Title: MATH 663 Analysis and Geometry of Nilpotent Groups

Prerequisites: MATH 653 Algebra I, MATH 608 Real Variables II, (MATH 622 Differential Geometry I will be useful but not required)

Course Description: One of the most celebrated results in geometric group theory is Gromov's theorem characterizing finitely generated groups of polynomial growth 
as exactly those that are virtually nilpotent. This course will cover, at variable depths, the main ingredients involved in the proof of Gromov's theorem. We then 
proceed to the fundamental theorems of Pansu on asymptotic cones of (finitely generated) virtually nilpotent groups and differentiation of Lipschitz maps between 
Carnot Lie groups. These theorems will be applied to study the quasi-isometric embeddability of virtually nilpotent groups into each other and Banach spaces. A 
large portion of the course will be dedicated to introductory material on geometric group theory and nilpotent Lie groups as needed for these topics. Time permitting, 
we will investigate representation theories and cohomology of nilpotent groups and applications to finer quasi-isometric embeddability results.

Average time dedicated per week (estimate): 3-4 hours