Principal Investigator: Michael Kapovich
This proposal is a continuation of Kapovich's research of previous years in the areas of geometry, topology and geometric group theory. Most subjects of the planned research revolve around geometry of group actions on various spaces and geometric structures on manifolds, as well as geometry of buildings. The research planned by Kapovich covers geometry of buildings with applications to representation theory, in particular, Kapovich intends to continue his study of the geometry of the moduli spaces of polygonal linkages in symmetric spaces and buildings in relation to the algebraic groups. Other topics of research include discrete groups in algebraic groups (in particular, the coherence problem for arithmetic lattices), real-projective and complex-projective structures.
Groups appear naturally as symmetries of mathematical and physical objects, like wall-patterns, minerals, snowflakes and, ultimately, the entire universe. This project studies relation between algebraic properties of groups and geometry of spaces for which groups appear as symmetries. This two-way relation is beneficial for both group theory (which is a part of algebra) and geometry. An example of such relation is provided by the theory of "buildings." Comparing various ways to navigate in buildings can be mathematically described in terms of certain inequalities. Such inequalities in turn provide answers to various questions about symmetry groups of buildings.
