The endpoint case of the Bennett–Carbery–Tao multilinear Kakeya conjecture

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We prove the endpoint case of the multilinear Kakeya conjecture of Bennett, Carbery and Tao. The proof uses the polynomial method introduced by Dvir.

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Aspects of Multilinear Harmonic Analysis Related to Transversality (2014, background): “Perhaps the most striking example is the algebraic-geometric approach to the endpoint multilinear Kakeya conjecture developed by Guth in [33].”
The proof of the l 2 Decoupling Conjecture (2015, methodology): “Indeed, the Multilinear Restriction Theorem 6.1 that we use repeatedly in the proof of our main Theorem 1.1 relies at its core on the multilinear Kakeya phenomenon, which has some topological ﬂavor (see [24], [16]).”
Minimal 2-spheres in 3-spheres (2017, background): “The following vanishing lemma is the essential topological ingredient needed in the proof (for a short proof of the lemma, see the Appendix of [15]):”
Algebraic methods in discrete analogs of the Kakeya problem (2008, methodology): “Both our proofs are adaptations of Dvir’s argument for the finite field Kakeya problem.”