Topology and Geometry of Manifolds Unit (Anastasiia Tsvietkova)

The interests of the group are centered around topology, geometry and complexity of low-dimensional manifolds. Informally, a manifold is a topological space that locally resembles Euclidean space near each point. It has been one of the main objects of study in topology since the beginning of 20th century. However only in 1970's it was noticed by several mathematicians that 3-dimensional manifolds can be studied from a new perspective: using geometry.

On a large scale, the geometric picture is now well-understood for 3-manifolds due to the Geometrization Theorem involving work of Hamilton, Perelman, Thurston, and many others. In particular, many 3-manifolds have hyperbolic metric or can be decomposed into pieces with hyperbolic metric. However, on a small scale, i.e. for a particular manifold, the intrinsic connections between its combinatorial, topological and geometric properties are still often a mystery. This is one of the main topics of our research. While the questions lie in the area of pure mathematics, many of them are well-suited for a computer study, and often lead to open problems concerning complexity or existence of certain algorithms, overlapping with complexity theory, computable analysis, and other areas of theoretical computer science.

Additionally, we are interested in establishing the connections between geometric invariants of 3-manifolds and invariants coming from other areas of mathematics, such as quantum topology, number theory or algebraic geometry.

Latest Posts