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I'm interested in understanding dynamics by computing algebraic invariants on the causal directionality latent in state spaces. Dynamics of interest include traffic flow, solutions to Einstein dynamics, and solutions to world problems on monoids. State spaces can range from directed graphs to spacetimes to classifying spaces.
Spaces often come equipped with directionality (e.g. vector fields, partial orders, simplicial orientations). Directed algebraic topology assigns algebraic invariants (e.g. fundamental categories, homology semimodules) to such spaces.
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 them [9]. 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].
Applications of Topology
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 of semigroups 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.
Formal verification is necessary to ensure the safety of critical software, like avionics and multiplayer videogames. Together with Eric Goubault and Emmanuel Haucourt, I've worked in the past on using directed homotopy theory to ignore point-set issues like size and detect essential system behavior (e.g. deadlocks, crashes) in theory [4] and practice.
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 currently working to develop target-tracking software for video imagery of vehicular traffic.
R. Ghrist, S. Krishnan, "A Topological Max-Flow Min-Cut Theorem," to appear in Proceedings of Global Conference on Signal and Information Processing (2013).
J. Feng, K. Martin, S. Krishnan, "A free object in quantum information theory," Electronic Notes in Theoretical Computer Science, (2010), vol. 265, pp. 35-47.
E. Goubault, E. Haucourt, S. Krishnan, "Future path-components in directed topology," MFPS 2010 Proceedings, Electronic Notes in Theoretical Computer Science, (2010), vol. 265, pp. 325-335.
S. Krishnan, "Criteria for homotopic maps to be so along monotone homotopies," Electronic Notes in Theoretical Computer Science (2009), vol. 230, pp. 141-148.