Past

Jointly with Number Theory

Consider a family of abelian varieties whose base is an algebraic variety. The union of all torsion groups over all fibers of the family will be called the set of torsion points of the family. If the base variety is a point then the family is just an abelian variety.

In this case the Manin-Mumford Conjecture, a theorem of Raynaud, implies that a subvariety of the abelian variety contains a Zariski dense set of torsion points if and only if it is itself essentially an abelian subvariety. This talk is on possible extensions to certain families where the base is a curve. Conjectures of André and Pink suggest considering "special points": these are torsion points whose corresponding fibers satisfy an additional arithmetic property. One possible property is for the fiber to have complex multiplication; another is for the fiber to be isogenous to an abelian variety fixed in advance.

We discuss some new results on the distribution of such "special points"

on the subvarieties of certain families of abelian varieties. One important aspect of the proof is the interplay of two height functions.

I will give a brief introduction to the theory of heights in the talk.

Meshless (or meshfree) methods are a relatively new numerical approach for the solution of ordinary- and partial differential equations. They offer the geometrical flexibility of finite elements but without requiring connectivity from the discretization support (ie a mesh). Meshless methods based on the collocation of radial basis functions (RBF methods) are particularly easy to code, and have a number of theoretical advantages as well as practical drawbacks. In this talk, an adaptive RBF scheme is presented for a novel application, namely the solution of (a rather broad class of) delayed- and neutral differential equations.

Using a large panel data set of Swedish pension savers (75,000 investors, daily portfolios 2000-2008) we show that active investors outperform inactive investors and that there is a causal effect of fund switches on performance. The higher performance is earned not by market timing, but by dynamic fund picking (within the same asset class). While activity is positive for the individual investor, there are indications that it generates costs for other investors.

In this talk we discuss the solution of the elastodynamic

equations in a bounded domain with hereditary-type linear

viscoelastic constitutive relation. Existence, uniqueness, and

regularity of solutions to this problem is demonstrated

for those viscoelastic relaxation tensors satisfying the condition

of being completely monotone. We then consider the non-self-adjoint

and non-linear eigenvalue problem associated with the

frequency-domain form of the elastodynamic equations, and show how

the time-domain solution of the equations can be expressed in

terms of an eigenfunction expansion.

A moduli problem in algebraic geometry is essentially a classification problem, I will introduce this notion and define what it means for a scheme to be a fine (or coarse) moduli space. Then as an example I will discuss the classification of coherent sheaves on a complex projective scheme up to isomorphism using a method due to Alvarez-Consul and King. The key idea is to 'embed' the moduli problem of sheaves into the moduli problem of quiver representations in the category of vector spaces and then use King's moduli spaces for quiver representations. Finally if time permits I will discuss recent work of Alvarez-Consul on moduli of quiver sheaves; that is, representations of quivers in the category of coherent sheaves.

What do historians of mathematics do? What sort of questions do they ask? What kinds of sources do they use? This series of four informal lectures will demonstrate some of the research on history of mathematics currently being done in Oxford. The subjects range from the late Renaissance mathematician Thomas Harriot (who studied at Oriel in 1577) to the varied and rapidly developing mathematics of the seventeenth century (as seen through the eyes of Savilian Professor John Wallis, and others) to the emergence of a new kind of algebra in Paris around 1830 in the work of the twenty-year old Évariste Galois.

Each lecture will last about 40 minutes, leaving time for questions and discussion. No previous knowledge is required: the lectures are open to anyone from the department or elsewhere, from undergraduates upwards.

I'll start with the definition of the first Grigorchuk group as an automorphism group on a binary tree. After that I give a short overview about what growth means, and what kinds of growth we know. On this occasion I will mention a few groups that have each kind of growth and also outline what the 'Gap Problem' was. Having explained this I will prove - or depending on the time sketch - why this Grigorchuk group has intermediate growth. Depending on the time I will maybe also mention one or two open problems concerning growth.

I will introduce Symmetric spaces via a result of Kleiner & Leeb, comparing the axioms in their definition of a Euclidean building with properties of symmetric spaces of noncompact type.

We show how to compute the symplectic homology of a 4-dimensional Weinstein manifold from a diagram of the Legendrian link which is the attaching locus of its 2-handles. The computation uses a combination of a generalization of Chekanov's description of the Legendrian homology of links in standard contact 3-space, where the ambient contact manifold is replaced by a connected sum of $S^1\times S^2$'s, and recent results on the behaviour of holomorphic curve invariants under Legendrian surgery.

The QCD axion is the leading solution to the strong-CP problem, a

dark matter candidate, and a possible result of string theory

compactifications. However, for axions produced before inflation, high

symmetry-breaking scales (such as those favored in string-theoretic axion

models) are ruled out by cosmological constraints unless both the axion

misalignment angle and the inflationary Hubble scale are extremely

fine-tuned. I will discuss how attempting to accommodate a high-scale axion

in inflationary cosmology leads to a fine-tuning problem that is worse than

the strong-CP problem the axion was originally invented to solve, and how

this problem is exacerbated when additional axion-like fields from string

theory are taken into account. This problem remains unresolved by anthropic

selection arguments commonly applied to the high-scale axion scenario.

I will recount progress regarding the robustness of solitary waves in
nonintegrable Hamiltonian systems such as FPU lattices, and discuss 
a proof (with Shu-Ming Sun) of spectral stability of small
solitary waves for the 2D Euler equations for water of finite depth
without surface tension.

Bourdon's building is a negatively curved 2-complex built out of hyperbolic right-angled polygons. Its automorphism group is large (uncountable) and remarkably rich. We study, and mostly answer, the question of when there is a discrete subgroup of the automorphism group such that the quotient is a closed surface of genus g. This involves some fun elementary combinatorics, but quickly leads to open questions in group theory and number theory. This is joint work with David Futer.

comonotonicity joint work with Carlier and Galichon Abstact This paper studies efficient risk-sharing rules for the concave dominance order. For a univariate risk, it follows from a \emph{comonotone dominance principle}, due to Landsberger and

Meilijson that efficiency is

characterized by a comonotonicity condition. The goal of the paper is to generalize the comonotone dominance principle as well as the equivalence between efficiency and comonotonicity to the multi-dimensional case. The multivariate case is more involved (in particular because there is no immediate extension of the notion of comonotonicity) and it is addressed by using techniques from convex duality and optimal transportation.

We consider valued fields with a distinguished isometry or contractive derivation, as valued modules over the Ore ring of difference operators. This amounts to study linear difference/differential

equations with respect to the distinguished isometry/derivation.

Under certain assumptions on the residue field, but in all characteristics, we obtain quantifier elimination in natural languages, and the absence of the independence property.

We will consider other operators of interest.