Infinitesimal Containment and Sparse Factors of IID

Explore advanced concepts in measured group theory and distributed algorithms through this mathematical seminar lecture from the Joint IAS/PU Groups and Dynamics Seminar. Delve into the intersection of probability theory, group theory, and geometric analysis as the speaker examines infinitesimal containment and sparse factors of independent identically distributed (iid) processes on Cayley graphs. Learn about distributed probabilistic algorithms running on vertices or edges of countable group Cayley graphs, where vertices communicate only with neighbors to accomplish tasks like selecting large independent sets or finding small connected subgraphs. Discover the geometric properties of sparse subsets distinguishable by such distributed algorithms, termed "sparse factor of iid subsets," and their connection to estimating the cost of group actions in measured group theory. Understand how measure-preserving group actions relate to this problem, particularly the concept of infinitesimal containment where statistics of group actions on small subsets can be projectively approximated. Examine the key result that Bernoulli shifts are always infinitesimally contained in group self-actions, leading to important geometric constraints on sparse factor of iid subsets in linear groups and lattices in semisimple Lie groups.