Krivine's theorem and related results

Sang Woo Ryoo, Princeton University
Fine Hall 110

In this talk, I will present Krivine's theorem and related statements. Given a Banach space, we wish to find subspaces that we can understand well. In particular, we wish to find subspaces that are close to $\ell_p$, $1<p<\infty$ or $c_0$. Krivine's theorem states that if we are given a non-degenerate basis (to be defined) of the Banach space, we can represent one of $\ell_p$, $1< p<\infty$ or $c_0$ on a block subbasis. I will sketch the proof of this theorem, which incorporates approximate eigenvalues and an application of Ramsey theory. If time permits, I will also mention quantitative bounds for a finite-dimensional version of Krivine's theorem, and a specific choice of $\ell_p$, $1<p<\infty$ for Krivine's theorem.