University of Oxford

It is well-known that a matrix $A$ is Hurwitz stable if and only if there exists a positive definite solution to the Lyapunov matrix equation $A X + X A^* = B$, where $B$ is Hermitian negative definite. We present a verified numerical algorithm to rigorously prove the stability of a given matrix $A$ in the presence of rounding errors.  The computational cost of the algorithm is cubic and it is fast since we can cast almost all operations in level 3 BLAS for which interval arithmetic can be implemented very efficiently.  This is a joint work with Andreas Frommer and the results are already published in ETNA in 2013.

“Narrative and Proof”, is an interdisciplinary discussion where one of the UK's leading scientists, Marcus du Sautoy, will argue that mathematical proofs are not just number-based, but also rely on narrative. He will be joined by author Ben Okri, mathematician Roger Penrose, and literature expert Laura Marcus, to consider how narrative shapes the sciences as well as the arts.

The discussion will be chaired by Elleke Boehmer, Professor of World Literature in English, University of Oxford, and will be followed by audience questions and a drinks reception.

The event will take place from 5 to 6:30 pm on Tuesday 20 January 2015 at the Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford. The event is free and open to all, but registration is recommended. 

Please click here to register.

This event is co-hosted by the Mathematical Institute and The Oxford Research Centre in the Humanities (TORCH), and celebrates the launch of TORCH’s Annual Headline Series 2014-15, Humanities and Science.
 

A key question to develop our understanding of turbulence in shear flows is: what is the smallest perturbation to the laminar flow that causes a transition to turbulence, and how does this change with the Reynolds number, R?  Finding this so-called ``minimal seed'' is as yet unachievable in direct numerical simulations of the Navier-Stokes equations. We search for the minimal seed in a low-dimensional model analogue to the full Navier-Stokes in plane sinusoidal flow, developed by Waleffe (1997). A previous such calculation found the minimal seed as the least distance (energy norm) from the origin (laminar flow) to the basin of attraction of another fixed point (turbulent attractor).  However, using a non-linear optimization technique, we found an internal boundary of the basin of attraction of the origin that separates flows which directly relaminarize from flows which undergo transient turbulence. It is this boundary which contains the minimal seed, and we find it to be smaller than the previously calculated minimal seed. We present results over a range of Reynolds numbers up to 2000 and find an R^{-1} scaling law fits reasonably well. We propose a new scaling law which asymptotes to R^{-1} for large R but, using some additional information, matches the minimal seed scaling better at low R.

Both sides of the geometric Langlands correspondence have natural Hecke
symmetries. I will explain an identification between the Hecke
symmetries on both sides via the geometric Satake equivalence. On the
abelian level it relates the topology of a variety associated to a group
and the representation category of its Langlands dual group.
 

Given a subset E of R^n with empty interior, what is the maximum regularity exponent s for which there exist non-zero distributions in the Bessel potential Sobolev space H^s_p(R^n) that are supported entirely inside E? This question has arisen many times in my recent investigations into boundary integral equation formulations of linear wave scattering by fractal screens, and it is closely related to other fundamental questions concerning Sobolev spaces defined on ``rough'' (i.e. non-Lipschitz) domains. Roughly speaking, one expects that the ``fatter'' the set, the higher the maximum regularity that can be supported. For sets of zero Lebesgue measure one can show, using results on certain set capacities from classical potential theory, that the maximum regularity (if it exists) is negative, and is (almost) characterised by the fractal (Hausdorff) dimension of E. For sets with positive measure the maximum regularity (if it exists) is non-negative,but appears more difficult to characterise in terms of geometrical properties of E.  I will present some partial results in this direction, which have recently been obtained by studying the asymptotic behaviour of the Fourier transform of the characteristic functions of certain fat Cantor sets.

It is well known that piecewise smooth signals are approximately sparse in a wavelet basis. However, other sparse representations are possible, such as the discrete gradient basis. It turns out that signals drawn from a random piecewise constant model have sparser representations in the discrete gradient basis than in Haar wavelets (with high probability). I will talk about this result and its implications, and also show some numerical experiments in which the use of the gradient basis improves compressive signal reconstruction.