# 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.

### Broken ray tomography

Broken ray tomography is a subfield of inverse problems. The basic problem of X-ray tomography applications is to figure out the (location dependent) attenuation coefficient of an object from measured changes of intensity of X-rays fired through the object. Elementary physics allows a mathematical formulation (of a good approcimation of the physical situation): Can one find a function if one knows its integrals over all lines? The answer is affirmative in many cases, but the question is not fully settled.

What changes in broken ray tomography compared to the standard X-ray tomography is the introduction of reflecting obstacles. Suppose part of the surface of the object you are trying to measure is covered with an inward pointing mirror off which X-rays reflect. Or maybe there are mirror-covered obstacles inside the object. (This is not physically feasible, but introduces the mathematical idea.) If you know the intensity change of any X-ray you fire through the object, can you figure out the attenuation coefficient? That is, broken ray tomography is X-ray tomography with reflections at surface or obstacles.

One can mathematically reduce some other inverse problems to the broken ray tomography problem: If we know how to solve this problem, we immediately know the solution some other ones. As waves reflecting from surfaces is not an uncommon phenomenon, the broken ray tomography question may also arise more directly.

Broken ray tomography is currently my main field of research.

## Neutrino oscillation

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.

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.