Title: Building 3-Manifolds from Hopf Algebras
Speaker: Thomas Kerler
Abstract: Around 1990 a remarkable new class invariants of knots and 3-manifolds arose from a series of discoveries in mathematical physics. Their constructions are mostly algebraic and combinatorial, but very different than those obtained via methods from algebraic topology. The so-called quantum invariants turn out to have stronger separating properties than many geometric invariants. They, further, resist generalizations to higher dimensions, and they all appear to derivable in some ways from certain types of Hopf algebras.
These properties can be explained from Hopf algebraic structures inherent to 3-dimensional topology itself. That is, every closed 3-manifold can be written as an abstract expression in the (categorical) structure maps of a ribbon Hopf algebra, and two expressions yield homeomorphic 3-manifolds if and only if they are related by a set of standard axioms. Recent proofs of this deep result are due to Piergallini, Bobtcheva, Beliakova, and De Renzi.
I will introduce the basic underlying concepts by sketching the derivation of the analogous result in 2-dimension, which is far less technical and features Frobenius algebras instead of Hopf algebras. I further plan to describe the building blocks for the 3-dimensional case, relating abstract algebraic structures to concrete 3-manifolds with boundaries. There will be plenty of diagrams and pictures.
