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14:00 to 15:00 |
Valeriy Bardakov (Sobolev Institute of Mathematics, Russia) |
Introduction to algebraic theory of quandles (Lecture 2) This is a short introduction to algebraic theory of quandles. We give definition and examples of racks and quandles; describe some their properties. Also, we consider extensions of quandles, groups which connect with quandles: group of inner automorphisms, full group of automorphisms, adjoint group. Prove the theorem of Joyce-Matveev, which says that any quandle cam be constructed from some group and a family of its subgroups. We give description of free quandles and some their quotients such as free abelian quandles, involutory quandles. Explain connection of quandles with set-theoretical solution of the Yang-Baxter equation.
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15:30 to 16:30 |
Seiichi Kamada (Osaka University, Japan) |
Braids, I This is an introduction to braids in knot theory. It includes, geometric and algebraic definitions of braids, the pure braid group and representations of the braid group to the permutation group, a graphical method of computing braid words, Artin automorphisms on the free group, racks and quandles and their automorphisms induced by braids.
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17:30 to 18:30 |
Abhijit Champanerkar (City University of New York, USA) |
Hyperbolic knot theory (Lecture 1) In this series of two talks we will introduce ideas, tools and examples in hyperbolic knot theory. In the first talk we will review basic hyperbolic geometry in dimensions two and three, important structure theorems for hyperbolic 3-manifolds and ideal tetrahedra which are building blocks for hyperbolic 3-manifolds. In the second talk we will discuss ideal triangulations, gluing equations, Thurston's Dehn surgery Theorem and give explicit examples of hyperbolic knots and links. We will also see computational tools like SnapPy to study and compute geometric invariants of hyperbolic knots and 3-manifolds.
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19:00 to 20:00 |
Jozef H. Przytycki (George Washington University, USA) |
From Fox 3-coloring to Yang-Baxter homology We start from naive invariants of arc colorings and survey associative and distributive magmas and their homology with relation to knot theory. We outline potential relations to Khovanov homology and categorification, via Yang-Baxter operators. We use here the fact that Yang-Baxter equation can be thought of as a generalization of self-distributivity. We show how to define and visualize Yang-Baxter homology.
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