Seminario di Analisi
Dipartimento di Matematica e Applicazioni
“Renato Caccioppoli”
Universit`a degli Studi di Napoli Federico II
Marted`ı 7 Novembre 2023 – Aula C, Dip. Matematica e Applicazioni, h. 15:45
KEITH ROGERS
IMPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE
USING SEMIALGEBRAIC GEOMETRY

ABSTRACT. The Kakeya problem considers tubes which point in different directions and
the extent to which they can be compressed by positioning them strategically. On the one
hand, we will see that the measure of any semialgebraic set that contains the tubes must
satisfy the expected bound. That is to say, the tubes cannot be compressed too much
if they are positioned in an algebraic way. The proof employs tools from real algebraic
geometry including the Tarski-Seidenberg projection theorem and the Gromov-Yomdin
algebraic lemma. On the other hand, the expected bound holds in the absence of algebraic structure, by polynomial partitioning. Balancing between the two extremes yields
improved bounds for the Kakeya maximal conjecture in higher dimensions. This is joint
work with J. Hickman, N. Katz and R. Zhang