Alex Barnett: Nontechnical introduction to my research
(updated Dec 2004)
My work centers around two main issues:
A lot of what I do involves getting computers to calculate things
(equations) that approximate real-world phenomena.
Here is an introduction to some of the motivation, methods, and questions.
- 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.
Imagine a drum stretched across a frame of a certain shape.
When you hit the drum, different frequencies are excited, each with its
own `mode' of vibration. (They rapidly decay, giving the drum its
characteristic sound). For the lowest frequency mode, the whole membrane
will move together (in phase), and higher modes will
have increasingly complicated
patterns, involving oscillations with decreasing wavelength.
What happens as you go to much higher modes---higher in the `spectrum' of
Different drum shapes give different behaviours, depending on whether
the shape is `chaotic' or not. Here `chaos' refers to a related problem,
that of a point particle (imagine a billiard ball)
a `billiard table' of that shape:
|| Lowest 10 modes of of a drum shape formed from
two straight lines and two arcs of circles. Red indicates upward (positive)
motion at some instant of time, and blue downward (negative).
(click to enlarge picture)
The motion (dynamics) depends on the shape, and can be regular,
chaotic, or some mixture (there are also some more exotic categories).
There is controversy as to how much the billiard problem (the `classical'
problem) influences the drum problem (the `quantum' problem).
The equation for drum modes (the Laplace eigenvalue equation) is the same
as the quantum-mechanics of a particle trapped in a cavity,
hence the term `quantum', and the strong connection to physics.
In fact, the drum problem emerged from classical physics (acoustics and
electro-magnetism), and already in 1917 Einstein
was concerned about the modes for chaotic systems
The field of quantum chaos has blossomed since the 1980's.
Questions I study include:
In the limit of high mode number, do the modes become uniformly distributed
across the shape, or do some modes cluster ('scar') along orbits where the
billiard ball repeats its motion?
What are the correlations between mode patterns?
What are the statistics of `matrix elements' (overlaps with each other)
of these modes?
These statistics have real-world consequences for the heating (dissipation)
rate in driven quantum systems, such as electrons trapped in tiny, cold
micro-chip layers called `quantum
Many of these issues have emerged from my doctoral work
with Rick Heller's group
at Harvard Physics Department.
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
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
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!).
There has been an explosion of research on how the human brain functions
in the last 15 years, driven largely by functional
Magnetic Resonance Imaging (fMRI)
via the discovery that when a part of the brain is active, it
`blushes', that is, has increased blood flow and oxygen level.
However, it is also possible to map blood oxygen levels in the cortex
(which lies mainly in the first centimeter of depth from the
inside of the skull) using near-infrared light, with sources and detectors
placed in contact with the scalp.
Shine a flashlight through the tissue of your fingers and you will see
only red light gets through, and that the light diffuses (otherwise
humans would be transparent!)
information from the detected light
signals is called
Diffuse Optical Tomography (DOT).
As you might imagine, this method is much cheaper and simpler than fMRI,
but also allows rapid imaging (hundreds of `frames' per second)
and can be sensitive to different chemicals (eg deoxy-hemaglobin and
The basic idea of Diffuse Optical Tomographic measurements
(click to enlarge)
However, the mathematics involved in some sense is harder, because
no direct 'image' is produced, Rather we have to solve for (that is,
adjust in a computer simulation)
some model of what optically is inside the head until we get similar
detected signals. This is called a nonlinear inverse problem,
and since there are many nearly equally-correct
answers, the problem is called `ill-posed'.
The image we are looking for is maps of absorption and scattering
strength of the tissues, at the various wavelengths of light used
(typically 700-900 nm).
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
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