In this lecture we give an introduction to the Grothendieck ring of algebraic varieties, and discuss Kapranov’s lifting of the Hasse-Weil zeta function to this Grothendieck ring. One interesting feature is that this makes sense over an arbitrary field. We will prove the rationality of Kapranov’s zeta function for curves by a variant of the argument used in Lecture 4 for the Hasse-Weil zeta func...

v det([1− ρ(Frobv)q v ]|W Iv )−1, W Iv = subspace of W fixed by inertia at v. Grothendieck gave a related description of ζ∗ X using continuous p-adic representations GF → GL(H ét,c(XF ,Qp)) =: GL(W ). These are unramified almost everywhere, including at all good places away from p. Here the ith cohomology group W i above vanishes for i > 2 dimX. Grothendieck proved that if we remove the contrib...

We give a general construction of rings graded by the conjugacy classes of a finite group. Some examples of our construction are the Hochschild cohomology ring of a finite group algebra, the Grothendieck ring of the Drinfel’d double of a group, and the orbifold cohomology ring for a global quotient. We generalize the first two examples by deriving product formulas for the Hochschild cohomology ...

We study the extendibility of integral vector-valued polynomials on Banach spaces. We prove that an X-valued Pietsch-integral polynomial on E extends to an X-valued Pietsch-integral polynomial on any space F containing E, with the same integral norm. This is not the case for Grothendieck-integral polynomials: they do not always extend to X-valued Grothendieck-integral polynomials. However, they...

We obtain semantic characterizations, holding for any Grothendieck site (C, J), for the models of a theory classified by a topos of the form Sh(C, J) in terms of the models of a theory classified by a topos [C,Set]. These characterizations arise from an appropriate representation of flat functors into Grothendieck toposes based on an application of the Yoneda Lemma in conjunction with ideas fro...

Pietsch Factorization and Grothendieck Factorization are the two landmark theorems in modern functional analysis. They were first introduced to the numerical linear algebra community by the work of Joel A. Tropp in the column subset selection problem, which seeks to extract from a matrix a column submatrix that has lower spectral norm. Despite their broad application in functional analysis, the...

in the Chow ring with rational coefficients CH(S)Q = ⊕nCH (S)Q. Here ch is the Chern character and Td(TX), Td(TS) stand for the Todd power series evaluated at the Chern classes of the tangent bundle of X, respectively S. Since both sides of (1.1) take values in CH(S)Q := CH (S)⊗Q, only information modulo torsion about the Chern classes of f∗[F ] can be obtained from this identity. The goal of o...

A general path-lifting theorem, which fails in the context of topological spaces, is shown to hold for toposes, for locales (a slight generalization of topological spaces), and hence for complete separable metric spaces. This result generalizes the known fact that any connected locally connected topos (respectively complete separable metric space) is path-connected. In [MW] we proved that every...

We caution that # Fix(ψ) has to be interpreted with some care: it really denotes a fixed point count with multiplicities. A better way to phrase it as follows. Inside M ×M we have two submanifolds: the diagonal ∆ and the graph Γψ. We can then try to take # Fix(ψ) := ∆ · Γψ = “intersection number of ∆ and Γψ”. When ∆ and Γψ intersect transversely this intersection number coincides with the naïve...