Most directions of the research supported by this grant revolve around geometry of group actions on various spaces and geometric structures on manifolds, as well as geometry of buildings. In more details, research conducted by M. Kapovich concerns: (1) Geometry of buildings with applications to representation theory. 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. Part of this study is establishing tropical structures on Euclidean buildings. (2) Kleinian groups in higher dimensions: Kapovich will study finiteness properties of higher-dimensional Kleinian groups. (3) Fundamental groups of complex-projective varieties: Kapovich will study fundamental groups of irreducible complex-projective varieties whose singularities are normal crossings. (4) Kapovich will study embeddings of Right-angled Artin Groups in the group of diffeomorphisms of the circle. (5) Semihyperbolicity of Teichmuller space: Kapovich will study coarse nonpositive curvature properties of Teichmuller space. (6) Kapovich will study classification of fundamental groups of closed 4-dimensional manifolds M with trivial 2nd homotopy group.
The goal of this research is to understand better interaction between geometry and groups. Groups appear naturally as symmetries of natural geometric objects. Simple examples of such symmetries come from wall-paper tilings (where geometry is Euclidean) or tilings appearing is some of the Escher pictures (hyperbolic geometry). Furthermore, groups appear as symmetries of geometric objects of physical nature, from elementary particles to the entire universe (treated as a geometric object). Algebra (group theory) allows one to encode the underlying symmetries, and, conversely, geometry allows one to approach successfully purely algebraic problems. One of the examples of such interaction of seemingly different mathematical fields is project (3), which aims to apply groups of symmetries of 3-dimensional hyperbolic space to complex-algebraic geometry (the latter deals with geometry of solution spaces of systems of polynomial equations with complex variables).
