Past

A Hyperkähler manifold is a riemannian manifold carrying three complex structures which behave like quaternions such that the metric is Kähler with respect to each of them. This means in particular that the manifold is a symplectic manifold in many different ways. In analogy to the Marsden-Weinstein reduction on a symplectic manifold, there is also a quotient construction for group actions that preserve the Hyperkähler structure and admit a moment map. In fact most known (non-compact) examples of hyperkähler manifolds arise in this way from an appropriate group action on a quaternionic vector space.

In the first half of the talk I will give the definition of a hyperkähler manifold and explain the hyperkähler quotient construction. As an important application I will discuss the moduli space of solutions to the gauge-theoretic "Self-duality equations on a Riemann surface", the space of Higgs bundles, and explain how it can be viewed as a hyperkähler quotient in an infinite-dimensional setting.

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.

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We define the edge reconnecting model, a random multigraph evolving in time. At each time step we change one endpoint of a uniformly chosen edge: the new endpoint is chosen by linear preferential attachment. We consider a sequence of edge reconnecting models where the sequence of initial multigraphs is convergent in a sense which is a natural generalization of the Lovász-Szegedy notion of convergence of dense graph sequences. We investigate how the limit objects evolve under the edge reconnecting dynamics if we rescale time properly: we give the complete characterization of the time evolution of the limiting object from its initial state up to the stationary state using the theory of exchangeable arrays, the Pólya urn model, queuing and diffusion processes. The number of parallel edges and the degrees evolve on different timescales and because of this the model exhibits “aging”.

This talk considers the problem of sampling an independent set uniformly at random from a bipartite graph (equivalently, the problem of approximately counting independent sets in a bipartite graph). I will start by discussing some natural Markov chain approaches to this problem, and show why these lead to slow convergence. It turns out that the problem is interesting in terms of computational complexity – in fact, it turns out to be equivalent to a large number of other problems, for example, approximating the partition function of the “ferromagnetic Ising model’’ (a 2-state particle model from statistical physics) in the presence of external fields (which are essentially vertex weights). These problems are all complete with respect to approximation-preserving reductions for a logically-defined complexity class, which means that if they can be approximated efficiently, so can the entire class. In recent work, we show some connections between this class of problems and the problem of approximating the partition function of the ``ferromagnetic Potts model’’ which is a generalisation of the Ising model—our result holds for q>2 spins. (This corresponds to the approximation problem for the Tutte polynomial in the upper quadrant

above the hyperbola q=2.) That result was presented in detail at a recent talk given by Mark Jerrum at Oxford’s one-day meeting in combinatorics. So I will just give a brief description (telling you what the Potts model is and what the result is) and then conclude with some more recently discovered connections to counting graph homomorphisms and approximating the cycle index polynomial.

Graphics processing units (GPU) are well suited to decrease the

computational in-

tensity of stochastic simulation of chemical reaction systems. We

compare Gillespie’s

Direct Method and Gibson-Bruck’s Next Reaction Method on GPUs. The gain

of the

GPU implementation of these algorithms is approximately 120 times faster

than on a

CPU. Furthermore our implementation is integrated into the Systems

Biology Toolbox

for Matlab and acts as a direct replacement of its Matlab based

implementation.

Topology can be generalised in at least two directions: pointless

topology, leading ultimately to topos theory, or noncommutative

geometry. The former has the advantage that it also carries a logical

structure; the latter captures quantum settings, of which the logic is

not well understood generally. We discuss a construction making a

generalised space in the latter sense into a generalised space in the

former sense, i.e. making a noncommutative C*-algebra into a locale.

This construction is interesting from a logical point of view,

and leads to an adjunction for noncommutative C*-algebras that extends

Gelfand duality.

We investigate a dynamic model of two dimensional crystal growth

described by a forward-backward parabolic equation. The ill-posed

region of the equation describes the motion of corners on the surface.

We analyze a fourth order regularized version of this equation and

show that the dynamical behavior of the regularized corner can be

described by a traveling wave solution. The speed of the wave is found

by rigorous asymptotic analysis. The interaction between multiple

corners will also be presented together with numerical simulations.

This is joint work in progress with Fang Wan.

Hamadène and Jeanblanc provided a BSDE representation for the resolution of bi-dimensional continuous time optimal switching problems. For example, an energy producer faces the possibility to switch on or off a power plant depending on the current price of electricity and corresponding comodity. A BSDE representation via multidimensional reflected BSDEs for this type of problems in dimension larger than 2 has been derived by Hu and Tang as well as Hamadène and Zhang [2]. Keeping the same example in mind, one can imagine that the energy producer can use different electricity modes of production, and switch between them depending on the commodity prices. We propose here an alternative BSDE representation via the addition of constraints and artificial jumps. This allows in particular to reinterpret the solution of multidimensional reflected BSDEs in terms of one-dimensional constrained BSDEs with jumps. We provide and study numerical schemes for the approximation of these two type of BSDEs

NO WORKSHOP - 09:45 coffee in DH Common Room for those attending the OCIAM Meeting

Elimination of wild ramification is used in the structure theory of valued function fields, with applications in areas such as local uniformization (i.e., local resolution of singularities) and the model theory of valued fields. I will give a survey on the role that Artin-Schreier extensions play in the elimination of wild ramification, and corresponding main theorems on the structure of valued function fields. I will show what these results tell us about local uniformization. I have shown that local uniformization is always possible after a separable extension of the function field of the algebraic variety (separable "alteration"). This was extended to the arithmetic case in joint work with Hagen Knaf. Recently, Michael Temkin has proved local uniformization by purely inseparable alteration.

Further, I will describe a classification of Artin-Schreier extensions with non-trivial defect. It can be used to improve one of the above mentioned main theorems ("Henselian Rationality"). This could be a key for a purely valuation theoretical proof of Temkin's result. On the other hand, the classification shows that separable alteration and purely inseparable alteration are just two ways to eliminate the critical defects. So the existence of these two seamingly "orthogonal" local uniformization results does not necessarily indicate that local uniformization without alteration is possible.

Interior-point methods for linear and convex quadratic programming

require the solution of a sequence of symmetric indefinite linear

systems which are used to derive search directions. Safeguards are

typically required in order to handle free variables or rank-deficient

Jacobians. We propose a consistent framework and accompanying

theoretical justification for regularizing these linear systems. Our

approach is akin to the proximal method of multipliers and can be

interpreted as a simultaneous proximal-point regularization of the

primal and dual problems. The regularization is termed "exact" to

emphasize that, although the problems are regularized, the algorithm

recovers a solution of the original problem. Numerical results will be

presented. If time permits we will illustrate current research on a

matrix-free implementation.

This is joint work with Michael Friedlander, University of British Columbia, Canada

This talk I present a study of BSDEs with non-linear terms of quadratic growth by using Girsanov's theorem. In particular we are able to establish a non-linear version of the Cameron-Martin formula, which can be for example used to obtain gradient estimates for some non-linear parabolic equations.

We consider a 3-dimensional elastic continuum whose material points

can experience no displacements, only rotations. This framework is a

special case of the Cosserat theory of elasticity. Rotations of

material points of the continuum are described mathematically by

attaching to each geometric point an orthonormal basis which gives a

field of orthonormal bases called the coframe. As the dynamical

variables (unknowns) of our theory we choose the coframe and a

density.

In the first part of the talk we write down the general dynamic

variational functional of our problem. In doing this we follow the

logic of classical linear elasticity with displacements replaced by

rotations and strain replaced by torsion. The corresponding

Euler-Lagrange equations turn out to be nonlinear, with the source

of this nonlinearity being purely geometric: unlike displacements,

rotations in 3D do not commute.

In the second part of the talk we present a class of explicit

solutions of our Euler-Lagrange equations. We call these solutions

plane waves. We identify two types of plane waves and calculate

their velocities.

In the third part of the talk we consider a particular case of our

theory when only one of the three rotational elastic moduli, that

corresponding to axial torsion, is nonzero. We examine this case in

detail and seek solutions which oscillate harmonically in time but

depend on the space coordinates in an arbitrary manner (this is a

far more general setting than with plane waves). We show [1] that

our second order nonlinear Euler-Lagrange equations are equivalent

to a pair of linear first order massless Dirac equations. The

crucial element of the proof is the observation that our Lagrangian

admits a factorisation.

[1] Olga Chervova and Dmitri Vassiliev, "The stationary Weyl

equation and Cosserat elasticity", preprint http://arxiv.org/abs/1001.4726

The aim of this talk is to get a feel for the Ricci flow. The Ricci flow was introduced by Hamilton in 1982 and was later used by Perelman to prove the Poincaré conjecture. We will introduce the notions of Ricci flow and Ricci soliton, giving simple examples in low dimension. We will also discuss briefly other types of geometric flows one can consider.