A Generalization of Martin's Axiom

We define the $\aleph_{1.5}$ chain condition. The corresponding forcing axiom is a generalization of Martin's Axiom and implies certain uniform failures of club--guessing on $\omega_1$ that don't seem to have been considered in the literature before.Comment: 36 page

Indestructibility of Vopenka's Principle

We show that Vopenka's Principle and Vopenka cardinals are indestructible under reverse Easton forcing iterations of increasingly directed-closed partial orders, without the need for any preparatory forcing. As a consequence, we are able to prove the relative consistency of these large cardinal axioms with a variety of statements known to be independent of ZFC, such as the generalised continuum hypothesis, the existence of a definable well-order of the universe, and the existence of morasses at many cardinals.Comment: 15 pages, submitted to Israel Journal of Mathematic

Cyclin A/B RxL macrocyclic inhibitors to treat cancers with high E2F activity

Cancer cell proliferation requires precise control of E2F1 activity; excess activity promotes apoptosis. Here, we developed cell-permeable and bioavailable macrocycles that selectively kill small cell lung cancer (SCLC) cells with inherent high E2F1 activity by blocking RxL-mediated interactions of cyclin A and cyclin B with select substrates. Genome-wide CRISPR/Cas9 knockout and random mutagenesis screens found that cyclin A/B RxL macrocyclic inhibitors (cyclin A/Bi) induced apoptosis paradoxically by cyclin B- and Cdk2-dependent spindle assembly checkpoint activation (SAC). Mechanistically, cyclin A/Bi hyperactivate E2F1 and cyclin B by blocking their RxL-interactions with cyclin A and Myt1, respectively, ultimately leading to SAC activation and mitotic cell death. Base editor screens identified cyclin B variants that confer cyclin A/Bi resistance including several variants that disrupted cyclin B:Cdk interactions. Unexpectedly but consistent with our base editor and knockout screens, cyclin A/Bi induced the formation of neo-morphic Cdk2-cyclin B complexes that promote SAC activation and apoptosis. Finally, orally-bioavailable cyclin A/Bi robustly inhibited tumor growth in chemotherapy-resistant patient-derived xenograft models of SCLC. This work uncovers gain-of-function mechanisms by which cyclin A/Bi induce apoptosis in cancers with high E2F activity, and suggests cyclin A/Bi as a therapeutic strategy for SCLC and other cancers driven by high E2F activity

Set Theory and Structures

Set-theoretic and category-theoretic foundations represent different perspectives on mathematical subject matter. In particular, category-theoretic language focusses on properties that can be determined up to isomorphism within a category, whereas set theory admits of properties determined by the internal structure of the membership relation. Various objections have been raised against this aspect of set theory in the category-theoretic literature. In this article, we advocate a methodological pluralism concerning the two foundational languages, and provide a theory that fruitfully interrelates a `structural' perspective to a set-theoretic one. We present a set-theoretic system that is able to talk about structures more naturally, and argue that it provides an important perspective on plausibly structural properties such as cardinality. We conclude the language of set theory can provide useful information about the notion of mathematical structure