A key tool in equivariant cobordism is an injective map $\nu$ from $\{\!$complex manifolds with torus action$\}$/(cobordism) to a less intimidating ring, which in the case of isolated fixed points is just given by listing of induced torus representations on tangent spaces to fixed points.
This map would be more useful if its image were actually known. Karshon, Ginzburg, and Tolman identified an upper bound: a classical localization theorem due to Atiyah–Bott/Berline–Vergne computes certain integrals $\int_M \omega$ to be polynomial expressions in weights of tangent representations, and such an $0$ integral is $0$ by definition when $|\omega| < \dim M$, the representations somewhat.
Somewhat comically, these are all the restrictions we know on the image of $\nu$.
Is it really possible this is the only constraint? In other words, is any conforming list of representations actually the fixed point data of some manifold?
Yes! (At least if every orbit is either free or a fixed point.) In this talk, we will use the localization theorem and 19th-century-style arithmetic manipulations to recover a characterization of semifree complex cobordism proved by Sinha in 2004.
Care will be taken to make this version accessible to everyone, but feel free to slow me down with questions. Alternatively, as I continue to suffer from extreme stage fright, you could always find other, more rewarding was to spend your Friday morning.