Homological algebra and topology in three and four dimensions

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The project aims to develop and understand homological invariants of three- and four-dimensional topological objects. Such invariants include Donaldson-Floer and Seiberg-Witten theories, and several bigraded homology theories of links. These theories have the Alexander and Jones polynomials as their Euler characteristics, as well as the quantum sl(3) link invariant. We would like to construct a bigraded homology theory of links for each complex simple Lie algebra g, with components of links colored by irreducible representation of g. The Euler characteristic of the theory should be the quantum invariant of colored links associated with the quantum deformation of g, and the theory should be functorial (extend to cobordisms of links). Our other goals include better understanding of the relations between existing theories, and an investigation of the categories that appear when link homology theories are extended to tangles. Topological objects in dimensions three and four have special properties and a number of connections to algebra and analysis that do not generalize to other dimensions. Three-dimensional objects, including knots, links, and three-manifolds (the latter are global objects glued out of three-dimensional spaces), admit combinatorial invariants, also known as quantum invariants, that come from algebraic structures and can also be recovered from two-dimensional conformal field theories. Quantum invariants of four-dimensional objects, for the most part, are not known to have combinatorial descriptions, and their definition and computation requires analytical tools. We would like to bridge this gap by constructing new four-dimensional invariants that are combinatorial, and, second, by finding combinatorial description of known analytical invariants of four-manifolds, including Donaldson-Floer and Seiberg-Witten invariants.