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One of my favourite family of conjectures (and one that has preoccupied a significant fraction of my own research) is the family of Kakeya conjectures in geometric measure theory and harmonic analysis. There are many (not quite equivalent) conjectures in this family. The cleanest one to state is the set conjecture:
Kakeya set conjecture: Let , and let contain a unit line segment in every direction (such sets are known as Kakeya sets or Besicovitch sets). Then E has Hausdorff dimension and Minkowski dimension equal to n.
One reason why I find these conjectures fascinating is the sheer variety of mathematical fields that arise both in the partial results towards this conjecture, and in the applications of those results to other problems. See for instance this survey of Wolff, my Notices article and this article of Łaba on the connections between this problem and other problems in Fourier analysis, PDE, and additive combinatorics; there have even been some connections to number theory and to cryptography. At the other end of the pipeline, the mathematical tools that have gone into the proofs of various partial results have included:
- Maximal functions, covering lemmas, methods (Cordoba, Strömberg, Cordoba-Fefferman);
- Fourier analysis (Nagel-Stein-Wainger);
- Multilinear integration (Drury, Christ)
- Paraproducts (Katz);
- Combinatorial incidence geometry (Bourgain, Wolff);
- Multi-scale analysis (Barrionuevo, Katz-Łaba-Tao, Łaba-Tao, Alfonseca-Soria-Vargas);
- Probabilistic constructions (Bateman-Katz, Bateman);
- Additive combinatorics and graph theory (Bourgain, Katz-Łaba-Tao, Katz-Tao, Katz-Tao);
- Sum-product theorems (Bourgain-Katz-Tao);
- Bilinear estimates (Tao-Vargas-Vega);
- Perron trees (Perron, Schoenberg, Keich);
- Group theory (Katz);
- Low-degree algebraic geometry (Schlag, Tao, Mockenhaupt-Tao);
- High-degree algebraic geometry (Dvir, Saraf-Sudan);
- Heat flow monotonicity formulae (Bennett-Carbery-Tao)
[This list is not exhaustive.]
Very recently, I was pleasantly surprised to see yet another mathematical tool used to obtain new progress on the Kakeya conjecture, namely (a generalisation of) the famous Ham Sandwich theorem from algebraic topology. This was recently used by Guth to establish a certain endpoint multilinear Kakeya estimate left open by the work of Bennett, Carbery, and myself. With regards to the Kakeya set conjecture, Guth’s arguments assert, roughly speaking, that the only Kakeya sets that can fail to have full dimension are those which obey a certain “planiness” property, which informally means that the line segments that pass through a typical point in the set must be essentially coplanar. (This property first surfaced in my paper with Katz and Łaba.) Guth’s arguments can be viewed as a partial analogue of Dvir’s arguments in the finite field setting (which I discussed in this blog post) to the Euclidean setting; in particular, both arguments rely crucially on the ability to create a polynomial of controlled degree that vanishes at or near a large number of points. Unfortunately, while these arguments fully settle the Kakeya conjecture in the finite field setting, it appears that some new ideas are still needed to finish off the problem in the Euclidean setting. Nevertheless this is an interesting new development in the long history of this conjecture, in particular demonstrating that the polynomial method can be successfully applied to continuous Euclidean problems (i.e. it is not confined to the finite field setting).
In this post I would like to sketch some of the key ideas in Guth’s paper, in particular the role of the Ham Sandwich theorem (or more precisely, a polynomial generalisation of this theorem first observed by Gromov).