In the 1980s, Thurston introduced a new (asymmetric) metric on Teichmüller space based on best Lipschitz maps between two homeomorphic hyperbolic surfaces, instead of quasi-conformal maps which are used in the original theory of Teichmüller. In this series of lectures, I will explain his theory and discuss recent progress in the field. Three lectures will cover the following topics, but I may also add other materials. (1) General theory of Thurston’s asymmetric metric (2) Geodesics with respect to Thurston’s metric (3) Infinitesimal structures of Teichmüller space with Thurston’s metric

In the 1980s, Thurston introduced a new (asymmetric) metric on Teichmüller space based on best Lipschitz maps between two homeomorphic hyperbolic surfaces, instead of quasi-conformal maps which are used in the original theory of Teichmüller. In this series of lectures, I will explain his theory and discuss recent progress in the field. Three lectures will cover the following topics, but I may also add other materials. (1) General theory of Thurston’s asymmetric metric (2) Geodesics with respect to Thurston’s metric (3) Infinitesimal structures of Teichmüller space with Thurston’s metric

In the 1980s, Thurston introduced a new (asymmetric) metric on Teichmüller space based on best Lipschitz maps between two homeomorphic hyperbolic surfaces, instead of quasi-conformal maps which are used in the original theory of Teichmüller. In this series of lectures, I will explain his theory and discuss recent progress in the field. Three lectures will cover the following topics, but I may also add other materials. (1) General theory of Thurston’s asymmetric metric (2) Geodesics with respect to Thurston’s metric (3) Infinitesimal structures of Teichmüller space with Thurston’s metric

An n-dimensional k-handlebody is an n-manifold obtained from an n-ball by attaching handles of index up to k, where n ≥ k. We will discuss that for any n ≥ 2k + 1, any n-dimensional k-handlebody is diffeomorphic to the product of a 2k-dimensional k-handlebody and an (n − 2k)-ball. For example, a 2025-dimensional 6-handlebody is the product of an 12-dimensional 6-handlebody and a 2013-ball. We also introduce (n,k)-Kirby diagrams for some n-dimensional k-handlebodies. Here (4,2)-Kirby diagrams correspond to the classical Kirby diagrams for 4-dimensional 2-handlebodies.

In this talk, we study several computational problems related to knots and links. We investigate lower bounds on the computational complexity of theoretical knot theory problems. Unknotting number is one of the most interesting knot invariants, and various research has been done to find unknotting numbers of knots. However, compared to its simple definition, it is generally hard to find the unknotting number of a knot, and it is known for only some knots. There is no algorithm for determining unknotting numbers yet. First, we show that for an arbitrary positive integer n, a non-torus knot exists with the unknotting number n. Second, we show that the computational complexity of the diagrammatic un-knotting number problem is NP-hard. We construct a Karp reduction from 3-SAT to the diagrammatic unknotting number problem. Third, we also prove that the prime sublink problem is NP-hard by making a Karp reduction from the known NP-complete problem, the non-tautology problem.