Gabriel Katz

Flows in Flatland: A Romance of Few Dimensions

This paper is about gradient-like vector fields and flows they generate on smooth compact surfaces with boundary. We use this particular 2-dimensional setting to present and explain our general results about non-vanishing gradient-like vector fields on n-dimensional manifolds with boundary. We take advantage of the relative simplicity of 2-dimensional worlds to popularize our approach to the Morse theory on smooth manifolds with boundary. In this approach, the boundary effects take the central stage.

THE BURNSIDE RING-VALUED MORSE FORMULA FOR VECTOR FIELDS ON MANIFOLDS WITH BOUNDARY

Let G be a compact Lie group and A(G) its Burnside Ring. For a compact smooth n-dimensional G-manifold X equipped with a generic G-invariant vector field v, we prove an equivariant analog of the Morse formula

Equivariant semicharacteristics and induction

{\rm Ind}^G(v) = \sum_{k = 0}^{n} (-1)^k \chi^G(\partial_{k}^{+}X)

USING SIMPLICIAL VOLUME TO COUNT MULTI-TANGENT TRAJECTORIES OF TRAVERSING VECTOR FIELDS

which takes its values in A(G). Here IndG(v) denotes the equivariant index of the field v,

Local formulae in equivariant bordisms

\{\partial_{k}^{+}X\}

THE STRATIFIED SPACES OF REAL POLYNOMIALS AND TRAJECTORY SPACES OF TRAVERSING FLOWS

the v-induced Morse stratification (see [10]) of the boundary ∂X, and

Causal Holography of Traversing Flows

\chi^G(\partial_{k}^{+}X)

CONVEXITY OF MORSE STRATIFICATIONS AND GRADIENT SPINES OF 3-MANIFOLDS

the class of the (n - k)-manifold