# The Geometry/Topology Group

**The members of the Geometry/Topology group pursue matters concerning the structures and properties of space in a general and abstract form.**

The means being used are called (co)homology and homotopy. Symmetries of spaces form a structure called a Lie group named after Norwegian mathematician Sophus Lie. Manifolds are an important class of such spaces and can be thought of as higher dimensional surfaces. Many important problems in physics and enigneering sciences lead to differential equations residing in such spaces. The number of solutions of these equations depends on the topological and geometrical structure of the space. This is called index theory which was the theme for the Abel prize awarded in 2004. The classification of 3-dimensional manifolds is connected with the Poincar? conjecture recently solved by Russian matematician Perelman, earning him the Fields medal in 2006. Differentiable structures on manifolds were studied by J. Milnor who in 2011 received the Abel prize for his contributions in this area.

The Geometry/Topology group conducts research within the areas of analysis on loop spaces, algebraic topology, dynamical and complex systems, Lie theory and many-body problems, algebraic geometry and topological measures, topology and data. Specific examples include the construction of elliptic cohomology, use of higher order categories in topology and hyperstructures, integrability of many-body problems, moduli spaces and topological measures, and genomic data.