REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: The object of study is a non-singular gradient vector field v on a smooth manifold X with boundary. Specifically, I study the ways in which the v-flow interacts with boundary dX. Here is a central question: What kind of residual structure on the boundary dX will allow for a reconstruction of X (perhaps, together with the flow)? If such a reconstruction of X (or of (X, v)) is possible, the structure on the boundary deserves the name "holographic". For each point x of the boundary dX where the field is pointing inwards of X, consider the closest point B_v(x) in dX where the trajectory through x exits X or is tangent to its boundary. We get a map B_v : d⁺X --> d^-X which we call "ballistic" (B_v is only semi-continuous). It turns out that the ballistic map B_v is an example of the desired holographic data.
Theorem. For a generic v on X, the piecewise smooth type of (X, v) is reconstructable from the ballistic map B_v.
So the map B_v is an example of a more general notion of holographic structure H(dX) on dX. Again, H(dX) allows for a reconstruction of the bulk of X, together with the dynamics of the v-flow. If time permits, I will talk about the characteristic classes of ballistic structures. They are intimately linked to an interesting combinatorics.