I am a Ph.D. student in the
Department of Mathematics at the University of Toronto. My supervisor is Dror Bar Natan.
You can reach me by emailing petervdovjak(location symbol)yahoo(punctuation)ca.
My interest is in using algebraic methods to understand knots. It is a source of endless surprise to me that such a discrete, non-geometric subject as algebra can be used to study such inherently geometric objects as knots.
One of the main areas of investigation for knot theorists is the discovery and study of `knot invariants'. A particularly useful and powerful kind of invariant consists of `universal finite type invariants'. Just proving that these objects exist is no easy feat - but it turns out that algebraic methods can be very useful for this purpose.
In particular, it has been known for some time that an understanding of a particular algebraic object - the cohomology of a certain complex - can be used to provide an iterative construction of a universal finite type invariant. Click here for a paper I have just written describing this particular complex and one way to analyze it using homological algebra.
One of the ingredients that can be used to construct a universal finite-type invariant is known as an "associator". Click here for a draft paper which derives closed-form formulae for an associator (and a related gadget known as a "braidor") in a particular simplified context.
Here are a couple of talks I have given: