Random geometry studies prioperties of random fractals that arise e.g. as scaling limits of discrete models of statistical physics, especially in connection with conformally invariant field theories. This includes the famous SLE (Schramm Loewner evolution) curves (or collections of curves) that describe the scaling limits of various 2-dimensional lattice models including the famous Ising model and percolation. Our interest here lies especially in the description of some of the models via conformal welding.
Our special interest here is the further development of the theory of Gaussian multiplicative (GMC) and its applications. GMC consists of a special class of random measures that appear e.g. as a building block for the Liouville quantum gravity and for the corresponding conformal field, theory, as well as for the weldings of SLE curves. GMC appears also as random distributions for complex values of the inverse temperature parameter, as well as statistical limits of natural quantities in random matrix theory and in analytic number theory.
Probabilistic number theory studies various number theoretic phenomena where the common feature is that some number theoretic quantities turn out to have an interesting statistical behaviour, or probabilistic methods appear as an important research method. Our group has a special interest in the intriguing statistical aspects of the Riemann zeta function in the critical strip or on the critical line. One of the surprises here is the appearance of GMC (or slight modification of it) as the (conjectured) scaling limit of random shifts of the Riemann zeta function.
Geometric analysis and related random phenomena has long been one of the cornerstones of our research. This includes quasiconformal geometry and mappings of finite distortion. Further, we are interested in applying methods of geometric analysis in random geometry.
We have been also studying probabilistic algorithms, especially MCMC (Markov Chain Monte Carlo methods). In the past we pioneered adaptive MCMC methods and their theory, as well as understanding infinite dimensional Bayesian theory. Today ouir interests include also validation of some of the most commonly used MCMC algorithms such as HMC and NUTS. Moreover, we make use of harmonic analysis methods in parameter infererence.