Talk

This talk will be a partial survey of topics in homological commutative local algebra. In particular, I will discuss results on depth, homological dimensions, and torsion properties of tensor products and powers, as well as characterizations of Gorenstein and regular rings via (co)homology vanishing. I will also mention related conjectures and open problems in this area.

Asymptotic counting methods will be used to study certain basic questions on the existence of rational points on surfaces. Joint work with Christopher Frei.

It is known that the partition function $p(n)$ obeys Benford's law in any integer base $b\ge 2$. A similar result was obtained by Douglass and Ono for the plane partition function ${\text{\rm PL}}(n)$ in a recent paper. In their paper, Douglass and Ono asked for an explicit version of this result. In particular, given an integer base $b\ge 2$ and string $f$ of digits in base $b$ they asked for an explicit value $N(b,f)$ such that there exists $n\le N(b,f)$ with the property that ${\text{\rm PL}}(n)$ starts with the string $f$ when written in base $b$. In my talk, I will present an explicit value for $N(b,f)$ both for the partition function $p(n)$ as well as for the plane partition function ${\text{\rm PL}}(n)$.