L3

An important observation in compressed sensing is the exact recovery of an l0 minimiser to an underdetermined linear system via the l1 minimiser, given the knowledge that a sparse solution vector exists. Here, we develop a continuous analogue of this observation and show that the best L1 and L0 polynomial approximants of a corrupted function (continuous analogue of sparse vectors) are equivalent. We use this to construct best L1 polynomial approximants of corrupted functions via linear programming. We also present a numerical algorithm for computing best L1 polynomial approximants to general continuous functions, and observe that compared with best L-infinity and L2 polynomial approximants, the best L1 approximants tend to have error functions that are more localized.

Joint work with Alex Townsend (MIT).

A holy grail of nano-technology is to create truly complex, multi-component structures by self assembly.
 

Most self-assembly has focused on the creation of `structural complexity'. In my talk, I will discuss `Addressable Complexity': the creation of structures that contain hundreds or thousands of
distinct building blocks that all have to find their place in a 3D structure.

Mining massive amounts of sequentially ordered data and inferring structural properties is nowadays a standard task (in finance, etc). I will present some results that combine and extend ideas from rough paths and machine learning that allow to give a general non-parametric approach with strong theoretical guarantees. Joint works with F. Kiraly and T. Lyons.

In this talk, I am reflecting on the last 8 extremely enjoyable years I spent in the department (DPhil, OCIAM, 2008-2012, post-doc, WCMB, 2012-2016). My story is a little unusual: coming from an Engineering undergraduate background, spending 8 years in the Maths department, and now moving to a faculty position at the Medical School. However, I think it highlights well the enormous breadth and applicability of mathematics beyond traditional disciplinary boundaries. I will discuss different projects during my time in Oxford, focusing on time-series, signal processing, and statistical machine learning methods, with diverse applications in real-world problems.