Abstract: This is a follow-up talk from my initial talk, where we introduced the definition and some properties of the Borel construction and the equivariant cohomology of $G$-spaces. In this talk, we aim to highlight several interesting properties of the equivariant cohomology of $G$-spaces through examples. It will also be discussed a couple of additional features for the case of a connected compact Lie group $G$.

Meeting ID: 997 4840 9440 Passcode: 911104

Abstract: Last time, we defined (possibly singular) foliations to be certain collections of vector fields. It was emphasized that, defined this way, singular foliations are not uniquely determined their leaves. In this sequel talk, I will discuss a class of singular foliations I considered in my PhD thesis. These foliations have only two or three leaves total: a closed hypersurface (the singular leaf) and the components of its complement. Depending which vector fields gave the partition, however, interesting holonomy can result along the singular leaf. It turns out this holonomy can be used to completely classify such foliations (localized around the singular leaf). If time permits, I will talk about a question I am currently thinking about: under a suitable orientation hypothesis, is the "fundamental class" of these foliations always nontrivial? The answer to this question hinges on a rather concrete index calculation.