# Master Thesis: possible topics

If you would like to write a Master Thesis under my supervision, here is a list of possible topics. This list is not complete, and both of us may bring other suggestions.

“Coxeter groups and finite state automata”
Finite state automata are a standard feature of computer science, and serve as a model of simple however functional computing device (i.e. a primitive computer). Coxeter groups (studied and classified by a British Canadian mathematician H.S.M. Coxeter) belong to the field of geometry, and are a key feature in understanding symmetries of geometric objects (as well as objects in other domains of mathematics, chemistry, biology and arts). What is the link between the two? Actually,  there is a whole theory of computing with groups and symmetries, which lies at the interface of discrete mathematics, geometry and computer science. This project assumes basics of group theory and linear algebra. Knowledge of programming in SAGE/Python is an asset.

“Conformal mappings and protein docking”
Given two metrics on a topological sphere, how do we find a conformal (or a reasonably close to conformal) map between them? This question seems rather theoretical, however it appears to be directly related to such an important biological process as protein docking, as shown by Joel Hass and Patricia Koehl. Moreover, there exists an approach  by using hyperbolic geometry (which would be another rather theoretical feature, however it appears surprisingly effective in solving certain practical questions) initiated by Igor Rivin. There has recently been more questions than answers related to this type of research, which makes it an interesting, however complicated, part of mathematics. This project assumes basics of group theory, linear algebra, and Riemannian geometry. Knowledge of programming in Mathematica/MAPLE/SAGE/Python is necessary for computer experiments.

“Is there a formula for it?”
One of the problems formulated by Donald Knuth (yes, THE Donald Knuth) in his famous book “The Art of Computer Programming” read as “Develop computer programs for simplifying sums that involve binomial coefficients”. Nowadays most of the computer algebra systems that you can get even for free can perform this task (with varying speed and varying degree of simplification of their final answer). What kind of algorithm will take a formula with an unbounded, in principle, number of terms, such as $\sum^n_{k=1} {n \choose k}^2 {{3n+k} \choose {2n}}$, and simplify it down to a clear-cut expression like ${3n \choose n}^2$ ? The answer is known as Wilf-Zeilberger theory. This project assumes basics of combinatorics, algebra and calculus. Knowledge of programming in Mathematica/SAGE/Python is a definite asset.

There are a few hyperlinks above pointing at some relevant material: hopefully making decision based on this information will be easier for you. Indeed, writing your Master Thesis is a serious project that requires time and effort, and your choice has to be an informed one (as well, there is still some randomness involved, since it’s hard to say ahead of time how things will work out).

Writing your thesis is also exciting, since you will learn many new things and may  contribute your own research. It may also let you understand better what type of professional activity you would like to carry out after your graduation. All in all, it is definitely a task worthy to do. However, please notice that this one comes with no warranty: it’s not promised to be neither easy nor rewarding (albeit this holds for both of us, so we are even as partners in business or trouble alike).