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Emmanuel Breuillard, Ben Green, and I have just uploaded to the arXiv our survey “Small doubling in groups“, for the proceedings of the upcoming Erdos Centennial.  This is a short survey of the known results on classifying finite subsets $A$ of an (abelian) additive group $G = (G,+)$ or a (not necessarily abelian) multiplicative group $G = (G,\cdot)$ that have small doubling in the sense that the sum set $A+A$ or product set $A \cdot A$ is small.  Such sets behave approximately like finite subgroups of $G$ (and there is a closely related notion of an approximate group in which the analogy is even tighter) , and so this subject can be viewed as a sort of approximate version of finite group theory.  (Unfortunately, thus far the theory does not have much new to say about the classification of actual finite groups; progress has been largely made instead on classifying the (highly restricted) number of ways in which approximate groups can differ from a genuine group.)

In the classical case when $G$ is the integers ${\mathbb Z}$, these sets were classified (in a qualitative sense, at least) by a celebrated theorem of Freiman, which roughly speaking says that such sets $A$ are necessarily “commensurate” in some sense with a (generalised) arithmetic progression $P$ of bounded rank.   There are a number of essentially equivalent ways to define what “commensurate” means here; for instance, in the original formulation of the theorem, one asks that $A$ be a dense subset of $P$, but in modern formulations it is often more convenient to require instead that $A$ be of comparable size to $P$ and be covered by a bounded number of translates of $P$, or that $A$ and $P$ have an intersection that is of comparable size to both $A$ and $P$ (cf. the notion of commensurability in group theory).

Freiman’s original theorem was extended to more general abelian groups in a sequence of papers culminating in the paper of Green and Ruzsa that handled arbitrary abelian groups.   As such groups now contain non-trivial finite subgroups, the conclusion of the theorem must be  modified by allowing for “coset progressions” $P+H$, which can be viewed as “extensions”  of generalized arithmetic progressions $P$ by genuine finite groups $H$.

The proof methods in these abelian results were Fourier-analytic in nature (except in the cases of sets of very small doubling, in which more combinatorial approaches can be applied, and there were also some geometric or combinatorial methods that gave some weaker structural results).  As such, it was a challenge to extend these results to nonabelian groups, although for various important special types of ambient group $G$ (such as an linear group over a finite or infinite field) it turns out that one can use tools exploiting the special structure of those groups (e.g. for linear groups one would use tools from Lie theory and algebraic geometry) to obtain quite satisfactory results; see e.g. this survey of  Pyber and Szabo for the linear case.   When the ambient group $G$ is completely arbitrary, it turns out the problem is closely related to the classical Hilbert’s fifth problem of determining the minimal requirements of a topological group in order for such groups to have Lie structure; this connection was first observed and exploited by Hrushovski, and then used by Breuillard, Green, and myself to obtain the analogue of Freiman’s theorem for an arbitrary nonabelian group.

This survey is too short to discuss in much detail the proof techniques used in these results (although the abelian case is discussed in this book of mine with Vu, and the nonabelian case discussed in this more recent book of mine), but instead focuses on the statements of the various known results, as well as some remaining open questions in the subject (in particular, there is substantial work left to be done in making the estimates more quantitative, particularly in the nonabelian setting).

Van Vu and I have just uploaded to the arXiv our paper “Random matrices: The Universality phenomenon for Wigner ensembles“. This survey is a longer version (58 pages) of a previous short survey we wrote up a few months ago. The survey focuses on recent progress in understanding the universality phenomenon for Hermitian Wigner ensembles, of which the Gaussian Unitary Ensemble (GUE) is the most well known. The one-sentence summary of this progress is that many of the asymptotic spectral statistics (e.g. correlation functions, eigenvalue gaps, determinants, etc.) that were previously known for GUE matrices, are now known for very large classes of Wigner ensembles as well. There are however a wide variety of results of this type, due to the large number of interesting spectral statistics, the varying hypotheses placed on the ensemble, and the different modes of convergence studied, and it is difficult to isolate a single such result currently as the definitive universality result. (In particular, there is at present a tradeoff between generality of ensemble and strength of convergence; the universality results that are available for the most general classes of ensemble are only presently able to demonstrate a rather weak sense of convergence to the universal distribution (involving an additional averaging in the energy parameter), which limits the applicability of such results to a number of interesting questions in which energy averaging is not permissible, such as the study of the least singular value of a Wigner matrix, or of related quantities such as the condition number or determinant. But it is conceivable that this tradeoff is a temporary phenomenon and may be eliminated by future work in this area; in the case of Hermitian matrices whose entries have the same second moments as that of the GUE ensemble, for instance, the need for energy averaging has already been removed.)

Nevertheless, throughout the family of results that have been obtained recently, there are two main methods which have been fundamental to almost all of the recent progress in extending from special ensembles such as GUE to general ensembles. The first method, developed extensively by Erdos, Schlein, Yau, Yin, and others (and building on an initial breakthrough by Johansson), is the heat flow method, which exploits the rapid convergence to equilibrium of the spectral statistics of matrices undergoing Dyson-type flows towards GUE. (An important aspect to this method is the ability to accelerate the convergence to equilibrium by localising the Hamiltonian, in order to eliminate the slowest modes of the flow; this refinement of the method is known as the “local relaxation flow” method. Unfortunately, the translation mode is not accelerated by this process, which is the principal reason why results obtained by pure heat flow methods still require an energy averaging in the final conclusion; it would of interest to find a way around this difficulty.) The other method, which goes all the way back to Lindeberg in his classical proof of the central limit theorem, and which was introduced to random matrix theory by Chatterjee and then developed for the universality problem by Van Vu and myself, is the swapping method, which is based on the observation that spectral statistics of Wigner matrices tend to be stable if one replaces just one or two entries of the matrix with another distribution, with the stability of the swapping process becoming stronger if one assumes that the old and new entries have many matching moments. The main formalisations of this observation are known as four moment theorems, because they require four matching moments between the entries, although there are some variant three moment theorems and two moment theorems in the literature as well. Our initial four moment theorems were focused on individual eigenvalues (and later also to eigenvectors), but it was later observed by Erdos, Yau, and Yin that simpler four moment theorems could also be established for aggregate spectral statistics, such as the coefficients of the Greens function, and Knowles and Yin also subsequently observed that these latter theorems could be used to recover a four moment theorem for eigenvalues and eigenvectors, giving an alternate approach to proving such theorems.

Interestingly, it seems that the heat flow and swapping methods are complementary to each other; the heat flow methods are good at removing moment hypotheses on the coefficients, while the swapping methods are good at removing regularity hypotheses. To handle general ensembles with minimal moment or regularity hypotheses, it is thus necessary to combine the two methods (though perhaps in the future a third method, or a unification of the two existing methods, might emerge).

Besides the heat flow and swapping methods, there are also a number of other basic tools that are also needed in these results, such as local semicircle laws and eigenvalue rigidity, which are also discussed in the survey. We also survey how universality has been established for wide variety of spectral statistics; the ${k}$-point correlation functions are the most well known of these statistics, but they do not tell the whole story (particularly if one can only control these functions after an averaging in the energy), and there are a number of other statistics, such as eigenvalue counting functions, determinants, or spectral gaps, for which the above methods can be applied.

In order to prevent the survey from becoming too enormous, we decided to restrict attention to Hermitian matrix ensembles, whose entries off the diagonal are identically distributed, as this is the case in which the strongest results are available. There are several results that are applicable to more general ensembles than these which are briefly mentioned in the survey, but they are not covered in detail.

We plan to submit this survey eventually to the proceedings of a workshop on random matrix theory, and will continue to update the references on the arXiv version until the time comes to actually submit the paper.

Finally, in the survey we issue some errata for previous papers of Van and myself in this area, mostly centering around the three moment theorem (a variant of the more widely used four moment theorem), for which the original proof of Van and myself was incomplete. (Fortunately, as the three moment theorem had many fewer applications than the four moment theorem, and most of the applications that it did have ended up being superseded by subsequent papers, the actual impact of this issue was limited, but still an erratum is in order.)

Last month, at the joint AMS/MAA meeting in San Diego, I spoke at the AMS “Current Events” Bulletin on the topic “Why are solitons stable?“. This talk was supposed to be a survey of many of the developments on the rigorous stability theory of solitary waves in dispersive wave models (e.g. the Kortweg-de Vries equation and its generalisations, nonlinear Schrödinger equations, etc.), although my actual talk (which was the usual 50 minutes in length) only managed to cover about half of the material I had planned.

More recently, I completed the article that accompanies the talk, and which will be submitted to the Bulletin of the American Mathematical Society. In this paper I describe the key conflict in these wave models between dispersion (the tendency of waves of differing frequency to move at different speeds, thus causing any localised wave to disperse in space over time) and nonlinearity (which can cause any concentrated portion of the wave to self-amplify). Solitons seem to lie at the exact balancing point between these two forces, neither dispersing nor amplifying, but instead simply traveling at a constant velocity or oscillating in phase at a constant rate. In some cases, this balancing point is unstable; remove even a tiny amount of mass from the soliton and it eventually disperses completely into radiation, or one can add a tiny amount and cause the soliton to concentrate into a point and thence exhibit blowup in finite time. In other cases, the balancing point is stable; small perturbations to a soliton may end up changing the amplitude, position, and/or velocity of the soliton slightly, but the bulk of the solution still closely resembles a soliton in size, shape, and behaviour. Stability is sometimes enforced by linear properties, such as dispersive estimates or spectral properties of the linearised dynamics, but is also often enforced by nonlinear properties, such as nonlinear conservation laws, monotonicity formulae, and local propagation estimates for mass and energy (such as those provided by virial identities). The interplay between all these properties can be remarkably subtle, especially in the critical case when a key conserved quantity is scale-invariant (thus leading to an additional degeneracy in the soliton manifold). This is particularly evident in the remarkable series of papers by Martel and Merle establishing various stability and blowup properties near the ground state soliton of the critical generalised KdV equation, which I spend some time discussing (without going into too many of the (quite numerous) technical details). The focus in my paper is primarily on the non-integrable case, in which the techniques are primarily analytic rather than algebraic or geometric.