On the Knowledge of God and the Metaphysics of Aquinas

Thomas Aquinas argues in his seminal work, the Summa Theologiae, that one can come to know the existence of God through rational argumentation alone. As a theologian writing a work of theology, he makes his demonstrations concerning God’s existence from the point of view of his Christian faith. And in this, it will be argued, Aquinas is not necessarily mistaken. For his project is to present a grand scheme of reality and man’s place within it. Philosophers have often tried the same, and, like Aquinas, their attempts have been made from a certain point of view. That, it will be shown, is the difference. This paper will present how Aquinas accounts for man’s ability to know generally and then metaphysically, but also how he reasonably presents his theses within the purview of his Christian faith

Boolean functions with small spectral norm

Suppose that f is a boolean function from F_2^n to {0,1} with spectral norm (that is the sum of the absolute values of its Fourier coefficients) at most M. We show that f may be expressed as +/- 1 combination of at most 2^(2^(O(M^4))) indicator functions of subgroups of F_2^n.Comment: 17 pp. Updated references

Monochromatic sums and products

Suppose that $\mathbb{F}_p$ is coloured with $r$ colours. Then there is some colour class containing at least $c_r p^2$ quadruples of the form $(x, y , x + y, xy)$.Comment: 48 pages, accepted for publication in Discrete Analysis. Second version has minor changes arising from the referee report. Third version updated to DAJ format. in Discrete Analysis 2016:

A quantitative version of the idempotent theorem in harmonic analysis

Suppose that G is a locally compact abelian group, and write M(G) for the algebra of bounded, regular, complex-valued measures under convolution. A measure \mu in M(G) is said to be idempotent if \mu * \mu = \mu, or alternatively if the Fourier-Stieltjes transform \mu^ takes only the values 0 and 1. The Cohen-Helson-Rudin idempotent theorem states that a measure \mu is idempotent if and only if the set {r in G^ : \mu^(r) = 1} belongs to the coset ring of G^, that is to say we may write \mu^ as a finite plus/minus 1 combination of characteristic functions of cosets r_j + H_j, where the H_j are open subgroups of G^. In this paper we show that the number L of such cosets can be bounded in terms of the norm ||\mu||, and in fact one may take L <= \exp\exp(C||\mu||^4). In particular our result is non-trivial even for finite groups.Comment: 28 page

The essential ideal in group cohomology does not square to zero

Let G be the Sylow 2-subgroup of the unitary group $SU_3(4)$. We find two essential classes in the mod-2 cohomology ring of G whose product is nonzero. In fact, the product is the ``last survivor'' of Benson-Carlson duality. Recent work of Pakianathan and Yalcin then implies a result about connected graphs with an action of G. Also, there exist essential classes which cannot be written as sums of transfers from proper subgroups. This phenomenon was first observed on the computer. The argument given here uses the elegant calculation by J. Clark, with minor corrections.Comment: 9 pages; three typos corrected, one was particularly confusin