The proposal is to construct knot invariants using algebraic geometry. Recall that to each pair of a Lie algebra and a representation one can associate a Reshetikhin-Turaev invariant of knots. For example, the Jones polynomial is one such invariant. In the case of the general linear group and the standard representation the PI and Joel Kamnitzer categorify these invariants, meaning that to each knot they assign a (bi-graded) homology group whose Euler characteristic is the original invariant. This is analogous to the way (singular) homology of topological spaces "categorifies" the Euler characteristic. Their construction involves studying the (derived) category of coherent sheaves on certain flag-like varieties. All this parallels the earlier pioneering work of Khovanov and Rozansky who discovered such categorifications using algebraic and combinatorial constructions. Using the algebraic geometric approach the investigator and his colleagues have a conjecture for how to categorify some of the remaining Reshetikhin-Turaev invariants. Checking, for instance, invariance under Reidemeister move 2 involves proving that certain integral transforms between (derived) categories are equivalences. The existence of such equivalences, which include Seidel-Thomas spherical twists, is a lively area of research which is interesting in itself.
The simplest example of a knot is obtained by taking a tangled-up shoe lace and glueing the ends together. A fundamental question in low-dimensional topology is to determine when two such knots are different (you are allowed to move the knots around without cutting them). One way to do this is to assign a number (an invariant) to each knot so that if two knots are assigned different numbers then they must be different. How to assign such numbers is a deep problem which is related to various areas of mathematics in surprising ways. For example, the Reshetikhin-Turaev invariants are obtained from representation theory (which is the study of matrices). There are also constructions of such invariants using algebra and others that are inspired by physics and string theory. The proposed research suggests yet another approach using algebraic geometry (which is the study of systems of polynomial equations). This approach may extend, unify and shed more light on the other constructions.
