# Rigidity properties of precipitous ideals

## Brent Cody

### Virginia Commonwealth University

An ideal $I$ on a set $X$ is called *rigid* if forcing with $\mathcal{P}(X)/I$ produces an extension $V[G]$ in which there is a unique $V$-generic filter for $\mathcal{P}(X)/I$. Note that this implies that $\mathcal{P}(X)/I$ has only the trivial automorphism. As part of his analysis of $\mathbb{P}_{\text{max}}$ forcing, Woodin showed that under $\text{MA}_{\omega_1}$, every normal, uniform, saturated ideal on $\omega_1$ is rigid. Indeed, in all previously known models which have rigid precipitous ideals on $\omega_1$ one also has $\lnot\text{CH}$. This leaves open the question: is it consistent to have $\text{CH}$ along with a rigid precipitous ideal on $\omega_1$? I will discuss some recent work which shows that if $\kappa$ is almost huge then there is a forcing extension $V[G]$ in which $\kappa=\omega_1^{V[G]}$, there is a rigid presaturated ideal $I$ on $\omega_1^{V[G]}$ and $\text{CH}$ holds. The proof of this result involves a certain coding forcing first used in the Friedman-Magidor results on the number of normal measures. This is joint work with Sean Cox and Monroe Eskew.

Brent Cody is a visiting assistant professor at the Virginia Commonwealth University. His research is in the field of set theory and involves using forcing and large cardinals to prove that certain statements are independent of the axioms of mathematics, ZFC. He received his doctorate from the CUNY Graduate Center in 2012 under the supervision of Joel David Hamkins and subsequently held postdoctoral positions at the Fields Institute in Toronto and the University of Prince Edward Island.