Current Teaching
In the autumn of 2016 I am on sabbatical and not teaching, but in the past I have been known to teach MATH34032 and MATH36041. Here is the website for MATH36041.
Expository Talks
Slides from various expository talks I have given. For talks on current research, see the research tab.
- In the autumn of 2014 I gave two talks on microlocal analysis at the University of Manchester inverse problems seminar. The slides are here: part 1, part 2.
Research Interests:
I am generally interested in inverse problems, PDES, and differential geometry. Inverse problems is the area of mathematics which studies the recovery of parameters in a differential equation from some observations of solutions to that equation. One of the main themes of my past and ongoing research has been the application of methods from microlocal analysis to questions of stable invertibility for a variety of inverse problems in tomography and seismic imaging. A general heuristic for inverse problems is that stable inversion is possible if the singularities of the quantity to be reconstructed appear in the data, and I am interested in investigating applications of this general principle
Together with Marta Betcke from University College London, and Natalia Bochkina from the University of Edinburgh I am an organiser for the research group Inverse Problems: Theory and Applications. This is funded by an LMS scheme 3 grant, and we are organising a series of workshops throughout the UK. Links to the workshops can be found here.
Where links to pdf documents or arXiv are given below the linked versions are generally prior to peer review, and may not match the published articles.
Research Projects:
Integral geometry problems
One class of inverse problems, sometimes called
integral geometry, deals with the recovery of a function or tensor field from its integrals along a family of curves or surfaces. I am currently interested in particular in the geodesic X-ray transform which arises when the curves are geodesics of a Riemannian metric.
- Recovering a tensor on the boundary from polarization and phase measurements, Inverse Problems, 2009, 25(3).
- The weighted Doppler transform, with P. Stefanov. Inverse Problems and Imaging, February 2010, 4(1).
- Generic local uniqueness and stability in polarization tomography, Journal of Geometric Analysis, January 2013, 23(1).
- On the microlocal analysis of the geodesic X-ray transform with conjugate points, with G. Uhlmann, accepted by the Journal of Differential Geometry.
- Generic Uniqueness in Polarization Tomography. (Thesis)
Generalized Dix method for velocity estimation in seismic imaging
As part of the Geo-mathematical Imaging Group (GMIG) at Purdue university I studied a number of aspects of seismic imaging. One exisiting technique, known as the
Dix method, for estimating velocities uses the curvature of wavefronts produced by point diffractors measured at the surface of the earth to estimate the wave speeds in the subsurface. The traditional method is based on a layered model, and we have generalised this to more general, possibly laterally varying media.
- Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times, with M. de Hoop, E. Iversen, M. Lassas, and B. Ursin. To appear, preprint: www.arxiv.org 1211.6127.
- Reconstruction of a conformally Euclidean metric from local boundary diffraction travel times, with M. de Hoop, E. Iversen, M. Lassas, and B. Ursin. To appear, preprint: www.arxiv.org 1211.6132.
Muon tomography
Muons are particles created in the upper atmosphere which are constantly raining down on the surface of the Earth. They pass straight through most things, but are scattered by materials with high atomic number. Thus they provide a potential passive method for detecting nuclear material. Together with collaborators Oliver Dorn, Joshua Hellier, Nicola Wadeson, and Matt Stapleton, we are currently working towards calculating the sensitivity functions of detected muon scattering to material parameters. This was the industrial MSc project of Joshua Hellier during the summer of 2014. Computer code written in C++ as part of the project can be found here
here.
Wave propagation and inverse problems in rough or random media
Starting with my work on seismic imaging I have also become interested in models of wave propagation in complicated, possibly random, media. This can be combined with inverse problems by asking whether statistical properties of a random medium may be recovered based on observations of waves propagating through that medium.
- Regularity and multi-scale discretization of the solution construction of hyperbolic evolution equations with limited smoothness, with M. de Hoop, H. Smith, and G. Uhlmann. Applied and Computational Harmonic Analysis, November 2012, 33(3).
- Retrieval of a Greens function with reflections from partly coherent waves generated by wave packets using cross correlations, with M. de Hoop, J. Garnier, and K. Sølna, SIAM Journal on Applied Mathematics, 2013, 73(1).
- Scattering enabled retrieval of Green's functions from remotely incident wave packets using cross correlations, with M. de Hoop, J. Garnier, and K. Sølna Comptes Rendus Geoscience, September 2011, 343(8-9).
Papers:
These papers are also listed above according to the topic.
- Recovering a tensor on the boundary from polarization and phase measurements, Inverse Problems, 2009, 25(3).
- The weighted Doppler transform, with P. Stefanov. Inverse Problems and Imaging, February 2010, 4(1).
- Generic local uniqueness and stability in polarization tomography, Journal of Geometric Analysis, January 2013, 23(1).
- Regularity and multi-scale discretization of the solution construction of hyperbolic evolution equations with limited smoothness, with M. de Hoop, H. Smith, and G. Uhlmann. Applied and Computational Harmonic
Analysis, November 2012, 33(3).
- Retrieval of a Greens function with reflections from partly coherent waves generated by wave packets using cross correlations, with M. de Hoop, J. Garnier, and K. Sølna, SIAM Journal on Applied Mathematics, 2013, 73(1).
- Scattering enabled retrieval of Green's functions from remotely incident wave packets using cross correlations, with M. de Hoop, J. Garnier, and K. Sølna Comptes Rendus Geoscience, September 2011, 343(8-9).
- Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times, with M. de Hoop, E. Iversen, M. Lassas, and B. Ursin. Journal de Mathématics Pures et Appliquées, 2014.
- Reconstruction of a conformally Euclidean metric from local boundary diffraction travel times, with M. de Hoop, E. Iversen, M. Lassas, and B. Ursin. SIAM Journal on Mathematical Analysis, 2014 46(6), pp. 3705-3726.
- On the microlocal analysis of the geodesic X-ray transform with conjugate points, with G. Uhlmann, accepted by the Journal of Differential Geometry.
Thesis:
What you don't think looking at my teaching, research, or CV is enough fun?!
Well then you may be interested to know that in my spare time I like to play viola in the band Vanity Pages.