Linear algebra and topology

Linear Algebra and Topology

Two real entried (respectively: orthogonal) n x n matrices^! andB are said to be linearly similar if there is an invertible real (resp. orthogonal) n x n matrix C with CAC~ = B. Of course, A, B and C may be regarded as linear functions from R (resp. isometries from S~) to itself. The matrices A and B are said to be (resp. homogeneously differentiably) differentiably similar if there is a diffeo...

Completions in Algebra and Topology

Duality in Algebra and Topology

In this paper we take some classical ideas from commutative algebra, mostly ideas involving duality, and apply them in algebraic topology. To accomplish this we interpret properties of ordinary commutative rings in such a way that they can be extended to the more general rings that come up in homotopy theory. Amongst the rings we work with are the differential graded ring of cochains on a space...

Quantum Algebra and Quantum Topology

I work in an area at the intersection of representation theory, low-dimensional topology, higher category theory, and operator algebras which is often referred to as quantum algebra or quantum topology. A practical description of this field is that it consists of the mathematics which is descended from the Jones polynomial [Jon85]. The unifying idea behind quantum topology is to consider a func...

Incompatibility of algebra and topology