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Just a brief post to record some notable papers in my fields of interest that appeared on the arXiv recently.

- “A sharp square function estimate for the cone in “, by Larry Guth, Hong Wang, and Ruixiang Zhang. This paper establishes an optimal (up to epsilon losses) square function estimate for the three-dimensional light cone that was essentially conjectured by Mockenhaupt, Seeger, and Sogge, which has a number of other consequences including Sogge’s local smoothing conjecture for the wave equation in two spatial dimensions, which in turn implies the (already known) Bochner-Riesz, restriction, and Kakeya conjectures in two dimensions. Interestingly, modern techniques such as polynomial partitioning and decoupling estimates are not used in this argument; instead, the authors mostly rely on an induction on scales argument and Kakeya type estimates. Many previous authors (including myself) were able to get weaker estimates of this type by an induction on scales method, but there were always significant inefficiencies in doing so; in particular knowing the sharp square function estimate at smaller scales did not imply the sharp square function estimate at the given larger scale. The authors here get around this issue by finding an even stronger estimate that implies the square function estimate, but behaves significantly better with respect to induction on scales.
- “On the Chowla and twin primes conjectures over “, by Will Sawin and Mark Shusterman. This paper resolves a number of well known open conjectures in analytic number theory, such as the Chowla conjecture and the twin prime conjecture (in the strong form conjectured by Hardy and Littlewood), in the case of function fields where the field is a prime power which is fixed (in contrast to a number of existing results in the “large ” limit) but has a large exponent . The techniques here are orthogonal to those used in recent progress towards the Chowla conjecture over the integers (e.g., in this previous paper of mine); the starting point is an algebraic observation that in certain function fields, the Mobius function behaves like a quadratic Dirichlet character along certain arithmetic progressions. In principle, this reduces problems such as Chowla’s conjecture to problems about estimating sums of Dirichlet characters, for which more is known; but the task is still far from trivial.
- “Bounds for sets with no polynomial progressions“, by Sarah Peluse. This paper can be viewed as part of a larger project to obtain quantitative density Ramsey theorems of Szemeredi type. For instance, Gowers famously established a relatively good quantitative bound for Szemeredi’s theorem that all dense subsets of integers contain arbitrarily long arithmetic progressions . The corresponding question for polynomial progressions is considered more difficult for a number of reasons. One of them is that dilation invariance is lost; a dilation of an arithmetic progression is again an arithmetic progression, but a dilation of a polynomial progression will in general not be a polynomial progression with the same polynomials . Another issue is that the ranges of the two parameters are now at different scales. Peluse gets around these difficulties in the case when all the polynomials have distinct degrees, which is in some sense the opposite case to that considered by Gowers (in particular, she avoids the need to obtain quantitative inverse theorems for high order Gowers norms; which was recently obtained in this integer setting by Manners but with bounds that are probably not strong enough to for the bounds in Peluse’s results, due to a degree lowering argument that is available in this case). To resolve the first difficulty one has to make all the estimates rather uniform in the coefficients of the polynomials , so that one can still run a density increment argument efficiently. To resolve the second difficulty one needs to find a quantitative concatenation theorem for Gowers uniformity norms. Many of these ideas were developed in previous papers of Peluse and Peluse-Prendiville in simpler settings.
- “On blow up for the energy super critical defocusing non linear Schrödinger equations“, by Frank Merle, Pierre Raphael, Igor Rodnianski, and Jeremie Szeftel. This paper (when combined with two companion papers) resolves a long-standing problem as to whether finite time blowup occurs for the defocusing supercritical nonlinear Schrödinger equation (at least in certain dimensions and nonlinearities). I had a previous paper establishing a result like this if one “cheated” by replacing the nonlinear Schrodinger equation by a system of such equations, but remarkably they are able to tackle the original equation itself without any such cheating. Given the very analogous situation with Navier-Stokes, where again one can create finite time blowup by “cheating” and modifying the equation, it does raise hope that finite time blowup for the incompressible Navier-Stokes and Euler equations can be established… In fact the connection may not just be at the level of analogy; a surprising key ingredient in the proofs here is the observation that a certain blowup ansatz for the nonlinear Schrodinger equation is governed by solutions to the (compressible) Euler equation, and finite time blowup examples for the latter can be used to construct finite time blowup examples for the former.

I’ve just uploaded to the arXiv my paper “A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential“, submitted to Dynamics of PDE. This paper continues some earlier work of myself in an attempt to understand the *soliton resolution conjecture* for various nonlinear dispersive equations, and in particular, nonlinear Schrödinger equations (NLS). This conjecture (which I also discussed in my third Simons lecture) asserts, roughly speaking, that any reasonable (e.g. bounded energy) solution to such equations eventually resolves into a superposition of a radiation component (which behaves like a solution to the linear Schrödinger equation) plus a finite number of “nonlinear bound states” or “solitons”. This conjecture is known in many perturbative cases (when the solution is close to a special solution, such as the vacuum state or a ground state) as well as in defocusing cases (in which no non-trivial bound states or solitons exist), but is still almost completely open in non-perturbative situations (in which the solution is large and not close to a special solution) which contain at least one bound state. In my earlier papers, I was able to show that for certain NLS models in sufficiently high dimension, one could at least say that such solutions resolved into a radiation term plus a finite number of “weakly bound” states whose evolution was essentially almost periodic (or almost periodic modulo translation symmetries). These bound states also enjoyed various additional decay and regularity properties. As a consequence of this, in five and higher dimensions (and for reasonable nonlinearities), and assuming spherical symmetry, I showed that there was a (local) *compact attractor* for the flow: any solution with energy bounded by some given level E would eventually decouple into a radiation term, plus a state which converged to this compact attractor . In that result, I did not rule out the possibility that this attractor depended on the energy E. Indeed, it is conceivable for many models that there exist nonlinear bound states of arbitrarily high energy, which would mean that must increase in size as E increases to accommodate these states. (I discuss these results in a recent talk of mine.)

In my new paper, following a suggestion of Michael Weinstein, I consider the NLS equation

where is the solution, and is a smooth compactly supported real potential. We make the standard assumption (which is asserting that the nonlinearity is mass-supercritical and energy-subcritical). In the absence of this potential (i.e. when V=0), this is the defocusing nonlinear Schrödinger equation, which is known to have no bound states, and in fact it is known in this case that all finite energy solutions eventually *scatter* into a radiation state (which asymptotically resembles a solution to the linear Schrödinger equation). However, once one adds a potential (particularly one which is large and negative), both *linear *bound states (solutions to the linear eigenstate equation ) and *nonlinear* bound states (solutions to the nonlinear eigenstate equation ) can appear. Thus in this case the soliton resolution conjecture predicts that solutions should resolve into a scattering state (that behaves as if the potential was not present), plus a finite number of (nonlinear) bound states. There is a fair amount of work towards this conjecture for this model in perturbative cases (when the energy is small), but the case of large energy solutions is still open.

In my new paper, I consider the large energy case, assuming spherical symmetry. For technical reasons, I also need to assume very high dimension . The main result is the existence of a *global* compact attractor K: every finite energy solution, no matter how large, eventually resolves into a scattering state and a state which converges to K. In particular, since K is bounded, all but a bounded amount of energy will be radiated off to infinity. Another corollary of this result is that the space of all nonlinear bound states for this model is compact. Intuitively, the point is that when the solution gets very large, the defocusing nonlinearity dominates any attractive aspects of the potential V, and so the solution will disperse in this case; thus one expects the only bound states to be bounded. The spherical symmetry assumption also restricts the bound states to lie near the origin, thus yielding the compactness. (It is also conceivable that the localised nature of V also restricts bound states to lie near the origin, even without the help of spherical symmetry, but I was not able to establish this rigorously.)

This weekend I was (once again) in San Diego, this time for the Southern California Analysis and PDE (SCAPDE) meeting. I gave a talk on “The asymptotic behaviour of large data solutions to NLS”, which is based on two of my previous papers on what solutions to focusing nonlinear Schrödinger equations behave like as time goes to infinity. (Note that this is a specialist conference, and this talk will be a bit more technical than some of the general-audience talks that I have blogged about previously.)

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