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My current interests are algebraic topology and its applications to physics.
My recent work has been applying the theory of partial algebras over an operad to intersection of geometric chains in a manifold. I have proven a result for extending partial algebras over operads of complexes and that the geometric chains of a manifold fit into this context.
As a result, one obtains an E-infinity algebra on a complex quasi- isomorphic to the the geometric chains of a manifold. Determining what this structure says about a manifold is work in progress. Using similar techniques, I've shown that the cochains of a oriented manifold have the structure of an E-infinity algebra that is induced by intersection. By a recent theorem of Mandel, under mild assumptions, this structure determines the weak homotopy type of the space. Lastly, much of this work may be applied to yield chain level versions of Chas and Sullivan's operations in String Topology.
In another (not unrelated) project, I have developed several algebraic structures on the cochains of a triangulated smooth manifold which are analogous to previously known algebraic structure in the smooth setting. Examples of this are the wedge product of forms, the Hodge star operator, and the period matrix of a suface. I have shown that under subdivision of a triangulation, these combinatorial structures convergence to the smooth ones.
The applications of such structures are numerous. In particular, they allow one to make discrete approximations to smooth phenomena, such as fluid flow, so that meaningful computations and simulations may be carried out on a computer. Also, these structures may be of use in discrete physical theories such as statistical mechanics and lattice field theory. This is work in progress.