Nonhomogeneous analytic families of trees

The Pila-Wilkie theorem for subanalytic families: a complex analytic approach

We present a complex analytic proof of the Pila-Wilkie theorem for subanalytic sets. In particular, we replace the use of $C^r$-smooth parametrizations by a variant of Weierstrass division.

Motivic volume of families of polarized rigid-analytic tori

Let $k$ be a non-Archimedean rational valued field. We construct the moduli space of linearly rigidified polarized analytic tori over $k$ that admit rigid-analytic uniformization by an algebraic torus and observe that it is in definable rigid subanalytic bijection with a $PGL_N$-bundle over a polyhedral domain in an algebraic torus. We use this observation to prove that the Hrushovski-Kazhdan motivic volume of a non-Archimedean semi-algebraic family of Abelian varieties admitting such a uniformization fibrewise vanishes. This question is motivated by the conjectural geometric interpretation of…

Van Douwen 族、可乘性与超滤极大性

Van Douwen Families, Productivity and Ultrafilter Maximality

本文研究了理想化的极大最终不同族，建立了 Van Douwen 族与 MAD 族之间的联系，引入新的可乘性概念，并给出解析族的存在性与不存在性结果。

Analytic P-ideals and Banach spaces

We study the interplay between Banach space theory and theory of analytic P-ideals. Applying the observation that, up to isomorphism, all Banach spaces with unconditional bases can be constructed in a way very similar to the construction of analytic P-ideals from submeasures, we point out numerous symmetries between the two theories. Also, we investigate a special case, the interactions between combinatorics of families of finite sets, topological properties of the "Schreier type" Banach spaces associated to these families, and the complexity of ideals generated by the canonical bases in these…

On the definability of mad families of vector spaces

We consider the definability of mad families in vector spaces of the form $\underset{n<ω}{\bigoplus} F$ where $F$ is a field of cardinality $\leq \aleph_0$. We show that there is no analytic mad family of subspaces when $F=\mathbb{F}_2$, partially answering a question of Smythe. Our proof relies on a variant of Mathias forcing restricted to a certain idempotent ultrafilter whose existence follows from Glazer's proof of Hindman's theorem.

There is a Van Douwen MAD family

We prove in ZFC that there is a MAD family of functions in omega^omega which is also maximal with respect to infinite partial functions. This solves a 20 year old question of Van Douwen. We also strengthen a result of J. Steprans stating that strongly MAD families of functions cannot be analytic. We show that analytic MAD families of functions, if they exist, must satisfy some strong constraints.