"Using hyperbolic geometry to obtain information on knots, links and 3-dimensional manifolds" Dr. Anastasiia Tsvietkova

Abstract

A mathematical knot is a knotted string (imagine a shoelace), but with its ends connected so that the knot cannot be undone. A link consists of several such strings. A complement of a knot or a link in a three-dimensional sphere is an example of a three-dimensional manifold.

Knots, links and three-dimensional manifolds have been an object of study for mathematicians since the 19^th century. However, only in the 1970’s William Thurston noticed that three-dimensional manifolds can be approached using geometry. This new perspective made geometric topology one of the most rapidly developing fields of pure mathematics, and allowed to solve the long-standing Poincaré conjecture. Soon it was realized that one particular type of manifolds is prevalent and the least understood: the manifolds that have hyperbolic geometry. Moreover, the hyperbolic metric on many three-dimensional manifolds is unique and therefore it gives rise to powerful invariants.

The talk aims to explain these terms in a concrete way, and to show how the resulting invariants can be calculated. In particular, I will discuss Thurston’s method for computing hyperbolic structures, as well as our alternative method (developed jointly with Thistlethwaite) and its various consequences. I will also outline recent results leading to connections of the intrinsic geometry with invariants coming from other areas of mathematics, such as quantum topology, number theory and algebraic geometry.

Biography

Anastasiia Tsvietkova is Krener Assistant Professor at the University of California, Davis, working in topology, geometry and knot theory, particularly in dimension three. Anastasiia often uses a blend of pure mathematics and computational techniques in her research. She received her PhD degree in Mathematics at the University of Tennessee, USA, in 2012. After that, she was a Postdoctoral Fellow at the Mathematics Department of the Louisiana State University, and at the Institute for Computational and Experimental Research in Mathematics of Brown University.