Probabilistic Methods in Symplectic Geometry and Fluid Mechanics

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Our world appears differently at different scales! For example a fluid is a collection of an enormous number of molecules that collide incessantly and move erratically without any particular aim. How do these molecules then manage to organize themselves in such a way as to form a flow pattern on a large scale? As another example, how fast the clotting (coagulation) should occur for a liquid (such as blood) to turn to a gel? The investigator's research concerns the relationship between the microscopic structure and the macroscopic behavior of fluids. The analysis of the mathematical models consisting of a large number of interacting particles is proved to be useful in understanding the intricate behavior of our microscopic world. The investigator also emphasizes on applying geometric and probabilistic ideas in order to develop new insights into complex dynamics of fluids. The proposal addresses the scaling and collective behavior of various stochastic and deterministic models that are of interest in statistical mechanics and differential geometry. These models are either formulated as interacting particle systems, partial differential equations or variational problems. In particular, the proposal suggests applying geometric and probabilistic ideas to study fluid equations. Also, Optimal Transport type problems are proposed for symplectic and contact forms that could play a role in understanding fluid equations. As another application of probabilistic ideas in differential geometry, we propose to study the set of fixed points for stochastic symplectic maps that appear as the flows associated with models in celestial mechanics. The proposal also formulates precise conjectures for particle systems modeling the phenomena of coagulation and formation of gels.