**Presentation - **Juraj Foldes is an assistant professor at the Department of Mathematics at the university of Virginia. He obtained his PhD degree from the University of Minnesota under the supervision of Peter Polacik. Later, he had postdoctoral and research experience at the Vanderbilt University, Institute for Mathematics and its applications at the University of Minnesota, Universite Libre de Bruxelles, and at the IAS and Universite de Cergy-Pontoise.

Juraj Foldes’ research interests are broad and encompass different parts of mathematics such as Partial Differential Equations, Dynamical Systems, Fluid Mechanics, and Stochastic processes. Since the doctoral studies he has been interested in qualitative properties of solutions of parabolic and of elliptic equations, and over-determined problems, which model various natural phenomena such as chemical reactions, or biological systems. Currently, his main focus is on randomly forced equations of Fluid Dynamics and related problems in the theory of turbulence. More specifically, he studies ergodic, mixing, and qualitative properties of invariant measures, which encode statistical properties of flows.

**Research project**

Complex systems exhibit very complicated, often chaotic dynamics and sensitive dependence on initial conditions. Due to highly unstable character of solutions, it is merely impossible, to follow one trajectory either numerically or analytically. Also, explicit solutions are often practically invisible in experiments due to exterior influences or hidden parameters.

However, generic solutions develop finer and finer structures as time progresses, and the microscopic disorder give rise to statistical states that are stable. Rigorous proof of the existence, uniqueness, and properties of statistical solutions is a notoriously difficult problem which is closely related to ergodic hypothesis, a conjecture open for most problems. The problem becomes more accessible if one does not neglect exterior influences but rather treat them as stochastic perturbations.

The goal of the project is to investigate statistical states in the buoyancy driven flows created by the stochastic boundary forcing. It is natural to assume that the heating (cooling) through the boundary is not uniform and can have stochastic nature. We will establish the existence, uniqueness and ergodicity of statistically invariant states for the Boussinesq system with a stochastic perturbation to the bottom (hot) boundary. This result verifies the ergodic hypothesis for the randomly driven system. This is a different stochastic setting with many applications, which however did not yet receive an appropriate attention.