Title: Directed Algebraic Topology
Speaker: Sanjeevi Krishnan
Abstract: The deduction of qualitative properties of a dynamic process from homotopy invariants on an associated topological state space dates back to the origins of algebraic topology. However, state spaces have additional structure beyond mere topology. Directed algebraic topology is a refinement of algebraic topology for state spaces equipped with some directionality, like spacetimes or classifying spaces of small categories.
Directed invariants are sometimes necessary to make applications of topology useful in studying real-world dynamic processes. For example, the functionality of a microprocessor can be coded up as local, semigroup-valued coefficients over a directed graph. The directed cohomology of a directed graph equipped with these local coefficients classifies behavior of the asynchronous microprocessor invisible to classical cohomology, such as the presence of race conditions. While the calculation of ordinary cohomology involves linear algebra, the calculation of directed cohomology involves a blend of linear programming and rewriting theory. Given the applications of directed algebraic topology to engineered systems explored over the past 20 years, it is natural to ask about applications to other dynamical sorts of systems.
For example, what does the directed cohomology of a spacetime tell us about global asymptotic behavior? Some toy calculations on closed timelike surfaces can be made, but calculations of interest will require an expansion of techniques in directed homotopy.
Note: This is part of the Invitations to Mathematics lecture series given each year in Autumn Semester. Pre-candidacy students can sign up for this lecture series by registering for one or two credit hours of Math 6193, class #9226 (with Prof H. Moscovici).
