糖心探花

Is there a connection between the distribution of prime numbers, the structure of Mandelbrot sets, compression algorithms for data sets, the description of polymers and the analysis of heat flow in media?

Probability theory naturally appears in the description and analysis of all these different areas. Located between pure and applied mathematics, this field overlaps with many different branches of mathematics and provides a background, as well as tools, to properly formulate and solve problems from a range of other sciences.

From its inception as an analysis of "chance", modern probability theory is indispensable in mathematical fields as different as combinatorics, real and complex analysis, and group theory.

Probability theory is key to providing robust foundations for statistical mechanics, an interdisciplinary research area between mathematics and theoretical physics focussing on the study of the emerging properties of systems with many degrees of freedom. 

• Research
• People
• Publications

Our research group in 糖心探花 is focused on the following areas:

Point processes and interacting particle systems

Systems consisting out of a large number of point like subsystems arise in many areas, for example if the system is based on a large number of interacting individual agents as in economics, biology or sociology. A classical area of application for such systems is the description of solid and soft matter from microscopic principles. The aim is a rigorous derivation of cooperative effects. Main areas of research are the moment problem for point processes, geometry of configuration spaces and analyticity properties.

Markov processes (Dr Tobias Kuna, Dr Jochen Broecker)

Of enormous practical relevance in applications are memoryless time developments, known as Markov processes, it is a well studied area of probability theory having many deep connections to real and complex analysis, in particular ordinary differential equations, spectral theory, partial differential equations. In particular, we do research in derivation of scaling limits, effective description for large systems in terms of few low-dimensional equations.

Random dynamical systems and time series analysis (Dr Tobias Kuna, Dr Jochen Broecker)

The mathematics of random as well as deterministic dynamical systems is central in the description of processes appearing in many areas of science. Research in this area includes the investigation of mixing properties.

Rough path theory

Many standard stochastic processes, such as Brownian motion, have no-where differentiable sample paths. Rough path is a deterministic theory of calculus purposed built for such paths. It simplifies and strengthens classical results in Stochastic Analysis such as large deviation principle and stochastic flow. Rough path theory also gives a canonical way to define differential equations driven by non-semi-martingales. Originating from T. Lyons' study of stochastic differential equation, the theory has inspired Martin Hairer's (Fields Medal 2014) work on stochastic partial differential equations and subsequently the theory of regularity structures.

Extreme events (Dr Tobias Kuna)</