## August 9: Morse field theory

### Robin Koytcheff

I will talk about work of R. Cohen and P. Norbury, who construct operations on the
(co)homology of a manifold M. They consider graphs with metrics and some numbers of
"incoming" and "outgoing" leaves, and map them into M according to gradient flow lines of
functions on M. Such "metric graph flows" form a moduli space analogous to the moduli
space of holomorphic curves in a symplectic manifold in Gromov--Witten theory. The
operations can actually be constructed in equivariant (co)homology, where the automorphism
group of a graph acts on products of M. They also have the structure of a TQFT, where one
views the graphs as generalized (0+1)-dimensional manifolds. The cup product,
intersection pairing, Steenrod squares, and Stiefel--Whitney classes all arise as examples
of these operations.