Representation theoretic methods in geometry and topology

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The common theme of this proposal is the use of representation theoretic methods (often inspired by ideas in categorification) to study a) 3-manifold invariants (e.g. Reshetikhin-Turaev invariants), b) categories arising in algebraic geometry (e.g. categories of coherent sheaves or D-modules on moduli spaces), c) (categorified) quantum groups and vertex operator algebras. One of the main tools in low dimensional topology are invariants which can distinguish between different 3-dimensional (or 4-dimensional) manifolds. A series of such invariant were introduced by Reshetikhin and Turaev. These Reshetikhin-Turaev (RT) invariants have deep connections to representation theory and, in particular, to quantum groups. In some ways, these connections are even more fundamental than the original relation to topology. The simplest RT invariant for knots in the 3-sphere is called the Jones polynomial (since it was discovered earlier by Vaughan Jones). In 2001 Mikhail Khovanov showed that the Jones polynomial can be lifted to a more powerful homological invariant (now called Khovanov homology). In subsequent work, jointly with Joel Kamnitzer, I showed that Khovanov homology can also be defined using categories of coherent sheaves on certain iterated Grassmannian bundles. These varieties can be defined using the affine Grassmannian which in turn is related (via geometric Satake) to the representation theory of semisimple Lie algebras. This is part of a sequence of ideas and results highlighting certain deep relations between algebraic geometry, representation theory and topology. The main part of this proposal involves extending this story further. On the topology side one would like to lift the RT invariants to homological invariants of arbitrary 3-manifolds (not just knots in the 3-sphere). On the algebro-geometric side one would like to understand certain categories of coherent sheaves (and D-modules) on more complicated varieties appearing in geometric representation theory. This involves developing new techniques in representation theory of quantum groups as well as categorification (the study of their higher, homological analogues). On the representation theoretic side one has the rich theory of vertex operators. These operators form one way to define the RT 3-manifold invariants. In earlier work with Anthony Licata, we showed how these operators can be categorified by relating them to categories of coherent sheaves on Hilbert schemes of points on surfaces (generalizing work of Nakajima and Grojnowski). I hope to continue this study of "categorified" vertex operators and their relation to geometry. One of the main aims is to use this to lift the RT invariants to homological invariants of all 3-manifolds. The original introduction of vertex operator algebras by Borcherds and Frenkel-Lepowski-Meurman was for the purposes of proving the moonshine conjecture (which is a remarkable relationship between the largest sporadic finite group and the j-function from number theory). Subsequently, understanding the "higher" theory of vertex operators could also lead to a categorical version of moonshine theory. To conclude, the aim of this proposal is to relate certain aspects from several fields: algebraic geometry, representation theory, 3-manifold invariants and categorification. There is a promising, rich and largely unexplored interplay between these areas along the lines sketched out above.