What's new:
What was new in 2000?
What was new in 1999?
Dec 14, 2001

Added: Fifth
RiviereFabes symposium on Analysis and PDE (U. Minnesota, Apr 57).

Uploaded: "Puzzles
and (equivariant) cohomology of Grassmanians", joint with Allen
Knutson. This paper concerns Schubert calculus  the problem
of computing the number of subspaces of a certain dimension which intersect
a number of generic flags in a specified manner. A typical result:
given four generic lines l,m,n,o in C^3, there are exactly two lines which
intersect all four lines l, m, n, o. Equivalently, Schubert calculus
asks to find the multiplicative structure constants for the standard basis
of the cohomology of the Grassmannian given by (the Poincare dual of) "Schubert
cells". These structure constants are usually computed using
the LittlewoodRichardson rule, but we give a slightly different (but equivalent)
formula using a geometric object which we call a "puzzle" (it's sort of
the discrete version of a honeycomb). We also give a selfcontained
proof (not requiring any previous knowledge of the LittlewoodRichardson
rule) that these puzzles actually do compute Schubert calculus correctly.
Schubert calculus is most naturally a problem in intersection cohomology,
but it turns out that the easiest proof requires us to first generalize
puzzles and Schubert calculus to equivariant intersection cohomology
(in which there is a torus action that everything needs to be equivariant
under), and then prove a certain "Pieri" recursion rule in the equivariant
setting.
Dec 11, 2001
Dec 4, 2001

Uploaded: "Relative efficiencies of kernel
and local likelihood density estimators", joint with Peter
Hall. This is a statisics paper, comparing the bias effects of
kernel estimators with loglinear estimators, for the purposes of estimating
the density function f of a onedimensional random variable from a fixed
number of samples; the kernel estimates are shown to have better bias properties
in most circumstances. The bias of the two estimates turns out to
be nonlinear integral expressions of f, which can then be compared by
some elementary integration by parts arguments and the HardyLittlewood
inequality. One amusing result is that the behavior in the unimodal
case is different from that in the bimodal case when it comes to the relative
amounts of bias present.
Dec 1, 2001
Nov 23, 2001

Added: Thirteenth International Colloqium on Differential Equations (Plovdiv,
Bulgaria, Aug 1823 2002)
Nov 17, 2001
Nov 13, 2001
Nov 11, 2001

Added: 2nd
WSEAS Int.Conf. on Multirate Systems and Wavelet Analysis (MSWA 2002)
(Rethymo, Greece, Jul 1417). And from the TMR
harmonic analysis page: Conference
in Analysis (in honor of Yuri Brudnyi, Technion Haifa, Israel, May
2328), Thirteenth
International Workshop on Operator Theory and Applications (Virginia
Tech, Aug 69), and Infinite
Dimensional Function Theory (Pohang, Korea, Aug 1216)
Nov 8, 2001
Nov 4, 2001
Oct 31, 2001

Added: 2nd
International Gabor Workshop (Vienna, Dec 38), Emphasis period on
mathematics and general relativity (AIM, California, Apr 15June 15), and
the program Noyaux
de la chaleur, marches aléatoires et analyse sur les graphes et
les variétés (Heat kernels, random walks, analysis on
graphs and manifolds), Institut Henri Poincare, France Apr 16June 13.

Updated: The URLs for the 2002
ICMS Instructional Conference on Combinatorial Aspects of Mathematical
Analysis (Edinburgh, Scotland, Mar 25Apr 5) and the Nonlinear
and Geometric Analysis meeting in Toronto, Dec 7 (a satellite to the
annual
CMS meeting).
Oct 24, 2001
Oct 18, 2001

Updated: The expository note on
Arrow's theorem. Thanks to Gil Kalai for some interesting comments.
(Voting theory is of course somewhat outside of my field, but I find this
theorem particularly beautiful).

Added: 2002 AMS
and MAA Spring Southeastern Section Meeting (Georgia Institute of Technology,
Atlanta GA, Mar 810 2002), Special Session in Harmonic Analysis and in
Frames, Wavelets, and Operator Theory. Thanks to Gerd Mockenhaupt
for the link.
Oct 10, 2001
Oct 3, 2001

Updated: The URL for the International
Conference on Harmonic Analysis and Related Topics (in honour of the
60th birthday of Alan McIntosh; Jan 1418, Macquarie University, New South
Wales, Australia) and for the International
Conference on Nonlinear Partial Differential EquationsTheory and Approximation
(Hong Kong, Aug 29Sep 2). Thanks to Alan McIntosh for the links.

Updated: The U. Arkansas Spring
Lecture Series has been moved to April 1113 (from April 1820).

Uploaded: "Sharp global
wellposedness results for periodic and nonperiodic KdV and modified KdV
on R and T", joint with the "Iteam". This is the sequel to our
earlier paper on KdV. In this paper we show how the "correction
term" technique improves the "Imethod" in the earlier paper. Specifically,
we improve the known H^s global wellposedness results for KdV from s >
3/10 to s > 3/4 on R (which matches the local theory), from s >= 0 to
s >= 1/2 on T (also matching the local theory), for mKdV from s > 3/5
to s > 1/4 on R (matching the local theory up to an endpoint) and from
s >= 1 to s >= 1/2 on T (also matching the local theory).
This is further evidence to support the conjecture that local wellposedness
implies global wellposedness in the presence of a subcritical conserved
quantity. Because s=1/2 is the regularity of phase space on T we
expect to also be able to invoke the theory infinite dimensional symplectic
geometry (and in particular the nonsqueezing theorem) for KdV on T, but
there is a technical difficulty with trying to approximate the infinitedimensional
evolution by a finitedimensional one which we will address in a later
paper.

Uploaded: "Multilinear
estimates for periodic KdV equations, and applications", also joint
with the "Iteam". This paper is a companion to the above paper
on KdV and mKdV, but focuses on the periodic case, and on the generalized
KdV (gKdV) equation. The main purpose is to prove a sharp multilinear
estimate at the regularity of H^{1/2}, thus obtaining local wellposedness
for all periodic gKdV equations at H^{1/2}. We show that this is
in some sense sharp for the quartic and higher nonlinearities. As
byproducts of the estimate we prove the key oustanding estimate required
to conclude the periodic global wellposedness results in the above paper,
and also give a new global wellposedness result for the quartic gKdV equation
(gKdV_3). Specifically, we show global wellposedness for s > 5/6
in the defocussing case. One interesting technical feature of the
paper is that to obtain the endpoint s=1/2 one requires some nontrivial
but elementary number theory. Specifically, we have to improve upon
the fact that a large integer N has O(N^{0+}) divisors. Indeed when
we restrict ourselves to a small interval (of width O(N^{1/3})) away from
the origin, any given integer N can have at most 3 divisors. This
fact turns out to be key to obtaining the endpoint results at s=1/2 (or
s = 1/2 for KdV).
Oct 2, 2001

Uploaded: "A refined global
wellposedness for the Schrodinger equations with derivative", joint
with the "Iteam" (Jim Colliander,
Mark
Keel, Gigliola
Staffilani, Hideo Takaoka, and myself). This is the sequel to
our
earlier paper on the derivative nonlinear Schrodinger equation (DNLS).
In this paper we show how the "correction term" technique can be used to
improve the "Imethod" in the earlier paper. Specifically, we improve
the known H^s global wellposedness results for DNLS from s > 2/3 to s
> 1/2, which nearly matches the best possible local wellposedness theory
s >= 1/2. When combined with the similar
results we have for KdV and mKdV, it seems to suggest that local wellposedness
always implies global wellposedness when there is a subcritical conservation
law, although admittedly these two examples are not totally convincing
since they are one dimensional (and DNLS can even be completely integrable
like KdV for certain choices of parameters), and furthermore we did not
need to all the way down to the critical regularity in either case.

Updated: "A counterexample
to a multilinear endpoint question of Christ and Kiselev", joint with
Camil
Muscalu, and Christoph Thiele.
Some minor changes made for the final submitted version.
Sep 29, 2001
Sep 23, 2001
Sep 13, 2001

Monetary donations to the Red Cross can be made (for example) at www.amazon.com.
Blood donations can be made almost anywhere; contact the Red
Cross.
Sep 5, 2001
Aug 28, 2001

Well, it's been a good four years or so maintaining the list
of harmonic analysts, but it's time to outsource this to someone else.
Philippe Jaming at the European
Harmonic Analysis TMR network has kindly agreed to host
the list on the TMR site. Thanks to Philippe's CGI skills, this
list is evolving to a searchable database which people can update by themselves.
There are still some transitional issues, but in the long term it should
be the best place for the list. As such, I won't be updating my local
copy of the list any longer, and will be slowly migrating any links I have
to the TMR site. Of course, I'll keep the local copy here for people
who are reluctant to update their bookmarks. :)
Aug 27, 2001
Aug 23, 2001

Uploaded: "A counterexample
to a multilinear endpoint question of Christ and Kiselev", joint with
Camil
Muscalu, and Christoph Thiele.
In a recent series of papers, Christ and Kiselev have analyzed the eigenfunctions
of Schrodinger and Dirac equations whose potentials decay like L^p for
p < 2, essentially by expanding these eigenfunctions as a multilinear
series and estimating each term individually. In particular they
show that these eigenfunctions are bounded for a.e. energy k > 0.
This statement is also believed to hold in the endpoint case p=2, but in
this paper we show that the multilinear expansions are badly behaved at
p=2. Our opinion is that we cannot proceed in the endpoint case by
multilinear methods and instead must go through a genuinely nonlinear approach;
we have some progress in this direction for a simplified Walsh model of
the eigenfunction problem. The counterexample is quite simple (basically
a "chirp") and may have application to other spectral problems for Schrodinger
operators.

Uploaded: Lecture notes for "Local
and global wellposedness for nonlinear dispersive equations".
These are notes for a fourpart minicourse I will shortly give for the
ANU special
program in Spectral and Scattering Theory. In these notes I discuss
the notion of local and global wellposedness, discuss the recent developments
in obtaining global wellposedness for very low regularities (in particular
discussing the "Imethod" developed by Colliander, Keel, Staffilani, Takaoka,
and myself), as well as the recent global regularity for small data results
for wave maps.
Aug 21, 2001

Uploaded: "L^p improving
estimates for averages along curves", joint with Jim
Wright. It took a little longer than expected to write, but it's
finally here. We combine ideas from previous work of Christ, PhongStein,
NagelSteinWainger, ChristNagelSteinWainger, CarberyChristWright,
and Wolff to obtain local (L^p, L^q) mapping properties for a general class
of averaging operators on curves which are sharp up to endpoints.
The methods are purely geometric combinatorics (as in Christ's work on
convolution with (t, t^2, ..., t^n)) and do not involve oscillatory integral
methods; on the other hand it seems difficult to obtain endpoints in this
manner. The idea is to lift the problem up to one higher dimension,
at which point the problem becomes essentially a twoparameter isoperimetric
inequality induced by two vector fields X_1 and X_2. The argument
then hinges on a close study of the geometry of the twoparameter CarnotCaratheodory
balls induced by these vector fields (following NagelSteinWainger).
One then uses the restricted weaktype iteration techniques of Christ to
reduce the problem to obtaining lower bounds for a Jacobian on a certain
set. A direct approach to this problem seems very difficult  and
indeed there are counterexamples showing that a naive application of this
method cannot work in general  however by using the twoends reduction
of Wolff we can restrict the set to one of the abovementioned balls, and
then rescale that ball to the unit ball, at which point the set becomes
"sparse" in a certain sense and one can obtain good lower bounds rather
easily.

Uploaded: "Carleson measures,
trees, extrapolation, and T(b) theorems", joint with Pascal
Auscher, Steve
Hofman, Camil Muscalu,
and Christoph Thiele.
In this mostly expository paper we discuss the wellestablished theory
of Carleson measures, and the more recent theory of tiles and trees, in
a simple dyadic model. In this model we show that the two theories
are actually closely related, and that the concept of maximal size can
be used to unify many results in harmonic analysis (from BMO theory to
Hardy space decomposition to paraproduct and bilinear Hilbert transform
estimates). We also use this machinery to prove various T(b) theorems,
in particular improving the hypotheses the local T(b) theorem of Christ
from L^infty type conditions to L^2 ones. Our proof is slightly different
from the standard proof (instead of constructing a global b by cutting
and pasting together local b's, we reduce matters to estimates localized
to trees, and then use local wavelet systems) and we believe it is of some
interest (our results are stated for the simplified dyadic model but this
is not a serious restriction).
Aug 11, 2001

Added: ICM Satellite conference on Harmonic Analysis and Applications (Aug
1418, Hangzhou, 2002)

Milestone: 650th harmonic analyst added to the
list!
July 25, 2001
July 24, 2001

Uploaded: "Uniform estimates
for multilinear operators with modulation symmetry", joint with Camil
Muscalu and Christoph Thiele.
This is the sequel to our earlier
paper on uniform estimates on paraproducts. Here we prove L^p
estimates for multilinear operators whose multiplier has a onedimensional
singular set (e.g. the oneparameter family of bilinear Hilbert transforms)
where the estimates are uniform in the choice of singular set. This
is similar to previous work by Thiele, Li, and Grafakos. As in these
works, the result is highly technical, however, we have managed to move
most of the technical difficulty into the task of estimating a single tree
(the "tree estimate"). We do this by investing some effort in developing
a phase space projection operator onto the tree, at which point the tree
estimate reduces to the paraproduct estimate proven in the
previous paper. The phase space projection is quite technical
but could perhaps be useful for other applications.
July 8, 2001
July 3, 2001

Uploaded: "The honeycomb model
of GL(n) tensor products II: Puzzles determine facets of the LittlewoodRichardson
cone", joint with Allen
Knutson and Chris Woodward.
Yes, this is the paper we announced about two years ago; the paper has
been through several rewrites in order to reduce the amount of sheer combinatorial
computation involved. (It's now gotten to the point where one only
needs to check at most four cases at a time, down from the sixteen or so
in the original). Anyway, the main purpose of this paper is to characterize
which of the inequalities conjectured by Horn to solve the sumofHermitianmatrices
problem are actually essential (some of them are superfluous). To
do so we characterize these inequalities by a geometric device which we
call a puzzle; these are in some sense the discrete version of a
honeycomb. It turns out that the rigid puzzles give essential inequalities,
while the nonrigid puzzles (those which can be altered without affecting
the boundary) are superfluous (the latter result was already discovered
by Belkale) . There is also an identification between rigid puzzles
and rigid honeycombs which answers a question of Fulton, characterizing
the triples of weights for which the LittlewoodRichardson number is one.

One byproduct of the theory we develop here is that we can give a purely
combinatorial proof of many basic results about the sumofHermitianmatrices
problem when posed in the honeycomb formulation (e.g. we can reprove
Klyachko's theorem, which in honeycomb language links integer honeycombs
to inequalities on the boundary of honeycombs), thus bypassing a lot of
machinery about algebraic geometry, geometric invariant theory, etc.
However, this does not quite give a completely elementary proof of Horn's
conjecture yet, because one still has to prove by elementary means that
honeycombs do indeed solve the sumofHermitianmatrices problem.
This will be the subject of a future paper.

Some of the results of this paper have recently
been proven (using a completely different technology, namely the use
of quiver representations) by Harm
Derksen and Jerzy
Weyman. Their results apply to much more general situations,
but they do not prove Fulton's conjecture and must use it as an assumption.
July 2, 2001

Uploaded: The expository note "Finite
field analogues of the Erdos, Falconer, and Furstenburg problems".
Here I am giving an informal description of some
connections Nets and I found between the Erdos ring problem, the Falconer
distance problem, and the Furstenburg set problems in Euclidean space.
Unfortunately the Euclidean setting makes this paper quite messy, due to
the logarithmically infinite number of scales, issues with small angles
and/or small separations, and the inherent inability to perfectly discretize
both addition and multiplication at the same time. So in this note
I outline the same arguments, but in the finite field setting, where all
the above mentioned problems don't exist (Plus, since this is only an expository
note, I am allowing myself the luxury of being nonrigorous, which really
helps a lot in this business). In this case the arguments are quite
clean (and not very original; they come from earlier arguments of Elekes,
Gowers, Bourgain, Chung, Szemeredi, and Trotter). Basically, it all
comes down to resolving the following open problem, which is one of my
favourite unsolved questions in the field: Does there exist a subset E
of the finite field Z_p such that E, E+E, and E*E all have cardinality
comparable to sqrt(p)? The corresponding question for the finite
field F_{p^2} (with sqrt(p) replaced by p) is of course true, just take
E to be the embedded field F_p. So how is F_p different from F_{p^2}?

Added: Prairie Analysis Seminar
(Oct 1920, KSU Kansas)
June 26, 2001
June 20, 2001
June 12, 2001

Uploaded: "A physical approach
to wave equation bilinear estimates", joint with Sergiu Klainerman
and Igor Rodnianski, in submission to the special Wolff memorial issue
of the Journal D'Analyse de Jerusalem. In this paper we give a somewhat
new proof of two bilinear estimates for wave equations  these are of the
type very useful in the study of low regularity behaviour of nonlinear
wave equations such as YangMills, wave maps, or MaxwellKleinGordon.
The main novelty is that instead of proving these estimates by using Plancherel's
theorem to convert matters to a weighted L^2 convolution, as is standard
practice, we follow a "wave packet approach" in the spirit of Bourgain
and especially Wolff. This approach works almost entirely in physical
space and seems better adapted to very rough metrics (we plan to apply
these methods in particular to solutions of the Einstein equation with
H^2 metrics). Daniel Tataru has done similar work on proving these
estimates in rough metrics using more Fourierbased methods; it will be
interesting to compare the strength of these different techniques in low
regularity situations.

Specifically, we decompose the wave into "photons" or wave packets (using
physical space type techniques instead of Fourier ones), and divide into
parallel and transverse interactions. Parallel interactions are handled
by classical vector fields methods, while transverse interactions exploit
an inductiononscales argument of Wolff.

We also give a proof of a different type of bilinear estimate when there
is a singularity at the spatial frequency origin \xi = 0. This singularity
is usually dealt with by chopping frequency space up into small squares,
but this seems rather expensive in low regularity situations. It
appears a more robust approach is to decompose physical space instead
into the dual squares, and this is the strategy of the argument in the
paper.

Uploaded: "Uniform estimates
on paraproducts", joint with Camil Muscalu and Christoph Thiele, also
in submission to the special Wolff memorial issue of the Journal D'Analyse
de Jerusalem. This is part I of a project to understand how estimates
on multilinear operators with singularities behave as the singularity becomes
degenerate. In the case of the bilinear Hilbert transform this was
studied extensively by Thiele, and later Grafakos and Li. Our aim
is to understand the general multilinear case. Fortunately we can
split the problem into two pieces: a "tilecombinatorics" part which reduces
the uniform estimate on singular multiplies to uniform estimates on paraproducts
(which are multipliers which are only singular at the origin), and a "LittlewoodPaley"
part in which one uses the combinatorics of interactions of LittlewoodPaley
pieces to estimate the paraproducts. It is the latter problem which
is addressed in this paper; specifically, we prove Holder estimates for
paraproducts with arbitrary frequency separation assumptions on the inputs,
which are uniform in the frequency separation parameters. The proof
is quite short  only 13 pages  and relies on quite simple combinatorics
and linear algebra, encoded using the convenient language of graph theory.
It also indicates the power of LittlewoodPaley theory when studying these
multipliers; it may not always be the quickest approach, but it is very
systematic and intuitive, and usually doesn't cost too much to apply.
May 31, 2001
May 17, 2001
May 14, 2001
May 2, 2001
Apr 17, 2001
Apr 8, 2001

Uploaded: "Fuglede's conjecture
for convex bodies in the plane", joint with Alex
Iosevich and Nets
Katz. This paper concerns Fuglede's conjecture, which asserts
that a bounded open set in R^n has an orthonormal L^2 basis consisting
of pure exponentials if and only if the set tiles R^n by translations.
We show that the conjecture is true when the set is a convex body in the
plane. The tools are: a reduction of Kolountzakis to the symmetric
case; an approximate computation (using stationary phase) of the zero set
of the Fourier transform of the body; some combinatorial arguments to dispose
of the nonpolygon case (using the fact that the zero set resembles a bunch
of parallel lines at certain places); and some further combinatorial arguments
to deal with the polygon case (using the fact that the zero set resembles
a Cartesian grid at certain places). The argument may well extend
to higher dimensions, but I do not see any reasonable way to significantly
relax the convexity hypothesis.
Apr 5, 2001
Apr 2, 2001

Updated: The URL for the Second
Pacific Northwest PDE Seminar (May 19, U. Washington / PIMS)

Uploaded: "Tree chopping,
and Carleson measures" (expository note). This is a timefrequency
version of a Carleson measure extrapolation lemma which is used in the
recent work on the Kato problem, and arose after discussions with Steve
Hoffman, Loukas Grafakos, Alex Iosevich, and Xiaochun Li. Basically,
it says that a tree of large size (in the Carleson measure sense) can always
be chopped into a manageable number of trees of small size, plus a manageable
number of excess tiles. I've tweaked the original proof (which was
a stopping time argument followed by an iteration) to become an induction
on scales argument, following the general philosophy of "whenever uniformity
breaks down, apply the induction hypothesis instead".

Added: Harmonic
analysis since the Williamstown conference of 1978 (Williamstown, Oct
1314)
Mar 29, 2001

I'll be on the road for the next three months or so. Updates will
be variable, depending on my internet access.

Added: Harmonic Analysis and Approximation II (Armenia, Sep 1118)
Mar 23, 2001
Mar 13, 2001

Uploaded: "The Kakeya conjecture"
(+ Figures 1 2).
Yes, it's another expository article. This is a 15minute presentation
of my research for incoming grad students to UCLA. Don't expect too
much from it.
Mar 12, 2001
Mar 8, 2001
Mar 5, 2001

After recieving some feedback on my survey article "From
rotating needles to stability of waves: emerging connections between combinatorics,
analysis, and PDE" in the March 2001 issue of the Notices, I feel that
I should make some additional remarks to ward off any misconceptions that
might have arisen from the original article. (An abridged version
of these remarks has been sent to the Editors of the Notices).

Firstly, the results mentioned therein are only a small fraction of the
large amount of work and progress accomplished on these problems, and due
to space constraints I was only able to give a few representative results
on each problem. (The Notices guidelinesask
that the reference list be kept to under 10 papers, while the article should
be 69 pages in length.) As such, some authors and their results
were mentioned only briefly or not at all, for which I apologize.
A larger version of the article (32 pages long), which contains somewhat
more detail, especially on the connection with nonlinear wave equations,
can be found here.
Also, after the El Escorial conference in July 2000 I wrote (with Nets
Katz) an article for the El Escorial conference proceedings which was part
survey, part announcement, on progress on the Kakeya problem. This
article was not subject to the same kinds of space and reference constraints
as the Notices article, had a narrower focus, and was aimed at a more specialist
audience. As such, I believe it better reflects my views on the field.
It can be viewed here.
At some point in the future I hope to make a similar survey on the modern
oscillatory integral and PDE aspects of the field; the brief pages in the
Notices article devoted to these topics fail to do them any sort of true
justice.

Secondly, the main point I was hoping to emphasize in the article was that
the open problems posed there appear to be extremely difficult, and that
deep ideas from other fields could be needed to make substantial new progress.
However, this is not to disparage the considerable amount of progress and
insight that has already been achieved; in recent years, the breakthroughs
of Jean Bourgain and Tom Wolff in particular have revolutionized the field.
The arguments and ideas coming from these breakthroughs continue to yield
further progress on these problems today. Nevertheless, it is my
opinion that even with these powerful new techniques, we have roughly half
of the pieces of the puzzle required to solve even the Kakeya problem (which
should be the easiest of all the problems listed), and that one must combine
these existing techniques with further ingenious ideas or insights in order
to obtain a complete resolution. I certainly do not claim that I
am capable of coming up with these ideas, but there may well be talented
people from other fields whose fresh perspective can build upon the work
that has already been done. It is a testament to the extraordinary
difficulty of these problems that so many of the best people in the field
have expended enormous amounts of energy on these problems, and still we
are only halfway to a full solution.

As part of my brief I was advised to keep the presentation as elementary
as possible, so as not to scare off any nonspecialist readers. This
inevitably meant that I could not do justice to the most recent and deepest
results (which are inevitably technical and somewhat raw in presentation).
For similar reasons, I was forced to present some of the most powerful
arguments we have in the field in a caricature form. Unfortunately
this had the unintended effect of making it look as if I was trying to
trivialize the work of the leaders of the field, and in particular the
late Tom Wolff. This was the polar opposite of my intentions.
At their heart, the innovative ideas of Tom, Jean, and others are based
on very beautiful and simple (but highly nontrivial without hindsight)
ideas, and I was hoping to convey this beauty and simplicity to the readers
in the short space given to me. Of course, figuring out how to transform
these ideas into quantitative progress on these extremely difficult problems
requires a phenomenal amount of technical skill, as well as a deep and
accurate intuition. Unfortunately this side of the story is not easy
to communicate in such a nontechnical setting as a Notices article.

There are some other points I wish to make regarding specific sentences
in the article which may be misconstrued as an attempt to diminish previous
work in the field, but they are rather technical mathematically and I have
placed them here.

The timing of the article, regrettably, was not favourable at all.
The article was solicited from me in February 2000, and originally based
on the talk I gave at the Clay Millennium Event in May. The first
draft was submitted at the beginning of June, with an initial report
on the draft returned by midJune, and a more detailed report in early
July (which recommended, among other things, significantly shortening the
article and removing some of the more technical material). I had
essentially finalized the article in late July, although due to some technical
issues the final acceptance did not come until midAugust. In early August
I heard the shocking news of Tom's death, but at that time I could not
think of an appropriate postscript to add to the article to incorporate
this. (Admittedly, I would probably have written a very different
article after August than the one I wrote in May and June). I can
see how this has led to the article being viewed as being insensitive to
recent events, and I am deeply sorry for any hurt that this has caused.
I myself was greatly upset by Tom's untimely death, despite the insulation
of being ten thousand kilometers away in Australia at the time. His
death is a severe loss for the entire field. With Tom, I would have
expected the Kakeya family of problems to be very close to resolution within
five years. Without him, it may take ten years or more. I hope
to honour his achievements by contributing, in my own small part, to the
continuing development of his ideas and techniques to these problems.

I have one final thing to say, in the hope that future misunderstandings
can be avoided. It is my personal preference in mathematical writing
to focus on the ideas (and to some extent techniques) rather than on the
results, authors, or the actual details of the argument. Unfortunately,
ideas are a more nebulous thing to assign credit to than, say, a Theorem;
a prototype of the idea may be implicit in an early work by X, a more developed
version may be used in a specific application by Y, then generalized to
several other applications by Z, but the final explicit enunciation of
the idea may only be made in a textbook by W some time later. And
often the idea is not developed in the printed pages of journals at all,
but rather is passed along via blackboards and email as informal "folklore".
Because of this, it is quite possible that I will not fully assign the
credit to other authors that they deserve. I will attempt to rectify
this to the best of my ability in the future, but I cannot promise a flawless
accreditation of previous workers in the field every time. On the
other hand, in most of my papers I will have no restriction on the number
of references allowed, which should make the job of attribution easier
than in the Notices article.
Feb 23, 2001
Feb 16, 2001

Added: Summer school
on nonlinear partial differential equations Jul 1620, Portugal.

Linked: Izabella Laba's page on The
Kakeya problem and connections to harmonic analysis.

Uploaded: "New bounds on
Kakeya problems", joint with Nets
Katz. This represents some further progress that Nets and I achieved
on the Kakeya family of problems last year, extending the sumsdifferences
technology pioneered by Gowers and Bourgain, and also used by Nets and
myself in a previous paper. (Recently Michael Christ has also applied
this technology to multilinear operators in L^p for very small p).
Not only do we improve the best possible sumsdifferences exponent (the
trivial exponent is 2, the optimal is 1. 21/13 is due to Bourgain,
21/4 is by Nets and myself. The best exponent currently is 1.675...,
the solution to a cubic)  thus improving the Minkowski bound on Besicovitch
sets  but we also make certain sumsdifferences lemmas more flexible,
so that they can be applied efficiently to the Hausdorff and maximal problems.
In the Hausdorff setting we have also figured out how to "deslice" Bourgain's
"slices" argument, and thus squeeze about half a dimension more out of
the argument (so that these bounds become quite competitive in low dimensions
such as 5).
Feb 8, 2001

Uploaded: "L^p estimates
for the "biest" I. The Walsh model", joint with Camil
Muscalu and Christoph Thiele.
In this paper we begin the study of the "biest", a trilinear variant of
the bilinear Hilbert transform (although not exactly the trilinear Hilbert
transform) which comes up as (a simplified version of) the third term in
WKB expansion of the eigenfunctions of onedimensional Schrodinger operators
with a potential V, in the critical case when V is just in L^2. We
prove a large range of L^p estimates for the Walsh model of operator.
As a byproduct we have a short, selfcontained treatment of the Walsh
bilinear Hilbert transform which should be of independent interest.
The techniques are the usual BHT techniques (somewhat polished from the
first papers on this topic, now that we have the benefit of hindsight),
together with a "biest trick" which allows one to decouple the two frequency
parameters at a crucial juncture.

Uploaded: "L^p estimates
for the "biest" II. The Fourier model", joint with Camil
Muscalu and Christoph Thiele.
We continue the study of the biest, this time studying the more important
Fourier model for the biest rather than the toy Walsh model. The
results and methods are the same, except the usual technical difficulties
arising from the fact that one cannot localize perfectly in both space
and frequency simultaneously in the Fourier model. This causes a
special problem here in that the "biest trick" which works very neatly
in the Walsh case, has only restricted application in the Fourier case,
and one must pursue a slightly different approach.

Added: International
Conference on Complex Analysis and Related Topics (Brasov, Romania,
Aug 2731). Thanks to Camil
Muscalu for pointing out this conference.
Feb 4, 2001
Jan 31, 2001

Uploaded: "Global wellposedness
result for KdV in Sobolev spaces of negative index", joint with the
"Iteam" (Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka,
and myself). This paper is a short application of the "Imethod",
which allows one to obtain longtime control on a dispersive equation even
when the solution is very rough (so rough that the natural conserved quantities
are infinite). In this case the equation is the KdV equation, the
conserved quantity is L^2, and global wellposedness is obtained for H^s
down to s >3/10. One interesting thing about this argument is that
no new bilinear estimates are needed; one just uses "offtheshelf" estimates
of KenigPonceVega and CollianderStaffilaniTakaoka. In a later
paper we shall push these results much further, down to s > 3/4, thus
matching the local wellposedness theory.

Uploaded: "Global wellposedness
for the Schrodinger equations with derivative", joint with the Iteam.
This paper is a slightly more involved application of the Imethod to the
cubic derivative nonlinear Schrodinger equation in one dimension.
In this case the conserved quantity is the Hamiltonian (which is
equivalent to the H^1 norm) and global wellposedness is obtained for H^s
down to s > 2/3 assuming a smallness condition on the L^2 norm. The
argument combines the Imethod with the gauge transformations of Ozawa
to move the derivative into a favorable location (at the cost of introducing
a quintic term in the nonlinearity). Unlike the KdV paper, some
new multilinear estimates need to be developed. In a later paper
we shall push this down to s > 1/2, thus matching the local wellposedness
theory.
Jan 23, 2001

Uploaded: "Some connections
between Falconer's distance set conjecture, and sets of Furstenburg type",
joint with Nets Katz. Over the last one or two years, Nets and I
have been looking at a number of geometric combinatorics problems such
as the distance set problem, the Furstenburg set problem, and Erdos's ring
problem. Somewhat frustratingly, we were not able to solve any of
these problems, but we made the somewhat surprising discovery that they
were all equivalent to each other in a certain sense. In particular
it seems that 1/2dimensional rings should play an important role in these
types of problems. It would be nice to understand this phenomenon
more deeply.
Jan 17, 2001
Jan 14, 2001