Center for Advanced Intelligence Project

We are planning a Mathematical Seminar via Zoom meeting.
The schedule is the following :

Date & Time: August 6, 2021 11:00-16:00 (JST)

11:00-12:00 Hiroshi MATSUZOE
Geometry of quasi-statistical manifolds
13:30-14:30 Yuka KOTORII
On random link models and those linking numbers
15:00-16:00 Jun YOSHIDA
Categorification of derivatives of knot invariants

These talks will be given in English.
Please join us if you are interested.

Speaker 1: Hiroshi MATSUZOE (11:00-12:00)

Title: Geometry of quasi-statistical manifolds

Abstract: A statistical structure is a pair of a semi-Riemannian metric and a torsion-free affine connection on a manifold of which the covariant derivative of semi-Riemannian metric is totally symmetric. A manifold with a statistical structure is called a statistical manifold, which plays an important role in information geometry. Statistical manifolds naturally arise in geometry of affine immersions.
A quasi-statistical manifold is a generalization of statistical manifold. An affine connection on a quasi-statistical manifold may not be torsion-free. Such an affine connection naturally arises in quantum information geometry and the geometric theory of non-conservative estimating functions. In addition, quasi-statistical manifolds arise in geometry of affine distributions, which are generalization of affine immersions.
In this talk, after summarizing the geometry of statistical manifolds, basic properties of quasi-statistical manifolds are discussed. Quasi-statistical manifolds in the theory of affine distributions are also discussed.

Speaker 2: Yuka KOTORII (13:30-14:30)

Title: On random link models and those linking numbers

Abstract: A link is a disjoint union of simple closed curves in $mathbb{R}^3$, up to equivalence generated by an ambient isotopy of $mathbb{R}^3$. The linking number is a numerical invariant that describes the linking of links consisting of two simple closed curves. In this talk, we introduce a random link and an invariant for it, in general. We then discuss a distribution of linking number of a model of random link called grid diagram model.

Speaker 3: Jun YOSHIDA (15:00-16:00)

Title: Categorification of derivatives of knot invariants

Abstract: Knot theory is one of the most classical subject for human beings.
Although its formal study began in the 18th century (particularly by Gauss), it remains important in considerably many areas in nowadays science; including representation theory, quantum physics, and molecular biology as well as topology.
A central problem in the knot theory is determining the minimum number of crossing-change to make a knot into the unknot.
In 1990, Vassiliev studied the crossing-change in view of the space of knots and developed a sort of knot invariants.
His invariants are nowadays characterized in terms of differentials of knot invariants.
In this talk, we explain the basics of Vassiliev theory and our recent attempts to categorification of it.