Mathematics Colloquium

Categorification lifts numbers to vector spaces and homology groups,
and vector spaces to abelian or triangulated categories. While a
rigorous definition is still lacking, there are plenty of examples
to demonstrate its properties. The talk will center on the classical
example of categorification of the Jones polynomial and its localized
version, the Kauffman bracket of tangles. Upon categorification,
the invariant of tangles takes values in a triangulated category,
the homotopy category of complexes of modules over a suitable graded ring,
and extends to an invariant of tangle cobordisms, assigning to a
cobordism a morphism between the objects for its two boundary
components. These categories appear in representation
theory, algebraic geometry, and symplectic topology.