Pre-talk: Foliations and the boundedness of algebraic varieties
The MMP organizes the classification of algebraic varieties, and a central theme is boundedness: varieties with fixed invariants should fall into finitely many families. After Hacon–McKernan–Xu and Birkar this is well understood for general type and Fano varieties, but the picture for fibered varieties is far less complete. Aimed at a general audience, this pre-talk explains what boundedness means and why it matters, introduces (algebraically integrable) foliations and adjoint foliated structures as a flexible language for studying fibrations, and sets up the questions of the main talk. No prior familiarity with foliations is assumed.
Main talk: Boundedness of stable families and algebraically integrable foliations
I will discuss recent joint work with Paolo Cascini, Calum Spicer, and Roberto Svaldi: normal projective stable families of maximal variation, of fixed dimension and bounded adjoint volume, are birationally bounded. This follows from a stronger intrinsic statement — algebraically integrable foliations of fixed dimension and bounded adjoint volume are log birationally bounded — so the birational geometry of foliations gives a systematic framework for boundedness of fibrations. A key ingredient is a proof of McKernan's ACC conjecture for interpolated log canonical thresholds of algebraically integrable foliations, the foliated analogue of the Shokurov ACC theorem of Hacon–McKernan–Xu. As applications I discuss boundedness criteria for Fano foliations and several ACC-type results. Builds on the MMP for adjoint foliated structures (arXiv:2408.14258, 2504.10737, 2510.02498, 2510.04419).