Pure Maths Seminar Archive - Abstracts 2008
Let Hb,a denote the vector space whose basis is the set of all partitions of the set {1,2,...,ab} into a parts of size b. The symmetric group Sab acts naturally on this space. In the 1950's Foulkes conjectured that you can inject Hb,a into Ha,b. In this talk we shall discuss some recent progress on this and a related conjecture.
The supercuspidal representations of a p-adic reductive group G are the building blocks of the theory of smooth representations – any representation can be obtained as a composition factor of a representation parabolically induced from a supercuspidal representation of a Levi subgroup. Once the supercuspidal representations are understood, one then needs to understand how these parabolically induced representations decompose. One approach to this is to use Shahidi's L-functions, but this relies on the supercuspidal representation being generic.
In this talk, I will try to explain what all these words mean, and how one can determine explicitly which supercuspidal representations of the symplectic group Sp(4) are generic and which are not. This is joint work with Corinne Blondel (Paris 7).
Domination was originally studied for countable functions as a cardinal characteristic of the continuum. In general, for any infinite cardinal κ, the dominating number for κ (denoted d(κ)) is the minimal size of a family of functions from κ to κ which eventually dominate all other functions from κ to κ. The usual axioms of set theory, ZFC, are not sufficient to decide these numbers, which may range from κ+ to 2κ. We may construct various models of set theory where ZFC holds and these numbers are all fixed simultaneously. In joint work with Sy Friedman, we have constructed inner models for global domination using new techniques which have other interesting applications.
Two-sided ideals in the group algebra kG of a polycyclic-by-finite group G over a field k were studied intensively by Zalesskii, Passman, Roseblade and others from around 1970 onwards; the prime ideals in kG are now quite well-understood. When k is the finite field of order p, there is another class of Noetherian k-algebras arising from certain groups G, the so-called Iwasawa algebras ΩG. In this case G is a closed subgroup of the group of n by n invertible matrices with entries in the ring of p-adic integers Zp, and ΩG is a certain completion of the usual group ring kG. These algebras are very important in number theory, but have not been extensively studied from the algebraic viewpoint; in particular very little is known about their prime ideals.
I will talk about my recent work with F. Wei and J. J. Zhang: we showed that when G is a torsion-free finite-index subgroup of SL2(Zp), ΩG only has two prime ideals: zero and the augmentation ideal. These algebras are therefore quite close to being simple.
We give an account of an improved dynamical Mertens' theorem for toral automorphisms and describe how the properties of the eigenvalues with modulus 1 determine the coefficient in the main term.
A theoretical and numerical approach to extend the work of Smith and Timoshin (1996), where the special case of multiple aerofoils all aligned at a global angle of attack to an incoming uniform flow, is considered.
Associated to any monic polynomial with integer coefficients, there is a sequence of integers called a Lehmer-Pierce sequence. We investigate the appearance of primes in these sequences. For certain (infinite) families of quadratic polynomials, we show that every term beyond the 12th has a prime factor which has not appeared in previous terms.
We consider the motion of a rigid particle convected in Stokes flow through a branching channel, with the ultimate goal of describing the motion of a red blood cell through a bifurcating capillary. The problem is formulated using the boundary integral method for Stokes flow and solved using the boundary element method.
I will give an informal introduction to the recent model-theoretic notion of thorn-forking. The focus will be on results from outside logic that ought to have been the historical roots of stability theory: Matroids, John von Neumann's continuous geometry, and semimodular lattices.
Let F be a non-archimedean local field of odd residual characteristic and let G be the unramified unitary group U(2,2) defined over F. In this talk, we give a classification of irreducible smooth representations of G using the Hecke algebraic method. By establishing Hecke algebra isomorphisms, this problem is reduced to the classification of irreducible smooth representations of smaller classical groups.
Let G be a finite group, and let p be a divisor of the order of G. The dimension of P1 – the indecomposable projective F[G]-module with the trivial module in its head - is a multiple - say cp(G) – of the maximal p-power dividing the order of G. If G has a p'-Hall subgroup, this value is 1. However, in general the value of cp(G) is quite mysterious.
Together with G. Malle we have classified all finite simple groups G and prime numbers p for which cp(G) equals 1. Further analysis shows that for p∈{2,3,5} a finite group G satisfies cp(G)=1, if and only if it has a unique G-conjugacy class of p'-Hall subgroups.
The Weil representation is a well-known tool to study arithmetic and cohomological aspects of orthogonal groups. We construct certain, "special", cohomology classes for orthogonal groups O(p,q) with coefficients in a finite dimensional representation and discuss their automorphic and geometric properties. In particular, these classes are generalizations of previous work of Kudla and Millson and give rise to Poincaré dual forms for certain, "special", cycles with non-trivial coefficients in arithmetic quotients of the associated symmetric space for the orthogonal group.
Furthermore, we determine the behavior of these classes at the boundary of the Borel-Serre compactification of the associated locally symmetric space. As a consequence we are able to obtain new non-vanishing results for the special cycles.
This is joint work with John Millson.
Siegel proved that there are finitely many rational points on an elliptic curve which have an integral coordinate. I will explain why finiteness still holds when the denominator of the coordinate is an integer raised to a fixed non-trivial power. For Mordell curves these power-integral points correspond to a Fermat type Diophantine equation. I will use results of Bennett and Skinner on the equation to show that certain subsets of power-integral points are empty.
Phylogenetic diversity (PD) is a measure of the extent to which different subsets of taxa span an evolutionary tree, and provides a quantitative tool for studying biodiversity conservation. Recently, it was shown that the problem of finding subsets of taxa of given size to maximize PD can be efficiently solved by a greedy algorithm. In this talk, we describe some extensions of this work showing that some measures related to PD fail to be optimized by a greedy algorithm, but that we can still sometimes get around this problem. For a little more background please see:
http://plus.maths.org/issue46/features/phylogenetics/index.html
Let E be an ellipticcurve defined over a number field K. The N-torsion points of E are then algebraic over K, and there is a natural action of the absolute Galois group GK on the N-torsion subgroup E[N], as well as on the projective limit T(E) of these (for various N). It is then natural to consider the image of GK in the automorphism group GL(T(E)). A Theorem of J.-P. Serre asserts that the image is of finite index, if End(E) is equal to Z.
In this talk we will consider the analogous question for one-dimensional formal groups. Such formal groups arise, for example, when one considers elliptic curves (over general base rings) in an infinitesimal neighborhood of the identity element. When the formal group is a so-called universal deformation, one can prove that the fundamental group (of the underlying deformation space) surjects onto the automorphism group of the Tate module.
On an elliptic curve in the form C: U3+V3=m with any cube-free integer m, we study divisibility sequences obtained from rational points on the curve. Some properties of this sequence are considered in this talk.
Previous analyses of oblique stagnation-point flow at a plane wall are discussed and unified with reference to a free parameter. The oblique flow consists of orthogonal stagnation-point flow to which is added a shear flow whose vorticity is fixed at infinity. Physically the free parameter may be viewed as altering the structure of the shear flow component by varying the magnitude of the pressure gradient. For large adverse pressure gradients, the shear component has a region of reversed flow near the wall. Remarkably, combining an orthogonal flow with shear flows featuring different levels of reversed flow always produces the same oblique flow but with the stagnation-point of attachment shifted along the wall by a predictable amount.
A q-simplicial complex (q-complex for short) is a natural generalization of an ordinary simplicial complex. These q-complexes are isomorphic to a system of subspaces of a finite vector space over a finite field, closed under taking subspaces. We introduce these and the related concept of q-posets and see some examples of these mathematical objects. An important example of a q-complex is the set of elementary abelian q-subgroups of some finite group G and we go on to look at what this q-complex tells us about G. Finally we consider a combinatorial result for simplicial complexes and the work towards a q-analogue.
Coupled numerical models of the atmosphere and ocean are routinely used to perform experiments on long-term climate. The HiGEM (High Resolution Global Environmental Modelling) project sets a new precedent for spatial resolution in a coupled climate model. Statistical and dynamical analysis techniques are used to assess the improvement made by moving to this higher resolution, and some initial results are presented.
I will present two recent results that mark differences between usual First Order Model Theory and Continuous FO Model Theory. I will also present (depending on time) recent examples of Metric AECs.
The study of classical Goppa codes is important since they are a very large class of codes, near to random codes. They are easy to generate and possess an interesting algebraic structure.
We consider the problem of finding an upper bound for the number of permutation non-equivalent irreducible maximal Goppa codes. This question was considered by several authors (for example Chen, Berger and Charpin, Fitzpatrick and Ryan Giorgetti). We describe the action of a group FG isomorphic to AΓL(1,qn) on the qn columns of a suitable parity-check matrix Hα. This action does not describe exactly the orbits of Goppa codes, since in some cases the number of permutation non-equivalent Goppa codes is less than the number of orbits of FG. It seems interesting to study if there is a proper subgroup of Sqn containing FG, acting on the set Ω of classical irreducible maximal Goppa codes of fixed parameters, and giving on Ω exactly the orbits of permutation equivalent codes.
From another point of view, it could be helpful to consider the automorphism group for a Goppa code This approach was considered by Berger.
In 1946 Erdos established upper and lower bounds on the minimum number of distinct distances that can be determined by n points in the plane. It is believed that this upper bound is the best possible, and that the function in which it is expressed in fact bounds the minimum number of distinct distances above and below asymptotically. Although much has been done to improve the lower bound, we are still far from our goal of proving this. In my talk I will be giving a brief account of Erdos' original result and the subsequent history of what's become known as the Erdos distance problem, focusing in particular on the role graph theory plays. To this end, we will cover a result due to Szekely and, time permitting, briefly discuss the best currently known construction (due to Solymosi and Toth) and its limitations.
The Marriage Problem asks if there is a finite set of boys, each of whom knows several girls, under what conditions can we ensure that all the boys marry a girl they know. In 1935, Phillip Hall proved that a necessary and sufficient condition for a solution to this problem is that each set of k boys collectivelly know at least k girls.
In this talk we discuss this well-known result of graph theory, its connection with Konig's Duality Theorem and Menger's Theorem, and how this connection extends to the infinite version of these theorems.
Roughly speaking, a difference scheme (variety) is a scheme (variety) with a distinguished endomorphism. We will explain how to extend the methods of etale cohomology to this context and, time permitting, we will show the calculation of difference etale cohomology in some interesting cases.
Classical extremal graph theory, typified by the theorem of Turan, asks how many edges a large graph G can have if it does not contain a copy of some fixed graph H. We look at a variant of this problem in which the edges of the graphs are painted with two colours (some edges might get both colours). This variant is motivated by applications of the fundamental Regularity Lemma of Szemeredi, in particular to Ramsey-type games, and to the study of edit distance and property testing in computer science.
We shall describe an approach to answering this question and to determining the structures involved, hence settling some outstanding issues concerning edit distance.
The representation theories of symmetric groups and of general linear groups are linked through Schur-Weyl duality. In 1937, Brauer asked the following question: which algebra has to replace the group algebra of the symmetric group in this situation if one replaces the general linear group by its orthogonal or symplectic subgroup? As an answer he defined the Brauer algebra. We will discuss this, and also see how a theory of Young modules leads to another Schur-Weyl duality for Brauer algebras. This is joint work with Robert Hartmann, Anne Henke and Steffen Koenig.
In the first part of the talk we intend to survey the role that the behaviour of discrete subspaces (and related other subspaces, like free sequences) play in determining the cardinality properties of topological spaces, in particular of compacta (i.e. compact Hausdorff spaces).
In the second part we review two recent results of ours that fall into this category. The first one gives a partial answer to a problem of A. Dow and the second yields a significant strengthening of a classical result of Cech and Pospisil.
Theorem 1. Under the generalised continuum hypothesis every countably tight compactum X has a discrete subspace whose closure is of the same cardinality as X.
(A space is countably tight if whenever a point belongs to the closure of a set it already belongs to the closure of a countable subset.)
Theorem 2. If in a compactum X the character of every point is at least k then X cannot be covered by fewer than 2k many discrete subspaces.
(The character of a point is the smallest cardinality of a neighbourhood base of the point.)
We give new proofs of two results on the structure of the groups of points of certain group schemes over local rings (e.g., GLn(A), where A is a local ring). The first result is a commutator relation for certain normal subgroups, and holds for any affine group scheme over a local ring. Its proof is based on properties of the corresponding Hopf algebra.
The second result is an Iwahori decomposition which holds for split reductive group schemes over local rings, and its proof is based on a geometric result, in contrast to the more traditional group theoretic approaches.
We survey some of the known rigidity results which show that objects defined in the measurable category are of algebraic origin and discuss a joint work with Bader, Furman, and Weiss that classifies measurable isomorphisms and factors for ergodic actions on homogeneous varieties.
We discuss some basic problems in Diophantine geometry regarding the structure of the sets of solutions of polynomial equations, and explain to approach these problems using ideas originating in the theory of dynamical systems.
A locally compact abelian group is said to be self-dual if it is isomorphic to its Pontryagin dual (the group of its unitary characters). It is said to be symplectically self-dual if the isomorphism with its dual can be chosen in such a way as to give an alternating bicharacter on the group. For example, the additive group of any finite dimensional real vector space is self dual, but only the even dimensional ones are symplectically self-dual. After explaining the relationship of symplectically self-dual groups with Heisenberg groups (in the sense of Mumford, Norman and Nori), I will discuss some examples and outline how some homological techniques of Fuchs and Hofmann can be adapted to their study.
One can think of a groups as a set G together with a family of mappings ('verbal mappings')
When G is abelian, each verbal mapping is a homomorphism: all fibres have the same size and the image is a subgroup. In general, neither of these holds. There are interesting results about the fibres, but I will concentrate on the images, and more specifically on the issue of width. We say that w has width m in G if every product of w-values or their inverses is equal to such a product of length m.
The question of which words have finite width in which groups is particualarly interesting in the case of profinite groups, because a word w has finite width in a profinite group G if and only if the verbal subgroup (generated by the w-values) is closed; when this happes, we can use algebraic information to get topological conclusions. For example, this is the technical device used to solve an old problem posed by Serre, which asks if every subgroup of finite index in a finitely generated profinite group is open.
I will discuss some of the background ideas, some more or less recent results, and some open problems around this topic.
I will talk about classification in algebraic geometry, aiming at the 3-dimensional case of Mori theory while giving some of the motivation coming from complex curves. Gorenstein rings appear throughout the theory in many different roles. Here I want to explain their role as the homogeneous coordinate rings (that is, the rings of polynomial forms) on Fano 3-folds, a basic chunk of the classification that I will explain. This involves detailed analysis of Kustin and Miller's method of constructing large Gorenstein rings from small ones (as reformulated by Papadakis and Reid). I hope to explain some of my recent joint work with Reid on Gorenstein rings in codimension 4.