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What was new in 2005?  2004? 2003? 2002? 2001? 2000? 1999?

Dec 29, 2006

  • Uploaded: “John-type theorems for generalized arithmetic progressions and iterated sumsets”, with Van Vu, submitted, Adv. in Math..  Here we elaborate on, correct, and improve some material from our book concerning how to compare generalized arithmetic progressions with proper counterparts.  In particular we obtain a new John-type theorem showing how every progression contains a proper progression and is in turn contained in a dilate of that progression.  If one uses “convex progressions” instead of progressions then the containment constants are much better (linear in the dimension rather than exponential).  We also extend the arguments (which are quite elementary, the deepest tool being Minkowski’s second theorem) to the case when the ambient abelian group has torsion, in which case progressions must be generalized to coset progressions.  As an application we obtain a similar John-type theorem for iterated sumsets, showing that a sufficiently large sumset in any abelian group is eventually comparable in the previous sense to a proper coset progression.  This extends earlier work of Szemeredi and Vu and also gives quantitative bounds (sadly, they are triple-exponential in the dimension).

Dec 15, 2006

  • Uploaded: “A priori bounds and weak solutions for the nonlinear Schrodinger equation in Sobolev spaces of negative order”, with Michael Christ and Jim Colliander.    This paper serves as a kind of counterpoint to our previous joint papers on the NLS, which have mostly focused on illposedness issues at low regularities (and in particular in Sobolev spaces of negative index).  Here, we revisit a specific equation (the cubic NLS on the real line), which we showed to not have a uniformly continuous local solution map at negative index Sobolev spaces (despite having global well-posedness at higher regularities).  Nevertheless, we show here that smooth solutions obey a local a priori bound in these norms, and as such one can construct strong solutions to these equations (in the sense that they lie in C^0_t H^s_x and solve the equation in an integral sense).  However we do not have any uniqueness or continuity results for such solutions (which is consistent with our previous result demonstrating lack of uniform continuity).  There are several ingredients.  The first is to exploit a certain smoothing effect which permits us to obtain H^s bounds (but not X^{s,b} bounds) even when the solution is only known to lie in a rougher space X^{r,b}; here we have to crucially use frequency localised versions of the mass conservation law.  The second is to estimate the X^{r,b} norm by a more standard iteration argument, but using a variant Y^{s,b} of the X^{s,b} norms which tolerates more of a deviation away from the characteristic surface tau=xi^2 than the usual X^{s,b} spaces.

Dec 6, 2006

  • Uploaded: the short story “The Lindenstrauss maximal inequality”.  Here I present an abstract version of a recent strengthening of the Hardy-Littlewood maximal inequality due to Elon Lindenstrauss, in which the usual doubling condition on balls is replaced by two weaker conditions.  One asserts, roughly speaking, that the set formed by attaching a lot of small balls to a large ball does not have volume significantly larger than the original large ball, and also that adjacent balls of the same radius have comparable volume.  Actually the metric structure is not used at all and one can phrase the question in terms of abstract maximal operators formed from integral operators with “narrow” support.  The key new innovation in Lindenstrauss’ work is to replace the greedy selection procedure of the Vitali covering lemma at any given radius scale by a randomized selection procedure.  As one application we almost recover the Stein-Stromberg bound of O(d) on the weak (1,1) constant in the Hardy-Littlewood maximal inequality, recovering the slightly weaker bound of O( d log d ) instead.

Nov 13, 2006

  • Uploaded: “A (concentration-)compact attractor for high-dimensional non-linear Schr\"odinger equations”, submitted, Dynamics of PDE.  This is a sequel of sorts to an earlier paper on the asymptotics of large data focusing NLS: this time we consider a general non-critical Hamiltonian NLS in five and higher dimensions, with and without the assumption of spherical symmetry, but with the assumption of bounded energy.  The main result is that even though there is no dissipation in this equation (only dispersion), one still has a compact attractor for the dynamics in the spherically symmetric case (once one subtracts off the linear radiation), and a concentration-compact attractor in the general case.  This is still some ways off from the notoriously difficult “soliton resolution conjecture” for these equations, but at least establishes the substantially weaker “petite conjecture” of Soffer for these equations.  The restriction to five and higher dimensions is related to the absence of resonances of Schrodinger operators in these dimensions, and at a technical level arises from the need to have a decay for the fundamental solution which is faster than t^{-2}, in order to exploit a “double Duhamel trick” to time-localise the NLS integral equation and turn it into a sort of dispersive analogue of the ground state equation.

Nov 4, 2006

  • Uploaded: the short story “The Christ cube construction”.  This is an exposition of a result of Michael Christ in 1990 obtaining dyadic cubes in homogeneous spaces.  I have worked out some explicit constants and rephrased things in slightly more “modern” language using ultrametrics and chaining arguments.

Oct 29, 2006

Oct 25, 2006

  • Uploaded: the short story “Quasi-metric spaces”.  These are some brief notes on the Aoki-Rolewicz theorem and variants, which assert that a quasi-metric is comparable to some power of a metric, and a quasi-norm is comparable to a p-power summable norm for some p > 1; in fact some very explicit constants are given (involving the golden ratio, of all things!).

Oct 19, 2006

  • Uploaded: “New bounds for Szemeredi's Theorem, II: A new bound for r_4(N)”, with Ben Green.  This is the second paper in a series in which we try to bring the quantitative bounds for Szemeredi’s theorem for progressions of length 4 to match (or come closer to) the known bounds for progressions for length three.  In the first paper in the series, we established these types of bounds in the “finite field model” (where Z/NZ is replaced by a vector space over a finite field), with two arguments, a “cheap” argument (using a somewhat efficient version of the Roth-Gowers argument, in which we gather many quadratic relations before linearising) and an “expensive” argument (in which we gather the quadratic relations, pass to a “nilmanifold” model, count progressions on that model, and then linearise).  In this paper we perform the “cheap” argument in Z/NZ, eventually concluding that any subset of {1,…,N} of density at least N exp( - c sqrt(log log N) ) will contain arithmetic progressions of length 4.  In the third paper in the series we shall perform the “expensive” argument in Z/NZ, improving the above result to N log^{-c} N.

Oct 9, 2006

  • Uploaded: The “short” story “Perelman’s proof of the Poincare conjecture – a nonlinear PDE perspective”.  This project grew out of a desire to understand the work of my fellow ’06 classmate Grisha Perelman on Poincare and geometrization, especially given how closely it resembled standard paradigms in nonlinear PDE; I was hoping that by reading Perelman’s papers concurrently with the expositions of Kleiner-Lott, Morgan-Tian, and Cao-Zhu, that I would be able to extract a short heuristic sketch of the argument that might be accessible to nonlinear PDE experts.  Rather naively I had thought that perhaps Perelman’s argument relied on only a handful of new ideas that could be easily presented in a short story; it was only after I was halfway through that I realized just how many subtle tricks and insights went into this incredibly impressive argument.  Anyway, at 51 pages (longer than most of my research papers, and comparable to the length of Perelman’s original work) this no longer qualifies as a “short story”, but perhaps it still has value as a high-level description of the various technical aspects of the argument (with various computations, particularly those of a geometric or topological nature, omitted, in order to focus on the nonlinear PDE aspects of the argument).  The material here is drawn both from Perelman’s original papers and from the three surveys listed above, so I have tried to also present a sort of “concordance” showing where various steps of the argument appear in all four of these sources.

Oct 1, 2006

  • Uploaded: “The primes contain arbitrarily long polynomial progressions”, with Tamar Ziegler, submitted, Acta Math.   Here we extend my theorem with Ben Green on arithmetic progressions x, x+m, …, x+(k-1)m in the primes to the more general pattern of polynomial progressions x + P_1(m), …, x + P_k(m), where P_1,…,P_k are integer polynomials with vanishing constant coefficient.  We basically follow the method of the previous paper, but with Szemeredi’s theorem replaced by a (suitably uniform version of the) Bergelson-Leibman theorem.  Several new technical issues arise.  Firstly, one must truncate the shifts m to a substantially smaller range than x.  This is not a major problem – one simply has to reduce the sieve level accordingly – but the more difficult issue arises when trying to estimate polynomial averages by Gowers-type norms.  The trick is to use van der Corput’s lemma repeatedly (via PET induction) rather than Cauchy-Schwarz, and to keep the shifts arising from that lemma bounded by an extremely small power of x (much smaller than m).  This turns out to be necessary for a certain “clearing denominators” step when one has to control some very large degree polynomial averages arising from the Weierstrass approximation theorem step of the argument.  Finally, one ends up having to control various polynomial correlation averages of the enveloping sieve nu which involve both long-range and short-range shifts.  Of course we compute the long-range averaging first; a new issue here is that we have to count points in algebraic varieties over F_p rather than affine subspaces, which is of course quite non-trivial in general.  Fortunately, the polynomials that cut out those varieties happen to be linear in at least one long-range variable, which allows us to avoid the use of any deep arithmetic geometry.

Sep 28, 2006

  • Uploaded: “The cosmic distance ladder” (with accompanying figures).  These are the slides I used for my education afternoon talk at the AustMS conference.  I would like to work with someone with experience in astronomy and in computer media (or perhaps an undergraduate REU with background in these areas) to create a better-looking and better-researched version of this presentation; please contact me if you are interested in collaborating on this.
  • Uploaded: “Long arithmetic progressions in the primes”.  These are the slides I used for my plenary talk at the AustMS conference.  They are of course based on earlier slides of myself on this topic, but have been updated.
  • Uploaded: “Minimal-mass blowup solutions of the mass-critical NLS”, with Monica Visan and Xiaoyi Zhang.  This paper represents the “standard” half of the general program that Monica, Xiaoyi, and I have to understand the large data global behavior of defocusing and focusing mass-critical NLS for both radial and non-radial data.  It is by now fairly well understood that the best way to establish these critical results is to first obtain some sort of compactness result, based almost entirely on harmonic analysis (e.g. Strichartz estimates and compensated compactness) and perturbation theory, which reduces matters to understanding “almost periodic” solutions, taking into account the symmetries of the equation of course.  The second (and significantly more difficult) step would then be to employ (suitably localized versions of) monotonicity formulae and other dynamical control on physical quantities to rule out various blowup scenarios relating to these almost periodic solutions.  The second step will probably require case-by-case treatment depending on the dimension, the sign of the nonlinearity, and the symmetry of the data.  However, the point of the present paper is that the first step is much more abstract (and better understood) and can be done in all cases simultaneously; our main result is that if there is a minimal mass for which blowup occurs, then there exists a minimal mass blowup solution which is almost periodic modulo the symmetries of the equation.  This improves upon previous results of Keraani and Begout-Vargas, and is also closely related to results on the energy-critical problem by Bourgain, Colliander-Keel-Staffilani-Takaoka-Tao, Ryckman-Visan, Visan, and Kenig-Merle.  In fact we follow the Kenig-Merle approach which synthesizes the induction-on-energy method of Bourgain and the concentration-compactness method of Lions to create a nicely abstract and clean framework which should in fact work for many other critical equations with a group of symmetries which somehow generates all the “defects of compactness” in the problem.
  • Uploaded: “Global well-posedness and scattering for the mass-critical nonlinear Schr\”odinger equation for radial data in high dimensions”, with Monica Visan and Xiaoyi Zhang.  Here we complete the second part of the mass-critical NLS program in the simplest case we could find, namely the defocusing mass-critical NLS for radial data in three and higher dimensions.  In this case the ordinary one-particle Morawetz inequality (after localizing to low frequencies to obtain bounded momentum) turns out to be sufficient (combined with the radial Sobolev inequality and some radial improved Strichartz inequalities of Vilela) to obtain a preliminary spacetime bound on the almost periodic solutions mentioned earlier.  A mass evacuation argument (which is in many ways an inversion of the energy evacuation argument used in the energy-critical theory), again based heavily on improved Strichartz estimates, then closes the argument.
  • Uploaded: “A counterexample to an endpoint bilinear Strichartz inequality”, submitted, EJDE.  This paper disproves a bilinear L^1_t L^infty_x Strichartz estimates in the “forbidden endpoint” case in which the L^2_t L^infty_x linear estimate is known to be false (e.g. for two-dimensional Schrodinger or three-dimensional wave). This question was asked independently (and at widely different times) by Ioan Bejenaru for the Schrodinger equation and Sergiu Klainerman for the wave equation.  As it turns out, a standard and brief random sign argument lets one convert any counterexample in the linear setting to the bilinear setting (it seems to have to do with the exponent 2 in L^2_t; for higher exponents the bilinear theory of course does offer improvements).  So it seems that the forbidden endpoint remains forbidden even with bilinearization.


Aug 27, 2006

Aug 25, 2006

Aug 22, 2006

  • Thanks to everyone who has called or emailed in congratulations, it means a lot to me… unfortunately I cannot reply to all of them, but I am certainly very touched by it all.  (Now I need to get some rest…)  Thanks also to my many wonderful co-authors, without which most of my work would not have been possible.
  • Just for the record, in my own personal opinion, the work of Grisha Perelman in which he solves the Poincaré Conjecture is by far the most significant mathematical work appearing in the last ten years, and I am truly humbled to have been selected to accompany his award.

Aug 11, 2006

Aug 7, 2006

  • Open for business: The DispersiveWiki!  This is the successor to the Dispersive page, which has not been maintained since 2004 and is now obsolete.  This is an experimental wiki in which we hope to tap the collective expertise of the nonlinear dispersive and wave equation community to create a useful and current reference on many topics of interest in that field, such as the Cauchy problem for various equations or descriptions of various key concepts.  Please don’t hesitate to make a contribution, no matter how small!

July 21, 2006

  • Uploaded: the short story “Gauges for the Schrodinger map”.  These are some informal notes – basically, algebraic and geometric computations - arising from discussions with Ioan Bejenaru on what various gauges for the Schrodinger map look like when the target is a sphere.  These are probably only of interest to the very specialized set of people who want to analyze Schrodinger maps, but I’ll place it here in case someone finds it useful.

July 17, 2006

  • Uploaded: the short story “Nash-Moser iteration”.  These are my notes on understanding the abstract Nash-Moser-Hamilton iteration scheme.

June 11, 2006

  • Uploaded: “A pseudoconformal compactification of the nonlinear Schrodinger equation and applications”, submitted, New York Journal of Mathematics.  Here I try to popularize the lens transform for the nonlinear Schrodinger equation, used recently by Carles but not widely known in the field.  The lens transform is a variant of the much more well-known pseudoconformal transformation, but rather than inverting time (t -> -1/t), it compactifies time (t -> arctan(t)) to the interval (-pi/2,pi/2), at the cost of adding a attractive harmonic potential |x|^2/2 to the linear part of the evolution.  This compactification clarifies much of the “pseudoconformal” theory of the NLS, including the scattering theory and the role of the pseudoconformal energy, in particular explaining why for the pseudoconformal NLS the global existence theory, the scattering theory, and the spacetime bound theory are equivalent (a recent result of Begout-Vargas and Keraani).  In this paper I also extract an “inverse Strichartz theorem” implicit in the recent work of  Begout and Vargas, which I believe will be useful for further applications in NLS (especially L^2-critical NLS).
  • Uploaded: “Two remarks on the generalised Korteweg-de Vries equation”, submitted, Discrete Cont. Dynam. Systems.  Here I collect two (mostly unrelated) new observations on the generalized Korteweg-de Vries (gKdV) equation.  The first is that the conjectural spacetime bounds for the L^2-critical gKdV equation in fact imply the corresponding conjectured spacetime bounds for the L^2-critical NLS equation (in one dimension).  Thus bounding solutions to the gKdV equation is at least as hard as bounding solutions to the NLS equation, which is still open (though there is some encouraging progress in higher dimensions).  The approach is based on an asymptotic embedding of NLS into gKdV used by Christ, Colliander, and myself in an earlier paper.  The second observation is that in the defocusing case, while the energy and mass were both known to propagate to the left, the energy in fact propagates to the left faster than the mass, leading to a dispersive estimate which is a weak analogue of the Martel-Merle “Liouville theorem”.  Whereas the Liouville theorem pertained to solutions close to a soliton in the focusing case, the dispersive estimate here pertains to arbitrarily large (but decaying) solutions in the defocusing case.  Thus this estimate may be a key ingredient in the ultimate proof of the scattering conjecture for L^2-critical gKdV, though the first observation shows us that we must tackle the simpler NLS problem first.

June 7, 2006

  • Uploaded: “Global regularity for a logarithmically supercritical defocusing nonlinear wave equation for spherically symmetric data”, submitted, J. Hyperbolic Diff. Eq..  It’s been quite a while since I wrote a six-page paper (compare with the two papers with Ben Green below).  This paper resulted from an interesting question of Patrick Gerard, as to whether good spacetime bounds for critical equations imply anything about supercritical equations.  The answer seems to be: yes, but barely.  In particular, we take the simplest critical large data result known – namely, global regularity for the 3D energy-critical defocusing NLW Box u = u^5 with radial data, for which a simple argument of Ginibre, Soffer, and Velo gives a very good bound – and show that the exact same argument extends to the slightly supercritical NLW Box u = u^5 log(2 + u^2).  This of course is nowhere near true supercritical equations such as Box u = u^7, but it does show that the critical theory can reach just a tiny little bit into the supercritical domain.

June 4, 2006

  • Uploaded: “Quadratic uniformity of the M\"obius function”, with Ben Green, submitted, Annales de l’Institut Fourier.  This is the long-delayed second paper in a trilogy (the first concerning an inverse theorem for the Gowers U^3 norm).  Here we use the methods of Vinogradov and Vaughan to show that the Mobius function mu(n) is asymptotically orthogonal to all “quadratic” sequences including the “genuinely quadratic phase functions” exp( 2pi i alpha n^2 ), but more generally including “bracket quadratic phase functions” such as exp( 2pi i [alpha n] beta n ), where [x] is the integer part of x.  Actually, the full statement of our result is easiest to phrase using 2-step nilsequences, though I won’t do so here.  This is a rather technical estimate, but it has direct application in counting the number of prime points in certain affine lattices (see below).  Our techniques are based around an “inverse Vaughan approach”; we assume for contradiction that mu has a significant correlation with one of these sequences, then deduce that either a “type I” or “type II” sum is large.  The phases in such sums turn out to behave “quartically”, and so a certain amount of van der Corput type trickery then shows that these phases must be “major arc” to top order.  One has to massage this condition a little bit using some “bilinear geometry of numbers”, but eventually one shows that the quadratic phase is essentially negligible, at which point one then turns to the linear component of the phase and argues again.  Eventually one is reduced to checking the uniform distribution of mu on periodic sequences, which follows from a classical result of Siegel.
  • Uploaded: “Linear equations in primes”, with Ben Green, submitted, Annals of Math..  This is the third paper in the trilogy, and the one with all the interesting applications (but with all the really technical computations stashed away in other papers).  It is roughly analogous to our paper on long arithmetic progressions in the primes (which also drew on another paper to do all the “heavy lifting”, namely the paper proving Szemer\’edi’s theorem).  In this paper we show that if one can verify two conjectures, which we call the “Gowers inverse conjecture” GI(s) and the “Mobius and nilsequences conjecture” MN(s) for some given s=1,2,…, then we can count the number of prime points in any affine lattice of “complexity” s (verifying Dickson’s conjecture for those lattices).  What complexity means is a little technical, but think of it as like codimension; take a (nondegenerate) affine sublattice of Z^d of codimension s, and you can count prime points in it.  Unfortunately, this little adjective “nondegenerate” means that we can’t get the truly prized cases of Dickson’s conjecture, such as the twin prime, even Goldbach, or Hardy-Littlewood prime tuples conjecture, but what we can do is count how many progressions of a given length lie in an interval, or show that the primes contain arbitrarily high-dimensional parallelograms with one vertex fixed at 1 (and not counted as prime, of course).  The above two conjectures are now known to be true for s <= 2 (this is the point of the other two papers in the trilogy), and we are currently working on getting them for general s (stay tuned).  Roughly speaking, GI(s) asserts that any (bounded) function will be randomly distributed with respect to complexity s averages so long as they do not correlate with s-step nilsequences; MN(s) asserts that the Mobius function mu(n) indeed has no such correlation.  To apply this to the primes, one has to replace mu(n) with the von Mangoldt function Lambda(n) (modulo some technical maneuvers such as the W-trick).  The key difficulties here are that (a) the von Mangoldt function is not uniformly bounded, but grows logarithmically, and (b) the bounds that come out of GI(s) are only of a qualitative o(1) nature and so cannot absorb logarithmic losses.  (On the other hand, the bounds coming out of MN(s) can absorb logarithms.)  To resolve these difficulties one needs to rather carefully apply the transference principle technology from our earlier paper; each of the individual tools (pseudorandom majorant, dual functions, Koopman von Neumann theorem, generalized von Neumann theorem, W-trick, smooth/rough decomposition of divisor sums) is a variant of one which is already in the literature, but it is the order in which they are applied which turns out to be the main technical issue.

May 18, 2006

  • Uploaded: The short story “The Kenig-Merle scattering result for the energy-critical focusing NLS”.  These are some notes I wrote concerning Kenig and Merle’s recent  result of global existence and scattering for radial solutions to the energy-critical focusing NLS with energy less than that of the ground state.  The key innovation in their work is a certain local virial identity which forces energy decay inside a spacetime cylinder; it turns out that this can then be placed into the existing theory for the defocusing equation to give an alternate derivation of their main theorem.

May 15, 2006

  • Uploaded: “Scattering for the quartic generalised Korteweg-de Vries equation”, submitted, J. Diff. Eq..  Here we study the global asymptotic behaviour of the quartic gKdV equation.  A previous result by Grunrock gives local wellposedness to within an epsilon of the critical regularity H^{-1/6}; by using now standard critical space technology (a hybrid of X^{s,b,q} spaces and L^1_t L^2_x type spaces) we push this to the critical regularity.  This is the first critical regularity result for a model nonlinear dispersive equation in a negative Sobolev space. We then revisit an asymptotic stability result for the ground state solution of this equation due to Martel and Merle, and use this critical space machinery to upgrade the estimates for the radiation term, in particular demonstrating that it scatters to a solution of the linear Airy equation.
  • Uploaded: The short story “An informal derivation of the Schr\"odinger equation”.  This is just the textbook undergraduate physics derivation of the Schrodinger equation, but it seems that not everyone in the mathematical study of PDE is familiar with it, so I briefly reproduce it here.

Apr 20, 2006

  • Uploaded: The short story “Symmetries, scaling, and dimensional analysis”.  I have found dimensional analysis to be a tremendously useful tool in my own work, but sometimes have difficulty explaining how I can use it to eliminate bad arguments and locate good ones when trying to prove an estimate, or in eliminating useless hypotheses and highlighting the most important conclusions of a result.  Here is my attempt at formalizing my intuition on this subject.
  • Uploaded: The lecture notes “The ergodic and combinatorial approaches to Szemer\'edi's theorem”.  These are based on the lectures I gave at the workshop on additive combinatorics at Montreal on Apr 6-12 2006.  Here we discuss the three known direct proofs of van der Waerden’s theorem (classical colour focusing, topological dynamics colour focusing, and Shelah’s proof), the correspondence between Szemer\’edi’s theorem and ergodic theory, some basic recurrence theorems, an analytic approach to the triangle removal lemma and its transference to a pseudorandom counterpart, and an informal discussion of Szemer\’edi’s original proof.  The intent here is to give a sampling of the various approaches to Szemer\’edi’s theorem (omitting completely the important Fourier-analytic approach of Roth and then Gowers), though we do not actually go so far as to provide a full proof of this theorem.

Mar 20, 2006

Feb 17, 2006

  • Uploaded: The short story “Modulation stability – a very simple example”.  This grew out of an attempt to understand the modulational stability of ground states of NLS, as analyzed by Weinstein and later authors.  The concepts behind these arguments are rather obscured when working in a complicated PDE context, so I decided to present them here in an extremely special case, namely that of understanding the nonlinear ODE phi’ = i |phi|^2 phi with initial data phi(0) = 1+eps, thus perturbing off of the “ground state” R_0(t) = exp(it).  While this ODE is so simple that it can be solved explicitly, it still contains the basic ingredients necessary to conduct a modulational stability analysis, namely a symmetry of the equation (in this case, phase rotation symmetry) and the ability to write the linearised operator in an autonomous form with well-understood spectral properties.

Feb 1, 2006

  • Uploaded: “A correspondence principle between (hyper)graph theory and probability theory, and the (hyper)graph removal lemma”.  This paper (which has been almost totally rewritten from some earlier privately circulated versions) is an attempt to answer a question of Tim Gowers, namely to find an infinitary analogue of graph theoretic results such as the triangle and hypergraph removal lemmas, just as the Furstenberg recurrence theorem provides an infinitary analogue of Szemeredi’s theorem on arithmetic progressions.  After a couple clumsy attempts, I finally hit upon what is probably the most natural correspondence principle between the finitary and infinitary approach to graph theory, namely a “statistical” or “property testing” correspondence principle that starts with a sequence of increasingly large dense graphs (or hypergraphs) and ends up with a random infinite graph (or hypergraph) as a limiting object.  Furthermore, this random graph has a very strong invariance (it is invariant under the infinite permutation group S_infty) and also enjoys a number of “partitions” (or more precisely sigma-algebra factors) with respect to which the graph is perfectly regular (or more precisely, these factors are relatively independent of each other).  In this setting it is possible to prove difficult results such as the hypergraph removal lemma (which implies, among other things, Szemeredi’s theorem on arithmetic progressions) in a surprisingly clean manner, in particular without the logistical headaches of organizing large armies of epsilons to attack other large armies of epsilons.  It begins to shed some light on exactly what the connection is between the hypergraph approach and the ergodic approach to arithmetic progressions.

Jan 18, 2006

  • Uploaded: “Breaking duality in the return times theorem”, with Ciprian Demeter, Michael Lacey and Christoph Thiele, submitted, Acta Math.  The Return Times Theorem of Bourgain asserts, roughly speaking, that the values of an L^p function f on a dynamical system will almost surely provide a set of weights for a universal ergodic theorem that then gives convergence for any L^q function g on an arbitrary second dynamical system, as long as we are in the duality range 1/p + 1/q <= 1; this generalizes a classical ergodic theorem of Wiener and Wintner.  Actually a simple application of Holder’s inequality allows one to reduce this theorem to the (quite nontrivial) case when f and g are both bounded, where the issue (from the analytical perspective rather than the dynamical one) is to control a sort of maximal operator norm inequality.  We show that we can go beyond the duality range, to p>1 and q>= 2, and similarly for bilinear Hilbert transform type averages.  The latter is sort of a operator-norm version of Carleson’s inequality, where one considers a maximal multiplier norm of the partial Fourier integrals rather than just their supremum.  To achieve these goals we need to revisit the time-frequency proof of Carleson’s inequality, and recast it in such a way that one does not need to understand particularly the norm that is being taken on the partial Fourier integrals until late in the argument, when one has mostly restricted to a single tree.  The multiplier norms involved are then controlled mostly by variational norms.
  • Uploaded: “Product set estimates for noncommutative groups”, submitted, Combinatorica.  This is a spinoff from my book with Van Vu, and also from some discussions with Jean Bourgain.  Here we address the question of to what extent the standard theory of sum set estimates in additive groups – in particular, the Plunnecke-Ruzsa inequalities, the Balog-Szemeredi-Gowers theorem, and Freiman’s theorem – carry over to product set estimates in nonabelian multiplicative groups.  As pointed out to me by Jean, in this case there is a decoupling between the discrete and continuous cases (where one deals with finite sets or open sets respectively), but it turns out that a sufficiently abstract formulation of the theory can handle both in a unified manner.  (The discrete noncommutative estimates were sketched in my book with Van.)  On Freiman’s theorem, we make much less progress (the problem seems to come perilously close to demanding a classification of all subgroups of a given group) but we do have a new classification in the case of Heisenberg groups.

Jan 16, 2006

  • Uploaded: the short story “The Baker-Campbell-Hausdorff formula”.  There is nothing new here; this is just some notes I wrote up while learning about the standard derivation of the explicit Baker-Campbell-Hausdorff formula.  This formula is perhaps not as well known as it should be, so I have made it publicly available.

Jan 8, 2006

  • Uploaded: “Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions”.  This is a spinoff from my CBMS book, which fell out after I started writing about the energy-critical NLS and energy-critical NLW in the same chapter.  Basically, it turns out that much of the machinery of the former can be applied to the latter, in particular giving a new proof of global wellposedness and scattering for the NLW with spacetime norm bounds which are of exponential type with respect to the energy (or more precisely of the form E^O( E^{105/2} )).  The main ingredients are quantitative versions of the energy decay estimates of Grillakis and Shatah-Struwe with the machinery of exceptional and unexceptional intervals from a previous paper of mine, combined with the induction on energy strategy of Bourgain (but we use the latter only lightly, to avoid messing up the bounds too much).  I also use a simple argument based on the fundamental solution and energy flux estimates to show that “long range” effects of the nonlinearity are quite regular and thus mostly negligible, as well as an “inverse Sobolev inequality” that shows that the potential energy can only get large (which is one of the main sources of nonlinear behaviour for this equation) if the mass and energy concentrate in a ball.