The finite real spectral triples can be classified up to unitary equivalence according to Krajewski diagrams, and if each data except the Dirac operator $D$ of a finite real spectral triple is fixed, which is called a "fermion space", then it is easy to show that the set $\mathcal{G}$ of all the Dirac operators over this fermion space forms a vector space. If $\mathcal{G}$ is enriched with some measure, then we can consider the integral over $\mathcal{G}$. Here I am going to consider only a special kind of finite real triple: the type $(p, q)$ fuzzy space. And I will try to compute the integral $\int_{\mathcal{G}}e^{-\mathrm{Tr}D^{2}}\mathrm{d}D$ for the easiest type $(1, 0)$ fuzzy space.

I will present Brion's proof that the equivariant cohomology of a (sufficiently) smooth toric variety is given by the piecewise polynomials on the associated fan or, equivalently, by the Stanley-Reisner ring of the fan. If time permits, I will discuss how to obtain the non-equivariant cohomology as a consequence, both in the equivariantly formal and the general case.

The study of quadratic forms seems far removed from the study of (central) simple algebras, and in general, this is indeed the case. However, if we assume the central simple algebra carries an involution (such as matrix algebras and quaternion algebras) then we have a different story. In this talk, we will study the relationship between algebras with an involution and quadratic forms (assuming characteristic is not 2). In particular, we will discuss the relevance of cohomological invariants as a classification tool in the study of quadratic forms, in hopes of trying to extend what we have learned to central simple algebras with involution.