graphing by connecting the dots

Being the pointwise limit of a sequence of continuous functions, every function of Baire class one can certainly be expressed as a pointwise limit of a sequence of continuous piecewise linear functions. However, can these continuous pointwise linear approximating functions be chosen to have their vertices on the graph of the original function, i.e., can the graph of the function be approximated by connecting the appropriate dots? We shall show that the answer is 'maybe'.

### Thursday 4th of October

Prof Darji
(University of Louisville, USA)
The dichotomy between meager and Haar null sets

In this talk we discuss meager sets and Haar null sets in Polish groups. It's often the case that the sets of importance in Polish groups are simultaneously meager and Haar null. However, we show that for some natural concepts in algebra and analysis meager sets and Haar null sets exhibit very different behavior. We discuss examples of such phenomenon in the symmetric group on the natural number, the Baer-Specker group and the the group of continuous real-valued functions on [0,1].