Past

In this talk, we introduce the (general) homotopy groups of spheres as link invariants for Brunnian-type links through the investigations on the intersection subgroup of the normal closures of the meridians of strongly nonsplittable links. The homotopy groups measure the difference between the intersection subgroup and symmetric commutator subgroup of the normal closures of the meridians and give the invariants of the links obtained in this way. Moreover all homotopy groups of any dimensional spheres can be obtained from the geometric Massey products on certain links.

Since the work of Feigenbaum and Coullet-Tresser on universality in the period doubling bifurcation, it is been understood that crucial features of unimodal (one-dimensional) dynamics depend on the behavior of a renormalization (and infinite dimensional) dynamical system. While the initial analysis of renormalization was mostly focused on the proof of existence of hyperbolic fixed points, Sullivan was the first to address more global aspects, starting a program to prove that the renormalization operator has a uniformly hyperbolic (hence chaotic) attractor. Key to this program is the proof of exponential convergence of renormalization along suitable ``deformation classes'' of the complexified dynamical system. Subsequent works of McMullen and Lyubich have addressed many important cases, mostly by showing that some fine geometric characteristics of the complex dynamics imply exponential convergence.

We will describe recent work (joint with Lyubich) which moves the focus to the abstract analysis of holomorphic iteration in deformation spaces. It shows that exponential convergence does follow from rougher aspects of the complex dynamics (corresponding to precompactness features of the renormalization dynamics), which enables us to conclude exponential convergence in all cases.

In this talk, we report some of our recent work on hybrid switching diffusions in which continuous dynamics and discrete events coexist. Motivational examples in singular perturbed Markovian systems, manufacturing, and financial engineering will be mentioned. After presenting criteria for recurrence and ergodicity, we consider numerical methods for controlled switching diffusions and related game problems. Rates of convergence of Markov chain approximation methods will also be studied.

Andrew Stewart -

The role of the complete Coriolis force in ocean currents that cross the equator

Large scale motions in the atmosphere and ocean are dominated by the Coriolis force due to the Earth's rotation. This tends to prevent fluid crossing the equator from one hemisphere to the other. We investigate the flow of a deep ocean current, the Antarctic Bottom Water, across the equator using a shallow water model that includes the Earth's complete Coriolis force. By contrast, most theoretical models of the atmosphere and ocean use the so-called traditional approximation that neglects the component of the Coriolis force associated with the locally horizontal component of the Earth's rotation vector. Using a combination of analytical and numerical techniques, we show that the cross-equatorial transport of the Antarctic Bottom Water may be substantially influenced by the interaction of the complete Coriolis force with bottom topography.

In the first part of my presentation, I plan to review several applications modelled by delay differential equations (DDEs) starting from familiar examples such as traffic flow problems to physiology and industrial problems. Although delay differential equations have the reputation to be difficult mathematical problems, there is a renewed interest for both old and new problems modelled by DDEs. In the second part of my talk, I’ll emphasize the need of developing asymptotic tools for DDEs in order to guide our numerical simulations and help our physical understanding. I illustrate these ideas by considering the response of optical optoelectronic oscillators that have been studied both experimentally and numerically.

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.