My research field, index theory, has its roots in vector calculus, where Green’s Theorem helps prove that an irrotational vector field (closed differential one-form) on a planar region D is guaranteed to be conservative (exact) on D if and only if D is simply connected (i.e., every loop in D can be contracted to a point in D). This result can be regarded as a manifestation in analysis of the topological invariant known as the winding number or as a topological constraint on the solution set of a system of partial differential equations.
The problem of finding a potential function for an irrotational vector field (or of integrating a closed differential one-form) is typical of the problems addressed by algebraic topology in that it involves the question of whether local solutions can be patched together to form a global solution. Because calculus operations are defined locally, the relationship between partial differential equations and topology extends to the locally Euclidean spaces known as smooth manifolds. I study index theory on spaces that are not manifolds. Part of the challenge is to find fruitful definitions of analysis on these singular spaces. Tools from algebraic topology, such as K-theory, guide the search for these definitions.