Patrick Brosnan

Steenrod operations in Chow theory

An action of the Steenrod algebra is constructed on the mod p Chow theory of varieties over a field of characteristic different from p, answering a question posed in Fultons Intersection Theory. The action agrees with the action of the Steenrod algebra used by Voevodsky in his proof of the Milnor conjecture. However, the construction uses only basic functorial properties of equivariant intersection theory.

Matroids motives, and a conjecture of Kontsevich

Let G be a finite connected graph. The Kirchhoff polynomial of G is a certain homogeneous polynomial whose degree is equal to the first betti number of G. These polynomials appear in the study of electrical circuits and in the evaluation of Feynman amplitudes. Motivated by work of D. Kreimer and D. J. Broadhurst associating multiple zeta values to certain Feynman integrals, Kontsevich conjectured that the number of zeros of a Kirchhoff polynomial over the field with q elements is always a polynomial function of q. We show that this conjecture is false by relating the schemes defined by Kirchhoff polynomials to the representation spaces of matroids. Moreover, using Mnevs universality theorem, we show that these schemes essentially generate all arithmetic of schemes of finite type over the integers.

Periods and Igusa local zeta functions

We show that the coefficients in the Laurent series of the Igusa local zeta functions I(s) = ∫ C fω are periods. This is proved by first showing the existence of functional equations for these functions. This will be used to show in a subsequent paper (by P. Brosnan) that certain numbers occurring in Feynman amplitudes (up to Gamma factors) are periods. We also give several examples of our main result, and one example showing that Euler’s constant γ is an exponential period.

On the algebraicity of the zero locus of an admissible normal function

We show that the zero locus of an admissible normal function on a smooth complex algebraic variety is algebraic.

Zero loci of admissible normal functions with torsion singularities

We show that the zero locus of a normal function on a smooth complex algebraic variety S is algebraic provided that the normal function extends to a admissible normal function on a smooth compactification of S with torsion singularity. This result generalizes our previous result for admissible normal functions on curves [arxiv:math/0604345 [math.AG]]. It has also been obtained by M. Saito using a different method in a recent preprint [arXiv:0803.2771v2].