In this page I wish to clarify some specific sentences in my recent article "From rotating needles to stability of waves: emerging connections between combinatorics, analysis, and PDE" in the March 2001 issue of the Notices which may have been mis-interpreted.   These are a collection of rather technical comments.  For a more general discussion of these issues, see the March 5 entry in my "What's new" page.
• After discussing Tom Wolff's (n+2)/2 Kakeya result, I made the comment "However, there appears to be a limit to what can be achieved purely by applying elementary incidence geometry facts and standard combinatorial tools".  The reason for this comment is as follows.  In R^3, Tom's result gives a lower bound of 5/2 for the dimension of Besicovitch sets.  This argument, and any other argument which uses as its fundamental ingredients incidence geometry and combinatorics, will also work for the complex analogue of the Kakeya problem in C^3, giving a lower bound of 5 for the real Minkowski or Hausdorff dimension of such sets.  However, this bound of 5 is in some sense sharp for the C^3 problem, as the Heisenberg group {(z_1, z_2, z_3): z_3 = Im(z_1 \overline{z_2})} has 5 real dimensions, and contains a 4-real-dimensional (or morally 2-complex-dimensional) family of complex lines, and thus almost qualifies to be a Besicovitch set example.  (Admittedly some of the lines in the Heisenberg group example are parallel; however, this shows that one must use the non-parallel-ness of the line segments in a non-trivial way in order to surpass 5/2, at least in the context of the complex field (or any other field which contains a sub-field of half the dimension).  In particular, arguments which are based solely on incidence geometry (which cannot easily distinguish between parallel and non-parallel lines) are extremely unlikely to improve upon Wolff's bound.  (Further discussion can be found in my paper with Izabella Laba and Nets Katz here).  I elected not to include the above detailed explanation in the Notices article as it was rather technical and could be viewed as a gratuitious reference to one of my own papers.

• In discussions with Tom Wolff in 1999 it was clear that Tom was already aware of this fundamental limitation to the geometric method.  To escape this limitation there are essentially only two routes.  The first is to exploit the non-parallelism of the line segments, as per the arithmetic approach pioneered by Bourgain.  The other is to use the structure of the real field to show that, unlike the complex field, R does not contain a sub-field (or any similar object) of dimension 1/2.  This latter type of approach (together with other considerations) led Tom to other closely related problems in geometric analysis such as the Falconer distance problem and the dimension problem for sets of Furstenburg type.  These problems deserve further attention and may well be the key to further progress in the field; indeed, Tom appears to have already seen this in his survey article for the 1996 Prospects in Mathematics conference.  A (rather technical) paper of Nets and myself continuing an exploration of these issues can be found here.

In trying to make the point about the limitation of the geometric method, I had the unintentional effect of appearing to dismiss Tom's landmark (n+2)/2 paper as mere "elementary incidence geometry and standard combinatorial tools".  This was not the desired impression I wished to give at all.  In fact, I hold to the (perhaps paradoxical) opinion that some of the best and most beautiful papers in mathematics are those which use extremely elementary observations in an unforeseen way to yield progress on a problem which was otherwise thought to require much more sophisticated and technical arguments.  Tom's (n+2)/2 paper does, in my opinion, fall into this category, although Tom also introduced valuable technical devices in this paper as well, such as the systematic use of pigeonholing, the hairbrush construction, and the two-ends reduction.  It would have been nice to discuss these tools in the article as well but I could not see a way to do this given the space provided and the non-specialist nature of the readership, and so I concentrated instead on the ideas of Tom's paper which could be easily grasped by non-specialists.  (My coverage of the arithmetic arguments of Bourgain and Gowers were also in this vein).  It is also remarkable, given the rapid progress in the field, that Tom's paper, now 6 years old, has only barely been surpassed in the three-dimensional case (with a very small improvement of 10^{-10} in the Minkowski problem, no improvement in the Hausdorff problem, and an x-ray refinement only in the maximal problem).  Some idea of how difficult it is to improve upon Tom's results in the low-dimensional case can be found in this paper.  Despite being one of the authors of this paper, I have the opinion (which appears in the Notices article) that the techniques in this paper are also "clearly insufficient to resolve the full conjecture".

• The oscillatory integral section, which focusses on Fefferman's negative result on the disk multiplier and on the Bochner-Riesz problem, is not intended at all to be an exhaustive survey of the vast topic of oscillatory integrals.  This section in particular suffered from the space restriction of the Notices article; for instance I would have loved to devote much more space to the restriction problem, and in particular to the Tomas-Stein theorem and its variants and improvements.  (Indeed, in the original longer version of the article I devoted more space to the Tomas-Stein theorem and the closely related Strichartz estimates for dispersive and wave equations).  There are also a number of other results of similar flavour to Fefferman's results (using Besicovitch sets to yield unboundedness results in harmonic analysis) which also appeared in the 1970s, but I chose to focus the reader here on one specific result (Fefferman's) rather than a broad array of results.  For similar reasons I focussed the discussion of the Bochner-Riesz problem on those results which were specifically connected to the Kakeya problem or to PDE; as such, I did not devote much space to other important, but more technical and less relevant to the Kakeya-restriction-Bochner-Riesz-local-smoothing axis of conjectures, Bochner-Riesz results such as maximal or weighted inequalities or to the behaviour at or above the critical index (n-1)/2.  I hope to do fuller justice to this aspect of the field at a later date.

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• In the PDE section I discuss the local smoothing conjecture, arguably the most difficult of all the conjectures in this field, and then state "This conjecture is far from settled; even in two dimensions, the conjecture is proven only for p > 74 (due to T. Wolff), and at the critical exponent p=4 the conjecture is only known for \eps > 1/8 - 1/88 [TaoVargas], [Wolff]".  Once again, this sentence could be misconstrued as an attempt to belittle the work of Tom, Ana, and myself.  Actually, the intent was to highlight the extreme difficulty of this problem, in that even a mathematician of the stature of Tom Wolff , attacking the problem on two different fronts, was only able to achieve a modest amount of progress on both (though even this amount of progress would have been beyond many lesser mathematicians such as myself).  In fact, I rate the two papers of Wolff cited above as two of the most impressive papers I have ever read.  They share a common theme, in that they invoke a powerful new technique, namely induction on scales.  Prototypes of this idea have arguably appeared in earlier papers by Bourgain (and to some extent in the work of Ana Vargas, Luis Vega, and myself) but Tom was the first one to demonstrate (twice!) that this technique can be used to obtain nearly-sharp results on extremely difficult problems.  In the original longer version of the article I spent a great deal of space describing this technique, as I believe it is one of the most exciting recent developments in the field (as it is seemingly the only tool we have that is capable of giving optimal results, as opposed to small fractional improvements on existing bounds).  Unfortunately the referee and the editor decided that this material was somewhat technical, and made the article too long for the Notices, and so my commentary on the induction on scales technique was compressed to a half-paragraph which does not do the idea justice at all.  I have attempted to describe this idea more fully in some other unpublished expositions (one on Wolff's bilinear cone estimate, and the other on Wolff's local smoothing estimate).