Pure Maths Seminar Archive - Abstracts 2007
The questions "What are the typical properties of a graph where decisions as to which edges arise are taken as random", "are there particular graphs which have many of the properties of a typical random graph" and "How do I find graphs on a large number of vertices which have only small cliques and only small independent sets" would not appear to have much to do with each other. In this talk (which is intended for a broad pure mathematical audience, and will to a large extent be discursive and of survey character) I intend to indicate the links between these questions and what is known about them.
We define and discuss the dynamics of cellular automata with the property that, at each site in the lattice, a random choice among finitely many local rules is made. We call these stochastic CA's and we set up a mathematical framework for them and discuss examples and dynamical properties in one dimension. They arise in mathematical models of the spread of viruses where the fast mutation of the virus leads to the appearance of a random choice of local rule. We conclude with a one-dimensional version of the virus model. This is joint work with Donna Molinek.
A long standing open conjecture in the representation theory of the general linear groups predicts the existence of a 'standard' basis for the Specht modules. Recent work has produced such a basis when the Specht module is indexed by a partition with two parts. The construction introduced a class of polynomials, now known as rank polynomials, with certain interesting properties. We will discuss the rank polynomials before showing their connection to the general linear groups. This is joint work with Marco Brandt, Richard Dipper and Gordon James.
We apply the criterion for the consistent existence of universals for a class of models to the class of triangle-free graphs (joint work of Dzamonja and Shelah). In our search for other graphs complying with universality criterion we turn to the countable homogeneous graphs as classified by Cherlin. We present a theorem about the existence of distinct classes of models satisfying these axioms of universality. Finally we apply our axiomatic framework to the class of some models of some theory of an countable categorical structure and give some examples of such a theory.
Isogenies are surjective homomorphisms of abelian varieties with finite kernel --- that is, ``almost isomorphisms''. When working over the complex field, an abelian variety is just a torus, and we may talk about isogenies in terms of lattices. However, over other fields the situation is more complicated, and few explicit examples of non-trivial isogenies are known. In this talk, we will discuss some of the computational issues, and describe some examples of isogenies of higher-dimensional abelian varieties. In particular, we will exhibit families of hyperelliptic curves of genus 3, 5, 6, 7, 10, and 15, all of which are equipped with an an explicit isogeny from their Jacobian to another hyperelliptic Jacobian.
Samson Abramsky promoted the idea to use Stone duality to connect denotational semantics for programming languages with program logics. He developed a fairly specialised theory that was applicable to so-called Scott-domains (and SFP objects). In joint work with Drew Moshier, we found that the theory becomes much more elegant if carried out for stably compact spaces. More recently, we discovered that our duality (and the resulting logic) can usefully be expressed in the setting of bitopological spaces. This sheds new light on the logical set-up but, surprisingly, also on the classical dualities of Stone for Boolean algebras and distributive
The planar periodic Lorentz gas with finite horizon was introduced by Sinai and can be viewed as a simple deterministic model for Brownian motion.
In this talk, we discuss the almost sure invariance principle (ASIP): for almost every initial condition, the position vector q(t) in the Lorentz gas approximates a two-dimensional Brownian trajectory.
This answers a question of Chernov and Dolgopyat (ICM, 2006). Previously Bunimovich and Sinai (1981) proved a distributional version of this result. The ASIP implies the distributional result, as well as the (functional) law of the iterated logarithm and its refinements.
More generally, we prove a vector-valued ASIP for Axiom A diffeomorphisms and flows as well as large classes of nonuniformly hyperbolic systems.
This is joint work with Matthew Nicol.
Joint work with R. Marsh (Leeds). I will describe m-cluster categories of type A (and D) using a category of diagonals (arcs) of a regular polygon. This generalises a result of Caldero, Chapoton and Schiffler for m=1.
We consider a question of Ono concerning which spaces of classical modular forms can be generated by sums of η-quotients. We give some new examples of spaces of modular forms which can be generated as sums of η-quotients, and show that we can write all modular forms of level Γ0(N) as rational functions of η products.
In projective space one may consider the set of k-dimensional subspaces of an n-dimensional vector space over a field of size q. This set can be taken to be the basis for a permutation module over a field which has prime characteristic p, where p does not divide q. By considering an inclusion map one can obtain sequences of maps on these modules. It turns out that these sequences have homological properties. We consider the corresponding homology modules and prove some interesting results including a decomposition formula. We also look at some irreducible representations of GL(n,q) associated to these modules.
The talk will give an overview of topological groups with particular emphasis on totally disconnected, locally compact groups. Topological Kac-Moody groups form an interesting class of examples. Some results on Kac-Moody groups and Hecke algebras obtained from this perspective are stated and placed into context.
This talk will concern a class of group endomorphisms exhibiting slow orbit growth. An associated dynamical Dirichlet series is introduced and this is found to have a closed rational form. Analytic properties of the Dirichlet series are related to orbit-growth asymptotics: depending on the location of the abscissa of convergence and the degree of the pole there, various orbit-growth asymptotics are found, all of which are polynomially bounded.
In classical complexity theory the class P is defined as the class of all problems over 0,1 that can be solved in polynomial time by a deterministic Turing Machine over the field with two elements. Similarly NP is the class of all problems over 0,1 that can be solved in polynomial time by a nondeterministic Turing Machine over F2, which is allowed to guess elements from F2 during its calculation. The famous question wether NP is actually equal to P is still open.
It is possible to generalize the concept of computability to an arbitrary algebraic structure M, where the Turing Machine calculates with elements from M and can use all functions and relations which are contained in the language of M. Then we can also define the classes P and NP over M.
We will do this for some algebraic structures like the group and the ring of integers or the field of rationals and give answers there to the P versus NP question, if possible. We will also see, that the P versus NP question over M has some connections with model theoretic properties of M.
The Alhambra complex in Granada, Spain is a rich treasury of Islamic design. Alongside a world-wide appreciation of the tilings and decorations of the Palace, an 'urban myth' has evolved - that examples of all of the 17 possible symmetry groups may be found there.
Question: Is that so?
Answer(s): (1) No. (2) Yes if you cheat. (3) Yes in one sense.
In 1970 Matiyasevic resolved Hilbert's Tenth Problem for the ring of integers. Currently the problem is unresolved for the field of rational numbers. I will report on recent work of Poonen which took a giant leap in that direction by resolving HTP for some large subrings of the rationals. Another interesting feature of his work is his use of elliptic curves. I will report too of my own modest efforts to give qualitative results of the same kind.
This talk is on joint work with John Britnell (Newcastle). If G is a finite group and H is a subgroup of G of index 2 then there is a well-known relationship between the conjugacy classes of G and H. In my talk I will present two related generalisations. The first has a combinatorial flavour, and includes an interesting application of Hall's Marriage Theorem to group theory. The second has connections with character theory.
We study the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding nonconformal map on the torus given by an integer-valued diagonal matrix. The Hausdorff dimension of a "general Sierpinski carpet" was found by McMullen and Bedford and the uniqueness of the measure of full Hausdorff dimension in some cases was proved by Kenyon and Peres. We extend these results by considering a general Sierpinski carpet represented by a shift of finite type. Applying results of Ledrappier-Young and Shin, we study the Hausdorff dimension of such a general Sierpinski carpet for the case when there is a saturated compensation function. We givesome conditions under which a general Sierpinski carpet has a unique measure of full Hausdorff dimension and study the properties of the unique measure.
This talk will look at what is known about tilting modules for SL3 over an algebraically closed field of prime characteristic. I will first explain what a tilting module is and how finding the characters of the tilting modules then tells us what the 3 part decomposition numbers for the symmetric group are. I will then look at specific calculations of tilting modules and what is thought to be true in general (and why this is such a hard problem!)
The aim of this talk is to introduce the audience to the theory of local Howe correspondence. For the dual pair of type (GL(n),GL(m)) we will show a new proof which allows us to describe the correspondence in terms of Langlands parameters. At the end, we will discus about the possibility of having such a correspondence for l-modular representations.
There are many triangulated categories that arise in the study of group cohomology: the derived, stable or homotopy categories, for example. In this talk I shall describe the relative cohomological versions and the relationship with ordinary cohomology. I will explain what we know (and what we would like to know) about these categories, and how the representation type of certain subgroups makes a fundamental difference.
For the integers a and b let P(ab) be all partitions of the set N={1,...,ab} into parts of size a. Further, let CP(ab) be the corresponding permutation module for the symmetric group on N. A conjecture of Foulkes says that CP(ab) is isomorphic to a submodule of CP(ba) for all a not larger than b. The conjecture goes back to the 1950's but has remained open. Nevertheless, for small values of b there has been some progress. I will discuss some proofs and further conjectures.
There is a close correspondence between the representations of the symmetric groups and those of the general linear groups, via Schur-Weyl duality. Foulkes' conjecture therefore has implications for GL- representations, a fact first observed by Roger Howe at Yale in the 1980's. I hope to have the time to mention some of these interesting connections to classical invariant theory.
Say that a family ƒ of subsets of a set A is ½-dense if every finite F ⊆ A has a subset F0 ∈ ƒ satisfying |F0| ≥ ½|F|. The notorious problem DU on Fremlin's list is: Does every ½-dense family on ω1 which is closed under subsets admit an infinite set all of whose finite subsets are in ƒ. Under CH the answer is known to be negative.
In the hope of getting closer to the solution of DU we try to understand the relevance of the two requirements on the family. We show that there is a ½-dense family ƒ of subsets of the continuum c with the property that every infinite subset of c has arbitrarily large finite subsets in ƒ but there is no infinite set all of whose finite subsets are in ƒ. Michalewski found an easy example which is closed under subsets but where ½-density is weakened. By a result of Fremlin, modulo a measurable cardinal it is consistent that for every ½-dense closed under subsets family on c there is an infinite set whose all finite subsets are in the family, therefore the above examples are closest that one can get to a counterexample in ZFC.
I will describe basic facts on p-adic symmetric spaces. In particular σ-split tori, σ-split parabolic subgroups related to results of A.Helminck with S.P.Wang and G.Helminck. I will describe the Cartan decomposition for p-adic symmetric spaces (Benoist-Oh, Delorme-Sécherre). I will describe results of Lagier, Kato-Takano and myself on H-fixed linear forms on smooth-modules and Jacquet modules. This leads to the definition of the constant term of smooth functions on a p-adic symmetric space and of cuspidal functions. The construction of Eisenstein integrals will be described (Blanc-Delorme).
Hochschild cohomology is an important invariant in the representation theory of algebras. In joint work with Solberg, we used the Hochschild cohomology ring to construct a support variety for any module over a finite-dimensional algebra. This was motivated by the work of Carlson, who introduced the support variety of a module over a group algebra; this is now a powerful invariant in the modular representation theory of finite groups. I will describe the construction and properties of support varieties constructed using Hochschild cohomology, with specific reference to certain finiteness conditions, under which we have analogues of many of the properties of the group ring situation.
This work motivated our conjecture that the Hochschild cohomology ring modulo nilpotence of a finite-dimensional algebra is always finitely generated as an algebra. The second part of my talk will focus on the current position of this conjecture.
This is a report on joint work with Monica Nevins (University of Ottawa). Types are an important tool for understanding the representation theory of p-adic groups and they have been extensively studied in the case of supercuspidal representations. We consider the principal series representations of GL(n) and, in particular, reexamine earlier work of Howe.
The evidence for the use of geometry in the planning of medieval cathedrals and in their constructional design is conclusive. Accordingly, this paper will consider what evidence there is for the presence of geometry in the planning of Norwich Cathedral.
Geometry was fundamental to an architect's training and to the working practices of the masons' lodge. This resulted in plans sometimes being based on grids of squares, known as designing ad quadratum, also in architectural details being produced from squares in the technique known as quadrature.
The modernist critique of medieval architecture argued that these techniques were the purely practical procedures of masons which signified nothing beyond themselves. On the other hand, geometry, as one of the disciplines of the liberal arts, was regarded as a tool of reason for observing the universe. God was repeatedly portrayed as the Divine Architect, compass in hand, creating order from chaos, and geometric figures commonly carried symbolic meaning.
Norwich Cathedral was chosen for the investigation because its Norman layout stands virtually intact and a digitised laser survey has recently been completed.
The prominence of the square in medieval architectural practice has led numerous studies to propose the ratio of the side to the diagonal of a square, as $1: \sqrt(2)$, as the basis of much medieval plan design. Although the system carries with it certain problems, including its application to Norwich Cathedral, such is the weight of circumstantial evidence, it represents a case that needs to be answered.
Another system that has been applied to Norwich involves the figures of Platonic geometry. As the plane figures of the regular polyhedra, with which Plato associated the elements and the universe, much importance was given to them in medieval design. When the geometry of these figures was applied to the plan of Norwich Cathedral, and tested mathematically against the survey, a correlation was found between the geometric system and the building to within 1.5%, a degree of accuracy which would have appeared exact to anyone restricted to medieval draughting and surveying techniques.