Probabilistic Methods in Symplectic Geometry and Fluid Mechanics

我们的世界在不同的尺度下看起来不同！例如，流体是大量分子的集合，这些分子不停地碰撞，不规则地运动，没有任何特定的目标。那么这些分子又是如何组织起来，形成大规模的流动模式的呢？ 作为另一个例子，对于液体（如血液）转变为凝胶，凝结（凝固）应该发生多快？研究者的研究涉及流体的微观结构和宏观行为之间的关系。由大量相互作用粒子组成的数学模型的分析被证明是有用的，在理解我们的微观世界的复杂行为。研究者还强调应用几何和概率思想，以发展对复杂流体动力学的新见解。该提案涉及统计力学和微分几何中感兴趣的各种随机和确定性模型的标度和集体行为。这些模型可以表述为相互作用的粒子系统，偏微分方程或变分问题。特别是，该提案建议应用几何和概率思想来研究流体方程。此外，最佳运输型问题提出了辛和接触形式，可以发挥作用，在理解流体方程。作为概率思想在微分几何中的另一个应用，我们建议研究随机辛映射的不动点集，这些映射表现为与天体力学模型相关的流。该提案还制定了精确的粒子系统建模的凝聚和凝胶形成的现象。

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.