What was new in 2002? 2001? 2000? 1999?

Dec 11, 2003

- Travel: Due to visa delays, I will now not be returning to the US until January 21. My teaching duties until then will be covered by Christoph Thiele (for Math 133) and John Garnett (for Math 245B) respectively.
- Uploaded: “A Strichartz inequality for the Schrodinger equation on non-trapping asymptotically conic manifolds”, with Andrew Hassell and Jared Wunsch. Here we extend the interaction (or “two-particle”) Morawetz inequality, which was previously derived for linear and non-linear Schrodinger equations on R^3 to obtain an L^4-type Strichartz bound, to more general manifolds, thus obtaining a new L^4 Strichartz inequality for all smooth asymptotically conic non-trapping manifolds (and in particular for asymptotically flat non-trapping manifolds). The assumption of no trapped geodesics is crucial, since otherwise there are pseudomode solutions to the Schrodinger equation which have no local smoothing and in particular must necessarily obey worse Strichartz estimates than in Euclidean space. Similar results have been obtained before by Burq, Gerard, Tzvetkov, Staffilani, and Tataru; our result is weaker in that it is restricted to L^4 on three dimensions, but we do not lose any derivatives even in the asymptotically conic case. Unlike previous methods, which rely (among other things) on a microlocal parametrix which obeys good dispersive estimates, we do not use a parametrix here. Instead our argument is primarily based on the positive commutator method applied to the product solution; in fact one can phrase the main argument as an integration by parts based primarily on understanding the convexity properties of the distance function d(x,y), even when x and y are far apart; thus the paper requires a certain amount of Riemannian geometry computations. The trouble comes near infinity and at far distances when the distance function begins to develop singularities; we must apply cutoffs to avoid these, which in turn generate further terms which need to be estimated. Most of these can be estimated by local smoothing estimates (or “one-particle” Morawetz estimates), but there is an error term which requires a new type of Morawetz estimate (one with only one angular derivative instead of the usual two), which may also be of independent interest.

Nov 17, 2003

- Uploaded:
“Instability of the
periodic nonlinear Schrodinger equation”, with Michael Christ and Jim
Colliander. This is third in
our sequence of papers analyzing the instability at low regularities of
model dispersive equations; this time we focus on one-dimensional periodic
non-linear Schrodinger equation, which has a particularly clear separation
into high frequency and low frequency modes due to the periodic Fourier transform. Here we find that while, as
expected, these evolutions become ill-posed and discontinuous once the
data is rougher than L^2 in Sobolev norms, the nature of this
ill-posedness depends on the power of the non-linearity. In the cubic case (which should be
much more stable anyway, being completely integrable), it is possible for
a small change in the initial data in a negative Sobolev norm to lead to a
large change in even the zeroth Fourier mode in an arbitrarily short time;
the problem here is that while the negative Sobolev norm is small, the
mass is large. Thus the
solution map is not continuous below L^2 at the origin. For the quintic and higher
equations this phenomenon occurs, but there is another mechanism for
ill-posedness too... arbitrarily small perturbations
*in C^infty norm*can lead to a large oscillation in the zeroth Fourier mode in arbitrarily small time if the solution is small in negative Sobolev spaces but not in L^2; the quintic nature of the nonlinearity allows small perturbations at low modes to interact with high modes to return to cause massive instability in low modes. We emphasize that in both cases we show ill-posedness but not blowup; indeed we suspect in the completely integrable cubic case there is no blowup in negative Sobolev spaces despite the presence of instabilities; this may be a renormalization issue, and we are continuing to look into this question. - Added:
23
^{rd}Annual Western States Mathematical Physics Meeting (Caltech, Feb 16-17).

Nov 12, 2003

- Updated “Recent progress on the restriction conjecture”. These Park City lecture notes have been revised; the most noticeable change is the addition of 9 new figures, and removal of several typos, especially from the exercises. Special thanks to Julia Garibaldi for her proofreading while TA’ing the Park City Course!

Nov 5, 2003

- Uploaded: “On multilinear oscillatory integrals, nonsingular and singular”, with Michael Christ, Xiaochun Li, and Christoph Thiele. Here we consider multilinear operators similar to that of the bilinear Hilbert transform, but with an additional polynomial phase oscillation (thus these are the multilinear analogue of the singular oscillatory integrals studied for instance by Ricci and Stein, and are also related to the polyomial phase Carleson maximal operators studied for instance by Lacey). It turns out that one can fairly quickly decouple the singular and oscillatory parts of this operator into a singular piece where the oscillation is not significant (and which can be dealt with by existing multilinear multiplier theorems), plus an oscillatory part where the singularity of the kernel is irrelevant. This oscillatory part is the main focus of the paper, and in many ways we found it more interesting than the original problem. The question is under what conditions one can obtain decay estimates for these multilinear oscillatory integrals analogous to what one can obtain using Van der Corput’s lemma for oscillatory integrals of the first kind, or via e.g. the analysis of Phong and Stein for oscillatory integrals for the second kind. There is a natural conjecture, which is that decay occurs whenever the polynomial phase is non-degenerate (i.e. it cannot be factored away into the various component functions). We prove this conjecture when the number of functions is less than twice the number of integration variables, which co-incidentally is the same range that is needed to handle the singular part. When there are very few functions one can use a Cauchy-Schwarz (or Van der Corput) trick to repeatedly eliminate functions until one is left with a plain oscillatory integral; however this technique does not always work once there are too many functions, and instead we rely on some polynomial Fourier analysis (inspired by the recent work of Gowers on arithmetic progressions), in particular dividing into the cases when a function has a large polynomial Fourier coefficient (which can be treated by an inductive hypothesis), and when all functions are “polynomially uniform” (in which case we can descend to a simple two-dimensional problem). The cases of higher functions is still pretty much open however. In analogy with the work of Christ, Carbery, and Wright, we also consider level set analogues of this oscillatory problem, for which we have a more satisfactory range of results.
- Uploaded: “Ill-posedness for nonlinear Schrodinger and wave equations”, with Michael Christ and Jim Colliander. This continues our previous work on low regularity ill-posedness for the KdV and cubic non-linear Schrodinger equation, but now extended to nonlinear Schrodinger (NLS) and wave (NLW) equations of arbitrary power (and in particular not relying on the specific feature of the 1D cubic NLS that it is on the borderline between long-range and short-range scattering behavior). We introduce a general technique for demonstrating ill-posedness of these equations (even in the defocusing case) below scale-invariant or Gallilean invariant regularities, by rescaling the problem to a small dispersion limit and showing that for sufficiently small dispersion, the non-linear part of the equation dominates and causes phase decoherence. The illposedness we obtain is in the sense of failure of uniform continuity of the solution map; there seem to be subtle distinctions between this and actual failure of continuity of the solution map (or of blowup), which we shall partially address in a forthcoming paper. Unfortunately the method does not work so well with the Lorentz-invariant regularity of the wave equation both because of the second-order nature in time of the wave equation and because the Lorentz transformation distorts time as well as space (so in particular does not preserve the concept of initial data), so this remains the major remaining obstacle towards a complete well-posedness and ill-posedness for NLW. We do however obtain a satisfactory one-dimensional NLW theory showing that the obstruction of needing enough regularity for the solution to be a distribution is a necessary one for uniform well-posedness. We also relate our NLW work with the instability work of Lebeau for the supercritical wave equation; the results are similar but not directly comparable.

Oct 23, 2003

- Uploaded: “Bi-parameter paraproducts”, with Camil Muscalu, Jill Pipher, and Christoph Thiele, submitted to Acta Math. Here we study paraproducts on R^2 where the multiplier m(xi_1, xi_2) is not of classical Coifman-Meyer type (in that each derivative in xi_1 or xi_2 gains a power of |xi_1| + |xi_2|) but is rather of product Coifman-Meyer type, where we split xi_1 = (xi’_1, xi’’_1) and xi_2 = (xi’_2, xi’’_2), and each derivative in xi’_1 and xi’_2 gains a power of |xi’_1| + |xi’_2|, and similarly for the xi’’ terms. This particular type of “bi-parameter paraproduct” arises if one wishes to do things like fractional differentiation in the x_1 and x_2 directions simultaneously. We obtain the expected bilinear L^p x L^q -> L^r estimates for this operator for p,q > 1; we allow the possibility that r is less than one. For r > 1 the argument is straightforward, proceeding by the usual Littlewood-Paley type decomposition and then using boundedness of “maximal-maximal”, “maximal-square function”, “squarefunction-maximal” or “squarefunction-squarefunction” operators (which are concatenations of the one-dimensional Hardy-Littlewood maximal or Littlewood-Paley square function operators in either direction) in L^p, p > 1 to conclude the result. For r <= 1 one must take a little more care, but one can still proceed without much difficulty by using the time-frequency analysis of Fefferman and Lacey-Thiele. Specifically, we introduce biparameter space-frequency tiles, gather them into “trees” of various “size” (a phase space localized Hardy-Littlewood maximal function) and “energy” (a phase space localized Littlewood-Paley square function), organize the trees in a sort of descending order, and sum them seperately. Despite using the product theory, the arguments are simpler than those for say, the bilinear Hilbert transform, because the nature of the paraproduct is such that the only frequency singularity is at the origin (or more precisely, the two co-ordinate axes) so there are far fewer tiles to deal with in the dyadic decomposition. Indeed, we also show in the paper that if we consider the simplest biparameter operator of bilinear Hilbert transform type, namely the double bilinear Hilbert transform, then this operator is in fact unbounded in any Lebesgue space (this may have been discovered earlier as “folklore”, but does not appear explicitly in print).

Sep 26, 2003

- Uploaded: “On the asymptotic behavior of large radial data for a focusing non-linear Schr\"odinger equation”, submitted to Journal of Partial Differential Equations and Dynamical Systems. Here we give some partial results toward the problem of soliton resolution for non-integrable focusing nonlinear Schrodinger equations. There seems to be a fundamental distinction between L^2 subcritical and L^2 supercritical equations here; I chose the best of both worlds, namely an L^2 subcritical equation with an a priori assumption of no blow up (i.e. H^1 norm stays bounded). Specifically, I chose the three-dimensional cubic focusing NLS assuming bounded energy. (One can remove this bounded energy assumption if one is willing to mollify the nonlinearity smoothly at infinity) Here we show that the solution splits into a linear part (radiation), plus a ‘weakly bound’ state which is asymptotically orthogonal to all linear solutions. Specializing then to the radial case, we show that this weakly bound state must either have mass and energy strictly bounded away from zero, or else decay to zero. Also, it converges (modulo an error which decays to 0 in homogeneous H^1 norm) to a smooth, decaying time-dependent object which obeys symbol bounds like <x>^{-3/2+}, and obeys an asymptotic Pohozaev identity. This smooth decaying object is supposed to be the soliton, although I was unable to get the exponential decay that one expects for solitons, and also I obtained no control on the long-term dynamics of this object (though in the short term it obeys an approximate NLS equation, of course). This seems to require some “low frequency” techniques, e.g. from stability analysis of periodic orbits or other machinery from dynamical systems or Hamiltonian mechanics; the argument here is mainly “high frequency” and is focused on showing that high frequency radiation must eventually leave the origin, thus leaving the residual bound state smooth. To obtain the (somewhat unsatisfactory) decay of <x>^{-3/2+}, we need to show that the further one gets from the origin, the lower the frequencies which can escape to infinity (basically at a distance R from the origin, any component with frequency R^{-1+} can escape to infinity). There is one novel trick, which is to distinguish between incoming waves (heading towards the origin) and outgoing waves (heading away from the origin), and to choose either Duhamel’s formula backwards in time or forwards in time appropriately (such a trick will also appear in a forthcoming paper with Igor Rodnianski concerning decay estimates for the Schrodinger equation on manifolds).
- Added: IPAM program – Multiscale geometry and analysis in high dimensions (Sep – Nov 2004, IPAM) and Recent Developments in Applied Harmonic Analysis: Multiscale Geometric Analysis (Apr 15-17 2004, U. Arkansas Fayetteville).

Sep 22, 2003

- Uploaded:
“Global regularity
for the Maxwell-Klein-Gordon equation with small critical Sobolev norm in
high dimensions”, joint with Igor
Rodnianski. In this paper
we obtain a small data critical regularity result for the
Maxwell-Klein-Gordon equation in the Coulomb Gauge (MKG-CG) in six and
higher dimensions. The
statement of this result is similar to recent work on wave maps, but the
techniques are slightly different.
As with wave maps, the problem is critical with derivatives and so
standard iteration methods do not work unless one works in a Besov space
instead of a Sobolev space.
For wave maps, the passage to the Coulomb gauge (either globally,
or microlocally) ameliorates the derivatives in the nonlinearities to an
extent that one can indeed close the argument by iteration methods;
however for MKG, the Coulomb gauge does not go far enough to eradicate
derivatives (though it is undeniably useful nevertheless, for instance, it
allows all the nonlinear terms in the equation for the connection A to be
treatable, at least in high dimension). In fact to close the argument what must be done is to
obtain global Strichartz estimates for the covariant wave equation
assuming that the connection A is in the Coulomb gauge and is small in a
certain critical norm. The
key idea here is to do this by constructing a global parametrix for the
covariant wave equation which is like the usual Fourier parametrix for the
free wave equation but distorted by what is basically a (microlocal)
Cronstrom gauge potential function for A. This introduces phase terms which look roughly like
exp(i nabla^{-1} A) which would normally be bad (because nabla^{-1} A is
in H^{n/2} which just barely fails to be bounded) however the fact that A
is real makes these phase terms bounded. One then has to carefully check that these parametrices
do make reasonably good Fourier integral operators, and that they solve
the covariant wave equation reasonably accurately, but this can be done by
stationary phase arguments.
One useful technical innovation, which may be useful elsewhere, is
a new decoupling device to get rid of annoying amplitude functions in
oscillatory integrals.

It seems likely that the techniques here extend from MKG to Yang-Mills (but now the connection evolves according to a covariant wave equation rather than the free wave equation, which may complicate some of the Fourier angular decompositions used in our paper). Of course the real prize here is the small energy implies regularity result for Yang-Mills in*four*dimensions, but in order to accomplish that one would need bilinear estimates for the covariant wave equation and not just Strichartz estimates. However the recent theory of bilinear estimates on rough metrics developed for the Einstein equations by Klainerman and Rodnianski may be useful here. - Updated once again: “An uncertainty principle for cyclic groups of prime order”. More feedback from Gerd Mockenhaupt and Melvyn Nathanson has been incorporated. A depressingly large number of typos for a six-page paper have also been detected and eradicated.
- Added:
52
^{nd}Midwest PDE (U. Minnesota, Nov 15-16). Thanks to Markus Keel for this link.

Sep 19, 2003

- Added: NFS/CMBS Regional Conference, May 23-28, 2004, Georgia Tech. Thanks to Gerd Mockenhaupt for this link.

Sep 12, 2003

- Uploaded: The expository note “Viriel, Morawetz, and interaction Morawetz inequalities”. In this note we gather some different ways to view the monotonicity formulae listed above for the non-linear Schrodinger equation; either as a conservation law arising from the mass and momentum density and current; or as a quantum version of a classical collision counting inequality; or as positive commutator estimates on the solution (or of the tensor product of the solution with itself). We also present variants of the Morawetz inequality derived by Bourgain, Grillakis, and Nakanishi, and also speculate on extensions to the Klein-Gordon equation.

Sep 10, 2003

- Updated: “An uncertainty principle for cyclic groups of prime order”. This new version incorporates some feedback from Robin Chapman, Roy Meshulam, Michael Cowling, and Gergely Harcos. In particular, a shorter proof of the Cauchy-Davenport inequality due to Robin Chapman is included, as well as mention of Roy Meshulam’s extension to (Z/pZ)^N.

Sep 2, 2003

- Created: Math 3228 page.
- Updated: “An uncertainty principle for cyclic groups of prime order”. It turns out that the main lemma of this result was already discovered by Chebotarev and independently by Dieudonne.

Aug 29, 2003

- Uploaded: “An uncertainty principle for cyclic groups of prime order”, submitted to Math Research Letters. In a finite abelian group G, the standard uncertainty principle asserts that the support supp(f) of a function, and the support supp(\hat{f}) of its Fourier transform, are related by the formula |supp(f)| |supp(\hat f)| >= |G|. Here we show that in the case of a group of prime order, G = Z/pZ, we can improve this to |supp(f)| + |supp(\hat{f})| >= p+1. Furthermore, this is absolutely sharp; given any two sets A and B with |A|+|B| >= p+1 we can find an f such that supp(f)=A and supp(\hat f) = B. The proof is based on computing minors of the Fourier matrix and uses a number of tricks from algebra (some basic Galois theory and Vandermonde determinants, to be precise). As a rather modest application of this uncertainty principle we can now give a Fourier-analytic proof of the Cauchy-Davenport inequality. It also seems to have application to the zeroes of sparse polynomials, although I do not know the literature in that area well enough to judge the ramifications.

Aug 6, 2003

- Added: ESI programs in Singularity formation in non-linear evolution equations (Jul 1 – Aug 15 2004), Geometric methods in analysis and probability (May-Aug 2005), and Modern methods of time frequency analysis (Spring 2005). After this I will no longer be maintaining my conference page due to lack of time; I will link instead to the automated TMR harmonic analysis conference page.

Jul
23, 2003

- Uploaded: “Global well-posedness of the Benjamin-Ono equation in H^1(R)”, submitted to J. Hyperbolic Diff. Eq. Here we improve the local well-posedness theory of the Benjamin-Ono equation u_t + Hu_xx = uu_x (a slightly less dispersive, but still completely integrable, variant of the KdV equation) from H^{5/4+} to H^1, and thus (by the H^1 conservation law) obtain global well-posedness. No new estimates are proven – in fact, one relies on plain old Strichartz estimates for the 1D Schrodinger equation; the trick here is to apply an algebraic gauge transformation, similar to the Cole-Hopf transformation, to eliminate the interaction between very low and very high frequencies.

Jun 13, 2003

- Uploaded: The expository note "Korner's Besicovitch set construction". This is a presentation of a recent argument of Tom Korner, who established that "most" Besicovitch sets have measure zero, in the sense of the Baire category theorem. It leads to one of the more intuitive explanations as to why Besicovitch sets exist, although the quantative bound obtained by this method is rather poor.
- Uploaded: “A positive proof of the Littlewood-Richardson rule using the octahedron recurrence”, with Allen Knutson and Chris Woodward. Here we give a combinatorial and self-contained proof of the Littlewood-Richardson rule for tensor product multiplicities of GL_n representations. Indeed we show that any rule which is associative and obeys the “Pieri rule” on generates must correspond to the puzzle rule (which is known to be equivalent to the Littlewood-Richardson rule). The key is to show that the puzzle rule is itself associative, which we do by means of the octahedron recurrence. An alternate geometric proof based on “scattering” of honeycombs is also provided.

Jun 9, 2003

- Uploaded: "Fuglede's conjecture is false in 5 and higher dimensions". We give an explicit example of a set in R^5 (a finite union of unit cubes, actually), which has an orthogonal basis of exponentials, but which does not tile R^5 by translations. This disproves (one direction of) a conjecture of Fuglede in higher dimensions. The example is based on Hadamard matrices (orthogonal matrices whose entries are +1 or -1) of order not equal to a power of 2; actually this only gives a counterexample in R^{11}, to get down to R^5 we need the corresponding concept for cube roots of unity instead of square roots of unity. There is still however a chance that the conjecture can be salvaged in one dimension. Also we do not know the status of the converse direction (are there sets which tile, but have no orthogonal basis? The difficulty here is that it is not so obvious how to find a set which tiles, but only in a non-lattice manner).
- Uploaded: "Park city notes on the Restriction problem". These are an expanded version of "Some recent progress on the Restriction conjecture" from the proceedings of the Fourier Analysis and Convexity workshop. In line with the Park City philosophy, many homework questions have been added, and also two additional lectures on the connection with Bochner-Riesz and PDE have also been added.
- Added: Conference on Classical Analysis in honor of Paul Koosis (Oct 23-26, 2003, CRM Montreal)

May 30, 2003

- Uploaded: The expository
note "The Fourier
transform on non-abelian finite groups". These are some
notes I made on the Fourier transform on a finite non-abelian group G;
this stuff is of course very basic to representation theorists and finite
group theorists, but does not seem to be as well publicized among
real-variable harmonic analysts as it could be. I restrict to the
finite case for simplicity, similar to how the theory of the finite
Fourier transform is technically much simpler than that of the Fourier
transform on R or Z or T, although algebraically of course they are almost
identical. The notes have a harmonic analysis bias and assume no
prior knowledge of representation theory. Incidentally, this topic
has nothing to do with the nonlinear Fourier transform; that deals with
the Fourier analysis of functions whose
*range*lies in a non-abelian group, whereas the above notes deal with the Fourier analysis of functions whose*domain*is a non-abelian group.

May 21, 2003

- Updated: The link for Minicourses in Analysis (Rozenbloum, Sogge, Soria) (U. Padova, June 23-27).
- Added: Conference on Partial Differential Equations and Applications (U. Notre Dame, IN Aug 14-17).

May 5, 2003

- Updated: The link for Additive Number Theory and Applications to Harmonic Analysis (CUNY Graduate Center, NY, May 15-17)
- Added: Equations aux dérivées partielles et quantification - Colloque en l'honneur de Louis Boutet de Monvel (Institut de Mathematiques de Jussieu, Jun 23-27) and Conference on Probability in Mathematics (in honor of Hillel Furstenberg), Jerusalem, Israel Jun 17-24, and Winter 2003 meeting of the Canadian Mathematical Society (special session in Harmonic Analysis), Simon Fraser University, Vancouver, Dec 6-8

May 2, 2003

- Added: AARMS-CRM - Workshop on singular integrals and analysis on CR manifolds (May 3-8 2004, CRM, U. Montreal Canada). Thanks to Galia Dafni for this link.

Apr 28, 2003

- Added: Conference on PDE and Applications (in honour of Aizik Volpert) (Haifa, Israel, June 11-16), Third Prairie Analysis Seminar (Kansas State University, Oct 17-18), 994th meeting of the American Mathematical Society (Special Session in Harmonic Analysis) (Florida State University, Mar 12-13), AIMS fifth international conference on dynamical systems and differential equations (Pomona, CA Jun16-19 2004)

Apr 9, 2003

- Added: Additive Number Theory and Applications to Harmonic Analysis (CUNY Graduate Center, NY, May 15-17)

Apr 1, 2003 (No joke!)

- Uploaded: Tom Wolff Memorial Lectures on non-linear dispersive equations. These lectures discuss some selected recent results on non-linear dispersive equations such as the Korteweg de Vries (KdV) equation, the nonlinear Schrodinger (NLS) equation, and the wave maps equation. In particular, some results on global well-posedness for rough data, on scattering for NLS, symplectic non-squeezing for KdV, and global regularity for wave maps is discussed. Some of this material has been taken from my earlier talks on this subject (notably my Chicago lectures and my Canberra lectures) but some material is new, particularly the section on Morawetz inequalities and the application to scattering.

Mar 28, 2003

- Created: Math 131BH (Honors Analysis B) class web page.
- Added: 7th International Conference on Harmonic Analysis and PDE (Jun 21-25 2004, El Escorial) and Conference in honor of Haim Brezis (Jun 21-25 2004, Paris)
- Added: Wavelets, Frames, Operator Theory Focused Research Group (FRG) home page. Thanks to Palle Jorgensen for this link.

Mar 21, 2003

- Added: Workshop on harmonic analysis "Young researchers days of the HARP network" (Orleans, France, Jun 16-17). Thanks to Phillipe Jaming for this link.

Mar 12, 2003

- Uploaded: "Some recent progress on the Restriction conjecture". This is for the proceedings of the Fourier Analysis and Convexity workshop from June 2001. Here I attempt to summarize the progress on the restriction conjecture, at least for the model hypersurfaces of the sphere, paraboloid, and cone. In particular I focus on giving a relatively informal description of some of the main ideas used in the modern theory, including: (a) using the decay of the Fourier transform of surface measure to localize a restriction estimate; (b) working in the bilinear setting instead of the linear one to eliminate parallel interactions; (c) using the wave packet decomposition to reduce the issue to that of estimating sums of oscillatory functions on tubes; (d) use of induction on scale arguments to eliminate "highly localized" interactions; (e) use of L^4 theory and the additive geometry of the surface to obtain further orthogonality conditions on the oscillations.

Mar 10, 2003

- Updated: "A sum-product estimate for finite fields, and applications", joint with Jean Bourgain and Nets Katz. The statement of Theorem 4.3 (the Freiman type theorem for sum-products) was incorrect as stated (as was the proof, of course); this has now been repaired, and some other minor corrections also made. (Fortunately, Theorem 4.3 was not used elsewhere in the paper).

Mar 2, 2003

- Added: Special Trimester on Phase Space Analysis of Partial Differential Equations (Centro de Giorgi, Pisa Feb 15-May 15 '04) and Special Trimester on Harmonic Analysis (Centro de Giorgi, Pisa Apr 12 - July 4 '04), and Conference on quadrature domains and applications (Mar 27-30, UCSB California).
- Updated: The link for PDE meeting (Jun 2-6, Forges Les Eaux, France)

Feb 4, 2003

- Updated: The link for Short program on Analysis and Resolution of Singularities (Aug 18-Sep 15, CRM Montreal, Canada)

Jan 31, 2003

- Updated: The link for the Park city program in harmonic analysis and PDE (Jun 30-18, Park City, UT)
- Added: The Twentyseventh Summer Symposium in Real Analysis (Jun 23-29, Silesian University, Czech Republic), Applicable Harmonic Analysis (Banff, Canada, June 7-12), Wavelet Theory and Applications: New Directions and Challenges (National U. Singapore, July 14-18), Analysis and Geometric Measure Theory (Banff, Canada, July 26-31)

Jan 28, 2003

- Uploaded: "A sum-product estimate for finite fields, and applications", joint with Jean Bourgain and Nets Katz. Here we extend the classical Erdos-Szemeredi estimate |A+A| + |A.A| >= |A|^{1+epsilon} from finite sets A of integers, to finite subsets A of the field Z/pZ, with p prime. The method is a variant of the recent paper of Edgar and Miller used to prove the Erdos ring conjecture, combined with some standard tools from additive number theory (in particular, sumset estimates, the Balog-Szemeredi theorem, and a recent lemma of Katz and myself). As applications we present a non-trivial incidence bound of Szemeredi-Trotter type in finite fields, some non-trivial progress on the Erdos distance problem in finite fields, and a new estimate for the size of Besicovitch sets (or Kakeya sets) in three-dimensional finite field geometries.
- Added: Analysis and geometry in Carnot-Caratheodory spaces (Mar 7-8, Fayetteville AR)

Jan 27, 2003

- Uploaded: "Global existence and
scattering for rough solutions of a nonlinear Schroedinger equation on R^3",
joint with Jim Colliander,
Mark Keel, Gigliola
Staffilani, and Hideo Takaoka, submitted to CPAM.
This is the full version of our earlier announcement paper, "Existence globale
et diffusion pour l'équation de Schrödinger nonlinéaire répulsive cubique
sur R^3 en dessous l'espace d'énergie". Here we flesh out
in full detail a new "interaction-Morawetz" inequality for
three-dimensional nonlinear Schrodinger equations, which by using nothing
more than an integration by parts, gives an a priori
*unweighted*L^4_{x,t} spacetime estimate on solutions to defocusing NLS. From this estimate one can obtain a new and simpler proof of scattering of the cubic defocusing NLS in the energy class, and also for the first time obtain scattering results for regularities below the energy class, and specifically in H^s for s > 4/5. In particular, we also obtain global well-posedness in this class (thus superceding our earlier global well-posedness result, which held for s > 5/6).

Jan 21, 2003

- Link updated: 22nd Annual Western States Mathematical Physics Meeting (Feb 17-18, Caltech)

Jan 14, 2003

- Added: Sixth Riviere-Fabes Symposium on Analysis and PDE (U. Minnesota, Apr 25-27), Perspectives in Analysis (KTH, Stockholm, May 26-28)
- Linked: Open problems in Harmonic Analysis

Jan 6, 2003

- Uploaded: Multiple Choice Quiz applet. This applet gives simple multiple choice quizzes. More quizzes will be added later, but currently I have a satisfactory list of quizzes on set theory, functions, and sequences, and plan to add more in the future.