The proposal is to construct 3-manifold invariants using tools and techniques from algebraic geometry and geometric representation theory. Given a Lie algebra one can define a numerical invariant of 3-manifolds called the Witten-Reshetikhin-Turaev invariant. It is an open problem to lift these to homological invariants of 3-manifolds. This is analogous to the way (singular) homology of topological spaces lifts the Euler characteristic. In earlier work of the PI jointly with Joel Kamnitzer they were able to do this for complements of links inside the 3-sphere when the Lie algebra is sl(n) and the link labeled by fundamental representations. To do this they used certain (derived) categories of coherent sheaves on flag-like varieties and the hope is to generalize this approach to other 3-manifolds. Their constructions were algebro-geometric but used many techniques from representation theory. Inspired by representation theory the PI plans to develop other tools to study varieties and their categories of coherent sheaves. For example, he plans to study actions of the Heisenberg Lie algebra on categories of sheaves on Hilbert schemes.
The simplest example of a 3-manifold is the 3-dimensional world we live in. A little more complicated is what you obtain by cutting out a doughnut or perhaps an object with more holes. However, there are even more complicated 3-manifolds which are harder to describe. A fundamental problem in low-dimensional topology is how to tell whether two given 3-manifolds are "the same" or not ("the same" has a very precise mathematical definition in this case). A 3-manifold invariant is a tool for doing this. Their construction leads one to study other fields of mathematics and theoretical physics such as representation theory (the study of matrices) and algebraic geometry (the study of systems of polynomial equations). This rich connection to other fields is one of the attractive features of this problem.
