Lattices and Harmonic Analysis on Some 2-Step Solvable Lie Groups

Clay Research Awards

In the 1990s, a series of dramatic results led to the completion of the Gromov program for lattices in semisimple Lie groups. The next natural class of examples to consider are lattices in solvable Lie groups, and even results for the simplest examples were elusive for a considerable time. Fisher’s joint work with Eskin and Whyte in which they proved the first results on quasi-isometric classif...

Lattices in contact Lie groups and 5-dimensional contact solvmanifolds

This paper investigates the geometry of compact contact manifolds that are uniformized by contact Lie groups, i.e., manifolds of the form Γ \ G for some Lie group G with a left invariant contact structure and uniform lattice Γ ⊂ G. We re-examine Alexander’s criteria for existence of lattices on solvable Lie groups and apply them, along with some other well known tools. In particular, we restric...

About Existence and Non–Existence of Lattices in some Solvable Lie Groups

Harmonicity and Minimality of Vector Fields on Lorentzian Lie Groups

‎We consider four-dimensional lie groups equipped with‎ ‎left-invariant Lorentzian Einstein metrics‎, ‎and determine the harmonicity properties ‎of vector fields on these spaces‎. ‎In some cases‎, ‎all these vector fields are critical points for the energy functional ‎restricted to vector fields‎. ‎We also classify vector fields defining harmonic maps‎, ‎and calculate explicitly the energy of t...

Reduction of Differential Equations by Lie Algebra of Symmetries

The paper is devoted to an application of Lie group theory to differential equations. The basic infinitesimal method for calculating symmetry group is presented, and used to determine general symmetry group of some differential equations. We include a number of important applications including integration of ordinary differential equations and finding some solutions of partial differential equa...