- Relativity Seminar
In this talk I will present three families of differential equations (SDEs, BSDEs and PDEs) and their links to each other. The novel fact is that some of the coefficients are generalised functions living in a fractional Sobolev space of negative order. I will discuss the appropriate notion of solution for each type of equation and show existence and uniqueness results. To do so, I will use tools from analysis like semigroup theory, pointwise products, theory of function spaces, as well as classical tools from probability and stochastic analysis. The link between these equations will play a fundamental role, in particular the results on the PDE are used to give a meaning and solve both the forward and the backward stochastic differential equations.
- Stochastic Analysis Seminar
Stochastic differential equations have Taylor expansions in terms of iterated Wiener integrals. The convergence of such expansion depends on the limiting behavior of the order-N iterated integrals as N tends to infinity. Recently, there has been increased interests in processes stopped at a random time. A breakthrough in the study of the iterated integrals of Brownian motion up to the exit time of a domain was included in the work of Lyons-Ni (2012). The paper leaves open an interesting question: what is the sharp rate of decay for the expected iterated integrals up to the exit time. We will review the state of the art in this problem and report some recent progress. Joint work with Ni Hao (UCL).
- Stochastic Analysis Seminar
1.00pm: Introductory Remarks by Camilla Serck-Hanssen, the Vice President of the Norwegian Academy of Science and Letters
1.10pm - 2.10pm: Andrew Wiles
2.10pm - 2.30pm: Break
2.30pm - 3.30pm: Irene Fonseca
3.30pm - 4.00pm: Tea and Coffee
4.00pm - 5.00pm: John Rognes
Andrew Wiles: Points on elliptic curves, problems and progress
This will be a survey of the problems concerned with counting points on elliptic curves.
Irene Fonseca: Mathematical Analysis of Novel Advanced Materials
Quantum dots are man-made nanocrystals of semiconducting materials. Their formation and assembly patterns play a central role in nanotechnology, and in particular in the optoelectronic properties of semiconductors. Changing the dots' size and shape gives rise to many applications that permeate our daily lives, such as the new Samsung QLED TV monitor that uses quantum dots to turn "light into perfect color"!
Quantum dots are obtained via the deposition of a crystalline overlayer (epitaxial film) on a crystalline substrate. When the thickness of the film reaches a critical value, the profile of the film becomes corrugated and islands (quantum dots) form. As the creation of quantum dots evolves with time, materials defects appear. Their modeling is of great interest in materials science since material properties, including rigidity and conductivity, can be strongly influenced by the presence of defects such as dislocations.
In this talk we will use methods from the calculus of variations and partial differential equations to model and mathematically analyze the onset of quantum dots, the regularity and evolution of their shapes, and the nucleation and motion of dislocations.
John Rognes: Symmetries of Manifolds
To describe the possible rotations of a ball of ice, three real numbers suffice. If the ice melts, infinitely many numbers are needed to describe the possible motions of the resulting ball of water. We discuss the shape of the resulting spaces of continuous, piecewise-linear or differentiable symmetries of spheres, balls and higher-dimensional manifolds. In the high-dimensional cases the answer turns out to involve surgery theory and algebraic K-theory.
- Special Lecture
- String Theory Seminar
Lein Applied Diagnostics has a novel optical measurement technique that is used to measure various parameters in the body for medical applications.
Two particular areas of interest are non-invasive glucose measurement for diabetes care and the diagnosis of diabetes. Both measurements are based on the eye and involve collecting complex data sets and modelling their links to the desired parameter.
If we take non-invasive glucose measurement as an example, we have two data sets – that from the eye and the gold standard blood glucose reading. The goal is to take the eye data and create a model that enables the calculation of the glucose level from just that eye data (and a calibration parameter for the individual). The eye data consists of measurements of apparent corneal thickness, anterior chamber depth, optical axis orientation; all things that are altered by the change in refractive index caused by a change in glucose level. So, they all correlate with changes in glucose as required but there are also noise factors as these parameters also change with alignment to the meter etc. The goal is to get to a model that gives us the information we need but also uses the additional parameter data to discount the noise features and thereby improve the accuracy.
- Industrial and Interdisciplinary Workshops
In this talk we describe an approach to approximate the truncated singular value decomposition of a large matrix by first decomposing the matrix into a sum of Kronecker products. Our approach can be used to more efficiently approximate a large number of singular values and vectors than other well known schemes, such as iterative algorithms based on the Golub-Kahan bidiagonalization or randomized matrix algorithms. We provide theoretical results and numerical experiments to demonstrate accuracy of our approximation, and show how the approximation can be used to solve large scale ill-posed inverse problems, either as an approximate filtering method, or as a preconditioner to accelerate iterative algorithms.
- Computational Mathematics and Applications Seminar
In our Oxford Mathematics Christmas Lecture Alex Bellos challenges you with some festive brainteasers as he tells the story of mathematical puzzles from the middle ages to modern day. Alex is the Guardian’s puzzle blogger as well as the author of several works of popular maths, including Puzzle Ninja, Can You Solve My Problems? and Alex’s Adventures in Numberland.
Please email email@example.com to register.
- Public Lecture
Using synchronized high-speed video camera, hydrophone and microphone we investigated flow patterns, the impact and secondary sound pulses emitted by oscillating bubbles. On the submerging drop found short capillary waves produced by small secondary impact droplets. Picturesque filament and grid structures produced by colour drop of mixing fluid registered on the surface of the cavity and crown. Physical model includes discussion of the potential surface energy effects.
- Industrial and Applied Mathematics Seminar