| Directed Topology
Spaces often come equipped with directionality, like vector fields, partial orders, and simplicial orientations. Directed algebraic topology studies algebraic invariants on such spaces that can capture information about the topology and directionality. | Directed (co)homology refines classical (co)homology to detect directed structure on a directed space. Recently, I've constructed such theories and shown that they exhibit a Poincare Duality between such cohomology and homology theories. Examples of directed homology and cohomology respectively include the flows and Poincare sections of dynamical systems. Together with Eric Goubault and Emmanuel Haucourt, I've implicitly used directed homology to identify a large class of locally preordered spaces whose universal covers have antisymmetric global orders [3]. | | Directed homotopy theory studies directed spaces up to deformations respecting directionality. In the past, I've developed cubical and simplicial approximations [6], shown how continuous lattices from domain theory have trivial directed homotopy types [2], and generalized such lattices to a convenient category for directed homotopy theory [1]. | |
| Applied Topology
| Data analysis, which often takes the form of global geometric inferences from sample points whose local metric structure only is known, naturally lends itself to sheaf-theoretic methods. Together with Rob Ghrist, Dave Lipsky, Michael Robinson, Hank Owen, and Mike Stein, I'm working on software that analyzes video imagery by means of homological invariants on sheaves. | | Optimization dualities sometimes make seemingly intractable computations simple and fast. Recently, I've been working on interpreting and generalizing flow-cut dualities as special cases of a Poincare Duality for sheaves on directed graphs [8]. Together with Rob Ghrist, I'm interested in applying such a generalized flow-cut duality to handle logical, stochastic, and multicommodity constraints on directed graphs [7] and higher dimensional spaces. | | Semantics and Homotopy The use of directed spaces to model computation dates back to early work on the Lambda Calculus. Recently, I've become interested in directed extensions of homotopy type theory. Together with Eric Goubault and Emmanuel Haucourt, I've worked in the past on using directed homotopy theory to automate the formal verification of large, concurrent programs [4]. | |