Last time we described the Weil model for circle actions and showed that it is isomorphic to polynomial ring with coefficients in invariant forms. This week, we will prove the Cartan-Weil Isomorphism for general connected Lie groups and determine the Cartan differential. Then we will determine the Cartan model for torus actions. Lastly, we will compute the equivariant cohomology for torus action on a point.

Meeting ID: 997 4840 9440 Passcode: 911104

In this talk we will return our attention to matrix integrals and maps. I will review the definition and compare to the Feynman integrals previously discussed.

Levi-flat hypersurfaces are important in the context of minimal sets of holomorphic foliations in complex projective spaces. It is well-known that closed smooth real analytic Levi-flats do not exist in $\mathbb C \mathbb P^n$ for $n>2$, and it is an open problem if such hypersurfaces exist in $\mathbb C \mathbb P^2$. On the other hand, singular real-analytic Levi-flats exist in any dimension, and we discuss some recent developments in this subject.