Research areas
This page is an exposition of the topics that I have done research on. These descriptions are aimed at a rather broad audience. A reader interested in more details may look at my publications or contact me personally.
Inverse problems
Inverse problems is a branch of applied mathematics that studies indirect measurements. What qualifies an indirect measurement problem as an inverse problem is, roughly, that it requires complicated reasoning. The need for complicated reasoning is what makes the topic mathematical, although the final goal is in practical applications.
Inverse problems are best understood when contrasted to direct problems. A direct problem asks to deduce the effect from a cause, and an inverse problem asks to deduce the cause from an effect. For example, if we know that a melon is ripe, we can figure out the sound it makes when hit (direct problem). But can we decide whether it is ripe by the sounds it makes (inverse problem)? This example is an inverse spectral problem: can one figure out properties of an object from the collection of all sounds it makes (its spectrum)? Another spectral inverse problem asks whether one can hear the shape of a drum.
Direct problems are often much easier to solve, although not necessarily trivial. A great difference between inverse and direct problems is in the "direction of determinisim". For any cause there is a unique effect. (This holds even in the quantum realm if one interprets the effect statistically.) Therefore a direct problem can always be solved, at least in principle. An inverse problem, however, need not have behave like this. Many causes may cause the same effect, but there are favourable situation where one can actually deduce the cause from the effect – or measure the cause by measuring the effect. And even if the cause may not be measured, it still makes sense to ask if some properties of it could be measured by measuring its effects.
Besides knowing that the indirect measurement works in principle, it is convenient to have a practical method to calculate the cause from the effect. And to be able to rely on the obtained results, one needs to know that small measurement errors lead to small errors in the conclusion.
Examples of inveres problems: Can one measure the structure of an object (a human patient, for instance) by measuring how X-rays attenuate when going through it? Can one deduce the structure of the Earth by measuring how long seismic waves take to travel through? Can one measure the electrical properties of an object by placing electrodes on its surfaces and measuring currents caused by different voltages?
These examples present an important general feature of inverse problems: One tries to figure out what is inside something by making measurements outside. In other words, the measurements are non-invasive or non-destructive. This is a desirable feature in many practical imaging scenarios, perhaps most obviously in medical ones.
See also Wikipedia.
Ray tomography
Ray tomography is a subfield of inverse problems. The basic question is: Can we find a function when we know its integrals over a collection of curves? If this collection is the set of all straight lines in a Euclidean space, this is the classical problem of X-ray tomography. Changing the collection of curves changes the nature of the problem. They could be geodesics on a manifold or perhaps just a subset of all the lines.
In broken ray tomography one introduces reflecting obstacles to X-ray tomography. While no realistic mirrors reflect X-rays, the this provides the simplest way to introduce the problem. X-ray tomography problems arise from some problems with partial differential equations. Any parts of the boundary where the boundary conditions are in some way loose become reflecting when one passes to the ray transform. Therefore broken ray tomography can be regarded as a partial data variant of X-ray tomography.
In light ray tomography the integrals are taken over all light rays on a Lorentzian manifold or perhaps simply in the Minkowski space. This arises in geometrical problems in general relativity but also connection to inverse problems for hyperbolic equations because the singularities of a wave-like equation follow light rays of some sort. The restriction of only having lightlike directions at one's disposal leads to interesting phenomena.
In addition to the set of curves one can also change the nature of the object being integrated. If instead of a scalar function it is a tensor field, a connection, or something else, it is common that the idealized measurements can never uniquely determine the unknown. But one can still analyze what exactly is hidden by these transforms and prove uniqueness up to natural gauge.
Geomathematics
Geophysics is a natural playground for inverse problems: We can only make measurements at the surface of our planet but we would like to see inside. To prove any mathematical statements about the unique determination of material parameters or to apply mathematical tools to physical problems a model is needed.
Proofs are easiest when the models are easiest, and there are a great number of results for different toy models. One of my interests is to untoy the models — to make the models more realistic so that the mathematical results have more physical meaning.
As one might expect, more complex models make work harder. But there are also situations where the model offers some additional useful structure that actually helps, but making use of it is not trivial. A key goal in modelling is striking the balance between mathematical rigidity and physical generality. If there is not enough mathematical structure, nothing can be proven. If the model does not cover known and relevant physical phenomena, it fails to describe the situation.
Neutrino physics
Neutrinos are fundamental particles whose existence has been confirmed by experiment but which are completely invisible to an unequipped human. They are created, for example, in many nuclear reactions. There are three types of neutrinos, and these types are known as flavours. The three neutrinos are the electron neutrino, the muon neutrino and the tau neutrino.
A curious properties of neutrinos is that they change flavour spontaneously: an electron neutrino created in the sun may turn into a muon neutrino by the time it reaches Earth. This change is known as neutrino oscillation. As is the general case in quantum physics, the flavour in which a particular neutrino is observed is not determined, but each flavour has a probability of appearance. These probabilities change as neutrinos fly trhough space.
Neutrinos give imaging possibilities that are otherwise impossible. Any kind of "X-ray tomography" of our home planet will have to rely on neutrinos rather than photons, whether based on attenuation or oscillation. Neutrinos are also an interesting tool to study cosmology, as it makes a big difference geometrically that they fall a little short of the speed of light.
The exact properties of this oscillation or probability change have been studied extensively. My bachelor's thesis contains a quantum mechanical treatment of neutrino oscillations, and a study with a more advanced physical model (out of equilibrium thermal quantum field theory) can be found in my master's thesis.