# Bachelor/Master Projects

*If you are interested in one of these subjects and want to do BA/MA thesis please contact me.*

The list below shows** some** possibilities, many more are possible!

#### 1. Recurrence, Van der Waerden's theorem and multiple recurrence.

This subject introduces you to the connection between randomness and patterns in subsets of the integers. It is an introduction to material that is used and sophisticated in Green and Tao's proof of the fact that the primes contain infinitely many arithmetic progressions of any length.

#### 2. Duality

#### Duality is one of the most powerful techniques in the study of interacting Markov processes. It allows to relate a Markov process to a simpler process (called the dual process). E.g. diffusion processes (Brownian motion-like Markov processes) can be linked to continous-time discrete jump processes, and systems with many particles or interacting components can be linked to systems with only a few interacting components.

#### In this paper a good introduction to the subject can be found. The aim of a masterproject can be e.g. to further explore duality in the context of wealth distribution models (as is started in this paper) or in stochastic models of transport (as in this paper where we study properties of non-equilibrium states in some exactly solvable models)

#### 3. Ricci curvature and Markov chains.

This subject is about a recent approach introduced by (among others) Y. Ollivier to define positive Ricci curvature in discrete spaces and general metric spaces, via Markov chains. The idea is that for a space with positive Ricci curvature, random walks starting at different points can be coupled exponentially fast in time, and the exponent is related to the Ricci curvature. For random walk on Riemannian manifolds which jumps uniformly in small spheres this exactly corresponds to the classical Ricci curvature, but the present notion can be defined in much more general spaces, including discrete spaces such as the hypercube. Here is a good introduction to this subject.
#### 4. Sandpiles

The sandpile model (also called chip-firing game) is a paradigm for self-organized criticality, i.e., a dynamics which ``drives itself towards a critical point". It is also an interesting mathematical object, with an abelian group associated to it, and related to combinatorial objects such as spanning trees, and a source of beautiful self-repetitive patterns as you can see in the two pictures on this website, and much more in the links below. In the project, you get acquainted with the model, and study an extension where upon a toppling some sites can loose grains. A good introduction to the sandpile model can be found in the notices of the AMS, september 2010 issue here
#### 5. Non-equilibrium steady states.

In equilibrium statistical mechanics, there is a well-defined natural probability measure to work with for computing macro-averages: the Maxwell-Boltzmann distribution. Out of equilibrium, things become very different. Imagine e.g. a system in contact with two heat reservoirs at different temperatures. The systems ``relaxes" to a stationary state in which there is a constant heat current. Besides, it builds long-range correlations. This is very different from equilibrium systems, where correlations decay exponentially (except at the critical point). The aim of the project is to study simple models where some correlation functions can be computed explicitely, and to test with these models general claims about non-equilbrium steady states. An introduction to this subject can be found here
#### 6. Adaptive dynamics

Adaptive dynamics is an area of mathematical biology that tries to understand evolution in more (i.e., more than Darwin) quantitative terms. E.g.: how can we understand the evolution of dominant traits of a population, how can we understand polymorphism, etc. One can work either with differential-equation based models, or with stochastic individual-based models. Especially the link between stochastic models and equations of adaptive dynamics (the so-called canonical equation) is of interest. The link between fitness and birth/death rates in more microscopic models and how to understand (derive) selection in ``large scale dynamics" of a population (Wright-Fisher diffusions with selection). Here you find a whole website dedicated to this fascinating subject.