The project concerns the behavior of objects in a high-dimensional setting, in search of phenomena that arise as the number of degrees of freedom of a system tends to infinity. The theory underlying these objects has recently seen a boost in applications, in accordance with the explosion of interest in data science and machine learning, two fields of which this theory is a cornerstone. The project puts emphasis on expanding the theoretical foundations relevant, for example, to the understanding of interacting particle systems in statistical mechanics or neural networks or reinforcement learning problems with a large decision-space.
The intuition derived from low-dimensional examples in various fields such as topology and partial differential equations may suggest that an attempt to understand the behavior of high-dimensional objects is futile, as a system's behavior quickly becomes complex and intractable as the dimension increases. This is manifested in a meta-phenomenon referred to as the "curse of dimensionality" which, simply put, refers to the exponential growth of the number of configurations of a system with respect to dimension.
Surprisingly, in many cases of interest which often include ones relevant to real-life applications, high-dimensional systems turn out to be well-behaved and tractable and sometimes it is even the case that the behavior becomes more regular as the dimension, or the number of degrees of freedom, increases. An exemplary mathematical result which illustrates this is the central limit theorem, which shows that when we average a growing number of independent random quantities, the Gaussian (normal) distribution emerges. Broadly speaking, this is the type of phenomenon that the project aims to reveal.
The project is concerned in particular with one facet of this theory we refer to as "dimension free" behavior, which alludes to the fact that many quantities of interest and many central inequalities and bounds related to high dimensional objects have no explicit dependence on the dimension. This phenomenon is observed in several important classes of distributions. In particular we focus on the Gaussian measure and on measures with a convex potential (called "log concave" measures). To this end, the project aims to make progress on several conjectures which predict the dimension-free behavior of high-dimensional objects. One example of such conjecture is the Kannan-Lovasz-Simonovitz conjecture which asserts that for high-dimensional measures with convex potentials, half-spaces (hence, sets which only depend on one direction) will be approximate minimizers.
The project revolves around an emerging method, which is based on a connection with the theory of stochastic calculus, the theory that describes the motion of diffusing particles. The "pathwise method" attempts to analyse a high dimensional system by associating with it a certain stochsatic evolution driven by a Brownian motion, in a way that quantities of interest of the system such as entropy or variance can be expressed in terms of the stochastic process.
