TBA

Despite its being an important universal object in equivariant homotopy theory, concrete generators-and-relations presentations for the coefficient ring of equivariant complex cobordism with respect to a compact abelian Lie group $G$ are still known only for finite $G$.

For $G$ a torus, Ginzburg--Karshon--Tolman observed that the well-known fixed-point integral localization formula of Atiyah--Bott--Berline--Vergne determines a naive upper bound on this ring, and the geometrically important case of a so-called GKM action, leaning on work of Darby, Carlson--Gamse--Karshon showed this bound is an equality.

The author has recently shown the same for semifree circle actions with isolated fixed points, unexpectedly recovering a 2004 result of Sinha with a new proof that is classical in the literal sense: it would have been accessible in the era of Beethoven. In this talk we will give background and sketch this proof.

The Spectral Action principle of Noncommutative Geometry has been extremely successful to understand the mathematical structure of the Standard Model of particle physics. However, as a classical framework, it does not yet encompass the corrections of renormalization, which are crucial to understand the quantized version of any field theory. In this talk, I will describe a first attempt to describe renormalization directly within the setting of Noncommutative Geometry, in the case of Yukawa interactions. In particular, I will show that Wetterich's theory of functional renormalization, when applied to a matrix model inspired from the Dirac operator of the Noncommutative Standard Model, reduces to the usual Feynman diagram approach.