Probability

Aside from games of chance and a handful of textbook topics (e.g. opinion polls) there is little overlap between the content of an introductory course in mathematical probability and our everyday perception of chance. In this mostly non-mathematical talk I will give some illustrations of the broader scope of probability.

Why do your friends have more friends than you do, on average? How can we judge someone’s ability to assess probabilities of future geopolitical events, where the true probabilities are unknown? Were there unusually many candidates for the 2012 and 2016 Republican Presidential Nominations whose fortunes rose and fell? Why, in a long line at airport security, do you move forward a few paces and then wait half a minute before moving forward again? In what everyday contexts do ordinary people perceive uncertainty/unpredictability in terms of chance?

The study of maps, that is of graphs embedded in surfaces, is a popular subject that has implications in many branches of mathematics, the most famous aspects being purely graph-theoretical, such as the four-color theorem. The study of random maps has met an increasing interest in the recent years. This is motivated in particular by problems in theoretical physics, in which random maps serve as discrete models of random continuum surfaces. The probabilistic interpretation of bijective counting methods for maps happen to be particularly fruitful, and relates random maps to other important combinatorial random structures like the continuum random tree and the Brownian snake. This course will survey these aspects and present recent developments in this area.

Using the combinatorial method for the 2D Ising model originating in the works of Sherman, Burgoyne and others we derive formulas for the correlation functions in terms of signed loops and paths. In the case of regular lattices we also identify the critical temperature for the phase transition in the long range behavior of these functions. Joint work with Wouter Kager and Ronald Meester.

The study of maps, that is of graphs embedded in surfaces, is a popular subject that has implications in many branches of mathematics, the most famous aspects being purely graph-theoretical, such as the four-color theorem. The study of random maps has met an increasing interest in the recent years. This is motivated in particular by problems in theoretical physics, in which random maps serve as discrete models of random continuum surfaces. The probabilistic interpretation of bijective counting methods for maps happen to be particularly fruitful, and relates random maps to other important combinatorial random structures like the continuum random tree and the Brownian snake. This course will survey these aspects and present recent developments in this area.

Particles attempt to follow a simple dynamic (random walk, constant flow, etc) in some space (interval, line, cycle, arbitrary graph). Add a simple interaction between particles, and the behaviour can change completely. The resulting dynamical systems are far more complex than the ingredients suggest. These processes (interchange process, TASEP, sorting networks, etc) have diverse to many topics: growth processes, queuing theory, representation theory, algebraic combinatorics. I will discuss recent progress on and open problems arising from several models of interacting particle systems.