Pure Maths Seminar Archive - Abstracts 2012
"Primitive and highly arc transitive digraphs of finite out-valency"
David Evans (UEA)
January 30th 2012
We give a survey of some recent work on directed graphs of finite out-valency and infinite in-valency which have a 'large' automorphism group. Suitable notions of largeness here are primitivity, high arc transitivity or descendant homogeneity. The questions of interest are the construction and classification of such digraphs, and the structure of their automorphism groups. Much of what I will talk about is joint work by various subsets of Daniela Amato, John Truss and myself.
"Mod p Langlands correspondences"
Fred Diamond (King's College London)
February 6th 2012
The Langlands program envisions a conjectural framework of compatible correspondences between objects from number theory (such as elliptic curves and Galois representations) and objects from representation theory (such as modular forms and automorphic representations). Recent advances (due to Colmez, Emerton, Kisin and others) have come via proving mod p and p-adic versions of these correspondences. I'll discuss how this works for GL(2) and what some of the obstacles are in trying to generalize the approach to other groups.
"Representations by quadratic forms and Siegel modular forms"
Lynne Walling (Bristol)
February 13th 2012
Quadratic forms impose "geometry" on vector spaces, capturing notions of length and orthogonality. In number theory, we are typically interested in discrete objects; thus we might consider a lattice L equipped with a quadratic form Q. A classical question is, given an integer t, how many vectors in L have length t? More generally, given another quadratic form T, on how many sublattices of L does Q restrict to T? Siegel introduced generalised theta series to encode the answer to this question (for all T), giving us our first examples of Siegel modular forms. I will give an introduction to this area, then discuss how we can use Siegel Eisenstein series and Hecke operators to find closed-form formulas that answer the latter question posed above.
"Decomposable Specht modules"
Matt Fayers (QMUL)
February 20th 2012
The Specht modules are important modules defined for the symmetric group in any characteristic, and a great deal of effort is devoted to finding their structure. It is known that if the underlying characteristic is not 2, then all Specht modules are indecomposable. In characteristic 2 there are decomposable Specht modules, but examples are hard to find. I will report on recent joint work with Craig Dodge, in which we have found new examples of decomposable Specht modules. I will keep this talk elementary and it will be mostly background, with some hands-on calculation.
"Modal logic of forcing"
Benedikt Löwe (Amsterdam)
February 27th 2012
Forcing extensions of a model of set theory can be seen as its 'possible worlds'; as a consequence, a modal interpretation of forcing is natural. Interpreting the modal possibility operator as "there is a forcing extension in which φ is true", we can define the Modal Logic of Forcing. Can this modal logic be axiomatized?
Jointly with Joel Hamkins, I showed that it can, and in fact, it is equal to the well-known modal logic S4.2. In this talk, I will sketch a proof of our main result and then talk about generalizations that give rise to very interesting structural questions about the technique of forcing.
"Common zeros for subspaces of hermitian forms over finite fields"
Rod Gow (University College Dublin)
March 5th 2012
Let M be a non-empty set of hermitian forms defined over a field L with an involutory automorphism, whose fixed point field is K. A non-trivial common zero for the forms in M is a non-zero vector v such that f(v,v)=0 for all forms f in M. For the purposes of investigating common zeros, we may as well assume that M is a subspace over K.
When L is a finite field, we discuss a formula which calculates the number of common zeros in terms of the ranks of the elements in M. This formula implies, in particular, that when all the ranks are even, there are non-trivial common zeros. This conclusion does not hold for arbitrary fields. We then investigate whether there are canonical forms for a subspace of hermitian forms over a finite field, all of whose non-zero elements have rank 2, and whose dimension is as large as possible.
"Zilber's perfect exponential fields and involutions"
Vincenzo Mantova (Oxford)
March 12th 2012
In an attempt to describe the model theory of complex exponentiation, which interprets Peano's Arithmetic and therefore lies on the far end of the stability hierarchy, Zilber showed that, provided with a certain strengthening of first-order language, one can axiomatize an exponential field that is "perfect", in the sense that it has exactly one model in each uncountable cardinal, while still interpreting PA. It is conjectured that complex exponentiation is actually the model of power of the continuum, but the problem is at the moment out of reach as it includes Schanuel's Conjecture.
While the conjecture remains open, we look for more accessible similarities between the perfect exponential fields and complex exponentiation. I will survey some of the recent results on this subject, and show some details about my work on finding involutions, i.e., automorphisms of order two analogous to complex conjugation.
"Triangulated categories generated by a spherical object: elementary properties, torsion pairs"
Peter Jørgensen (Newcastle)
March 19th 2012
A d-spherical object has self-extensions only in degrees 0 and d, and each of these Ext groups is equal to the ground field. The triangulated category generated by such an object has a rich homological and combinatorial structure which has been investigated intensively in the past few years. The seminar explains some of the elementary properties of these categories and shows how they can be used to classify certain types of torsion pairs and related objects: t-structures, co-t-structures, and d-cluster tilting subcategories.
"Modular representations of GL(n,F) distinguished by GL(n-1,F)"
Vincent Sécherre (Versailles)
March 26th 2012
Let p be a prime number and F be a p-adic field. Let n be an integer >1. The complex smooth irreducible representations of GL(n,F) that are distinguished by GL(n-1,F), i.e. that have a nonzero linear form invariant by GL(n-1,F) have been explicitely classified by Venketasubramanian in terms of Galois parameters. This classification appears to be related to the Howe correspondence from GL(2,F) to GL(n,F). In this talk we investigate the modular case, that is, the case when the field of complex numbers is replaced by an algebraically closed field of positive characteristic. This is a work in progress with Venketasubramanian.
"Primes in rapidly increasing sequences"
Stephan Baier (UEA)
April 23rd 2012
It is generally a difficult task to detect primes in rapidly increasing sequences. For example, the question whether n²+1 (n being an integer) presents infinitely many primes is still open. Conjecturally, the answer should be "yes". There is even a more precise conjecture on the density of primes presented in this form due to Hardy and Littlewood. However, current methods of analytic number theory fail to produce a proof of this. A lot of my research has been inspired by this open "n²+1 question". In my talk, I will discuss several approximations, reformulations and generalisations of this problem that I have looked into and attacked by different methods, ranging from small fractional parts of √p to the Lang-Trotter conjecture on elliptic curves. Special focus will be put on Piatetski-Shapiro primes, a topic which my current research is centred around.
"2-strand twisting and knot homologies"
Andrew Lobb (Durham)
April 30th 2012
We give an introduction to some quantum knot homologies and show how twisting up a pair of adjacent strands in a knot, combined with some straightforward homological algebra, allows us to deduce some interesting consequences.
"Reflection group presentations arising from cluster algebras"
Robert Marsh (Leeds)
May 14th 2012
Finite reflection groups are often presented as Coxeter groups. We give a presentation of finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type for which the Coxeter presentation is a special case. We interpret the presentation in terms of companion bases in the associated root system.
"L-packets for classical groups"
Shaun Stevens (UEA)
May 21st 2012
The local Langlands conjectures predict a canonical bijection between certain (L-)packets of representations of a reductive group G over a p-adic field F, and the representations of the Weil group of F in the (Langlands) dual group of G; the case of GL(1) is given by Local Class Field Theory. The conjectures are now proven for general linear groups and, more recently, by relating it to this case, for symplectic and special orthogonal groups (Arthur). On the other hand, the representations of these p-adic groups G have been constructed in an explicit way so one can ask how the data involved in these constructions transfer under the Langlands correspondence. For general linear groups, this has been the subject of an ongoing programme of Bushnell and Henniart. As well as trying to explain what some of this means, I hope to talk about work in progress with Blondel and Henniart, where we have been considering what can then be said for symplectic groups.
"Minimal projective modules and Steinberg-like characters for Chevalley groups"
Alexandre Zalesski (Università Milano-Bicocca)
May 23rd 2012
Let G be a finite group and S a Sylow p-subgroup of G. For an algebraically closed field of characteristic p let FG denote the group algebra of G over F. Then FG can be viewed as an FG-module with respect to the natural multiplication in G, and it is called the regular FG-module.
Projective indecomposable modules are indecomposable direct summands of the regular module. They were introduced in a classical paper of Brauer and Nesbitt in 1940, but very little is still known about their dimensions, apart from the easy fact that these are multiples of the order of S.
I shall discuss the minimal case, specifically, when the dimension is exactly the order of S, and only for Chevalley groups with p being the defining characteristic.
Projective modules for finite groups are in bijection with ordinary characters that vanish at all elements of G of order divisible by p, with certain additional condition. Dropping this condition, I shall discuss the ordinary characters vanishing at these elements, which are minimal in the sense that the degree is equal to the order of S.
"Galois representations and automorphic representations"
Toby Gee (Imperial College London)
May 28th 2012
I will discuss the conjectural Langlands correspondence between Galois representations and automorphic representations, and some recent progress towards it. I will not assume any prior knowledge of automorphic forms, Galois representations or the Langlands program.
"Products of finite nilpotent groups"
John Cossey (Australia National University)
June 11th 2012
Suppose A and B are subgroups of a group G. We say that G is the product of A and B if G = AB = { ab | a ∈ A, b ∈ B}. A natural question to ask is whether properties of G can be deduced from properties of A and B. There is an extensive literature on this question. Many properties have been considered- see for example the book of Amberg, Franciosi and de Giovanni and that of Ballester-Bolinches, Esteban-Romero and Asaad.
Many results concentrate on the case of A and B nilpotent. Most results are aimed at restricting the structure of non-nilpotent products G; for example, under appropriate restrictions, G will be supersoluble. However very little is known about the structure when G is itself nilpotent.
If G is nilpotent, there are many invariants we could consider: derived length, class, coclass, breadth and rank as examples. Very little is known about any of these. I will describe what is known.
"Models and their applications in representation theory"
Chufeng Nien (Tainan)
October 8th 2012
In this talk, we will introduce some models in representation theory and their applications in obtaining functional equations and defining twisted gamma factors. If time permits, we can also talk about classification of irreducible cuspidal representations of general linear group over finite fields through the set of twisted gamma factors.
"Categoricity of the j-function"
Adam Harris (Oxford)
October 15th 2012
I will introduce the model theoretic notion of categoricity, and then outline a proof for categoricity of a natural structure involving the modular j-function. It turns out that categoricity in this situation is related to deep geometric and number theoretic results, in particular the Mumford-Tate conjecture, and the theory of complex multiplication. I will give brief introductions to the objects under discussion (e.g. elliptic curves and the j-function) as I go along.
"Rational points of bounded height and the Weil restriction"
Daniel Loughran (Bristol)
October 22nd 2012
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.
"On isomorphisms of Banach spaces of continuous functions"
Grzegorz Plebanek (Wroccław)
November 5th 2012
We consider Banach spaces of the form C(K) of real-valued continuous functions on a compact space K. We address the problem what are necessary topological relations between spaces K and L for which C(K) and C(L) are isomorphic.
"Quantum homogeneous spaces for quantized enveloping algebras"
Stefan Kolb (Newcastle)
November 26th 2012
The coordinate ring C[G] of a complex affine algebraic group G is a commutative Hopf algebra. A coideal subalgebra B of C[G] describes an affine homogeneous space forG if and only if C[G] is faithfully flat as a B-module. In the theory of quantum groups one studies various families of non-commutative Hopf algebras. By analogy to the commutative setting, it is natural to define a quantum homogeneous space for a Hopf algebra H to be a coideal subalgebra of H over which H is faithfully flat. An important example of a non-commutative Hopf algebra is the quantized enveloping algebra Uq(g) of a simple, complex Lie algebra g. In this talk, I will report on recent progress in the classification of quantum homogeneous spaces for Uq(g). Classification results will be given in terms of Weyl group combinatorics. The talk is based on joint work with I. Heckenberger.
"How frequently does the Hasse principle fail?"
Tim Browning (Bristol)
December 3rd 2012
Counter-examples to the Hasse principle are known for many families of geometrically rational varieties. We discuss how often such failures arise for Chatelet surfaces and certain higher-dimensional hypersurfaces. This is joint work with Regis de la Breteche.