- the nature of resonant modes occuring in wave and quantum systems
- 'seeing' inside the human body (without surgery, that is, noninvasively) by modelling how light diffuses through tissue.

Motion of a billiard ball bouncing inside regular and chaotic tables (click to enlarge) |

Such questions require *semiclassics* (the theory of how waves behave
in the limit of short wavelengths), and also large-scale
numerical calculation
of the modes. I have focussed on devising new and very efficient algorithms
that allow a computer to be used to calculate these modes for a large
variety of drum shapes, including with corners.
This requires analysis of the properties of the modes themselves.
As the wavelength gets shorter, the problem
gets more challenging, however my methods become relatively more efficient
compared to any other known methods. I typically
compute many thousands of modes at around the 100,000th mode.
Here the efficiency increase is about a thousand times!
These methods have technological applications in designing
micro-cavity lasers,
less than a hundredth of a millimeter in size, for fiber-optic communication,
and more generally in a wealth of acoustic and waveguide problems.
I continue to collaborate with physicists, applied and semiclassical
mathematicians, and even number theorists
(who care about quantum ergodicity too!).

The basic idea of Diffuse Optical Tomographic measurements (click to enlarge) |

DOT is so versatile that functional brain imaging can be performed even on a baby who is moving around and reacting to stimuli |

I have worked on fast numerical methods to calculate how short pulses
of light diffuse through the tissue of the human head, in the
complicated geometry of the scalp, skull and brain.
However, there is much to be improved in these methods.
Speed is important since solving the inverse problem requires using this
`forward' simulation many times.
Since the measurements (detected light intensities as a function
of time, on the scale of a billionth of a second) are noisy, statistical
methods are appropriate.
I favour *Bayesian* methods, which tell you exactly what you
have learned about your image (parameters in your model), and no more,
in the sense of a probability distribution.
The price you pay is that these methods are quite slow.
Questions remain such as: what is the ultimate practical resolution of DOT
in the human brain? What is the
uncertainty in measured parameters of regions of the brain, given
a certain noise?
How much uncertainty can there be in calibration
parameters of the experiment?
What are optimal
patterns of sources and detectors on the head surface?

I collaborate with David Boas's Photon Migration Imaging lab at the Martinos Center (Radiology Department) at Massachusetts General Hospital and Harvard Medical School, where many DOT techniques are being developed.

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