Uniform bounds in F-finite rings and their applications

This dissertation establishes uniform bounds in characteristic p rings which are either F-finite or essentially of finite type over an excellent local ring. These uniform bounds are then used to show that the Hilbert-Kunz length functions and the normalized Frobenius splitting numbers defined on the spectrum of a ring converge uniformly to their limits, namely the Hilbert-Kunz multiplicity function and the Fsignature function. From this we establish that the F-signature function is lower semicontinuous. Lower semi-continuity of the F-signature of a pair is also established. We also give a new proof of the upper semi-continuity of Hilbert-Kunz multiplicity, a result originally proven by Ilya Smirnov.

Splitting Numbers of Links

The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering links and Alexander invariants. As an application, we completely determine the splitting numbers of links with nine or fewer crossings. Also, with these techniques, we either reprove or improve upon the lower bounds for splitting numbers of links computed by Batson and Seed using Khovanov homology.

The lower semicontinuity of the Frobenius splitting numbers

We show that, under mild conditions, the (normalized) Frobenius splitting numbers of a local ring of prime characteristic are lower semicontinuous.

Splitting Numbers of Grids

For a subset $S$ of a finite ordered set $P$, let $$S\!\uparrow\;=\{x\in P:x\ge s \hbox{ for some }s\in S\}\quad\hbox{and} \quad S\!\downarrow\;=\{x\in P:x\le s \hbox{ for some }s\in S\}.$$ For a maximal antichain $A$ of $P$, let $$s(A)=\max_{A=U\cup D}{|U\!\uparrow|+|D\!\downarrow|\over|P|}\ ,$$ the maximum taken over all partitions $U\cup D$ of $A$, and $$s_k(P)=\min_{A\in {\cal A}(P),|A|=k}s(A)$$ where we assume $P$ contains at least one maximal antichain of $k$ elements. Finally, for a class ${\cal C}$ of finite ordered sets, we define $$s_k({\cal C})=\inf_{P\in {\cal C}}s_k(P).$$ Thus $s_k({\cal C})$ is the greatest proportion $r$ satisfying: every $k$-element maximal antichain of a member $P$ of ${\cal C}$ can be "split" into sets $U$ and $D$ so that $U\!\uparrow\cup\; D\!\downarrow$ contains at least $r|P|$ elements. In this paper we determine $s_k({\cal G}_k)$ for all $k\ge 1$, where ${\cal G}_k=\{{\bf k}\times{\bf n}:n\ge k\}$ is the family of all $k$ by $n$ "grids".