Model theoretic forcing in analysis

We present a framework for model theoretic forcing in a non-first-order context, and present some applications of this framework to Banach space theory

Hyperarithmetic numerals

Within the framework of computable infinitary continuous logic, we develop a system of hyperarithmetic numerals. These numerals are infinitary sentences in a metric language $L$ that have the same truth value in every interpretation of $L$. We prove that every hyperarithmetic real can be represented by a hyperarithmetic numeral at the same level of complexity

On the undefinability of Tsirelson's space and its descendants

We prove that Tsirelson's space cannot be defined explicitly from the classical Banach sequence spaces. We also prove that any Banach space that is explicitly definable from a class of spaces that contain $\ell_p$ or $c_0$ must contain $\ell_p$ or $c_0$ as well