A computer-generated mathematical model

Research Activity

Probability theory is concerned with evaluation of chances where the central subjects include discrete and continuous random variables, probability distributions, and stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion.

Classical areas of research in probability are Limit Theorems and Large Deviations. It goes back to Jacob Bernoulli in 1713 that, if we repeat independently some stochastic experiment many times, then the frequency of some event stabilises in a long run, this law is known as the law of large numbers. Modern probability theory provides further developments in this direction, among them the central limit theorem which is one of the great results of mathematics. It explains the ubiquitous occurrence of the normal distribution in nature.

Another classical area of research concerns Markov Processes which goes back to the early 20th century. It is an extension of independent random sequences where predictions can be made regarding future outcomes based solely on the present state and—most importantly—such predictions are just as good as the ones that could be made knowing the process's full history. Research has reported the application and usefulness of Markov chains in a wide range of topics such as physics, chemistry, biology, medicine, music, game theory and sports.

Among other areas of research in our department is one related to random growth processes which describe growing objects that evolve over time according to some
underlying random structure. These processes occure often in nature. Examples
include tumoral growth, lightning strikes and mineral aggregation. We often want to study this growth in order to understand the underlying natural process.

Active research is also conducted in the area of Noncommutative Probability. In particular, Free Probability is a mathematical theory that studies non-commutative random variables. The "freeness" or free independence property is the analogue of the classical notion of independence, and it is connected with free products. This theory was initiated by Dan Voiculescu around 1986 in order to attack the free group factors isomorphism problem, an important unsolved problem in the theory of operator algebras.