Structure and randomness in the prime numbers

This week I am in Bremen, where the 50th International Mathematical Olympiad is being held.  A number of former Olympians (Béla Bollobás, Tim Gowers, Laci Lovasz, Stas Smirnov, Jean-Christophe Yoccoz, and myself) were invited to give a short talk (20 minutes in length) at the celebratory event for this anniversary.  I chose to talk on a topic I have spoken about several times before, on “Structure and randomness in the prime numbers“.  Given the time constraints, there was a limit as to how much substance I could put into the talk; but I try to describe, in very general terms, what we know about the primes, and what we suspect to be true, but cannot yet establish.  As I have mentioned in previous talks, the key problem is that we suspect the distribution of the primes to obey no significant patterns (other than “local” structure, such as having a strong tendency to be odd (which is local information at the 2 place), or obeying the prime number theorem (which is local information at the infinity place)), but we still do not have fully satisfactory tools for establishing the absence of a pattern. (This is in contrast with many types of Olympiad problems, where the key to solving a problem often lies in discovering the right pattern or structure in the problem to exploit.)

The PDF of the talk is here; I decided to try out the Beamer LaTeX package for a change.

Structure and Randomness: pages from year one of mathematical blog

The AMS has just notified me that the book version of the first year of my blog, now retitled “Structure and Randomness: pages from year one of a mathematical blog“, is now available.  An official web page for this book has also been set up here, though it is fairly empty at present.  A (2MB) high-resolution PDF file of the cover can be found here.

I plan to start on converting this year’s blog posts to book form in January, and hopefully the process should be a little faster this time.  Given that my lecture notes on ergodic theory and on the Poincaré conjecture will form the bulk of that book, I have chosen the working title for that book to be “Poincaré’s legacies: pages from year two of a mathematical blog“.

Concentration compactness and the profile decomposition

One of the most important topological concepts in analysis is that of compactness (as discussed for instance in my Companion article on this topic).  There are various flavours of this concept, but let us focus on sequential compactness: a subset E of a topological space X is sequentially compact if every sequence in E has a convergent subsequence whose limit is also in E.  This property allows one to do many things with the set E.  For instance, it allows one to maximise a functional on E:

Proposition 1. (Existence of extremisers)  Let E be a non-empty sequentially compact subset of a topological space X, and let be a continuous function.  Then the supremum is attained at at least one point , thus for all .  (In particular, this supremum is finite.)  Similarly for the infimum.

Proof. Let be the supremum .  By the definition of supremum (and the axiom of (countable) choice), one can find a sequence in E such that .  By compactness, we can refine this sequence to a subsequence (which, by abuse of notation, we shall continue to call ) such that converges to a limit x in E.  Since we still have , and f is continuous at x, we conclude that f(x)=L, and the claim for the supremum follows.  The claim for the infimum is similar. 

Remark 1. An inspection of the argument shows that one can relax the continuity hypothesis on F somewhat: to attain the supremum, it suffices that F be upper semicontinuous, and to attain the infimum, it suffices that F be lower semicontinuous.

We thus see that sequential compactness is useful, among other things, for ensuring the existence of extremisers.  In finite-dimensional spaces (such as vector spaces), compact sets are plentiful; indeed, the Heine-Borel theorem asserts that every closed and bounded set is compact.  However, once one moves to infinite-dimensional spaces, such as function spaces, then the Heine-Borel theorem fails quite dramatically; most of the closed and bounded sets one encounters in a topological vector space are non-compact, if one insists on using a reasonably “strong” topology.  This causes a difficulty in (among other things) calculus of variations, which is often concerned to finding extremisers to a functional on a subset E of an infinite-dimensional function space X.

In recent decades, mathematicians have found a number of ways to get around this difficulty.  One of them is to weaken the topology to recover compactness, taking advantage of such results as the Banach-Alaoglu theorem (or its sequential counterpart).  Of course, there is a tradeoff: weakening the topology makes compactness easier to attain, but makes the continuity of F harder to establish.  Nevertheless, if F enjoys enough “smoothing” or “cancellation” properties, one can hope to obtain continuity in the weak topology, allowing one to do things such as locate extremisers.  (The phenomenon that cancellation can lead to continuity in the weak topology is sometimes referred to as compensated compactness.)

Another option is to abandon trying to make all sequences have convergent subsequences, and settle just for extremising sequences to have convergent subsequences, as this would still be enough to retain Theorem 1.  Pursuing this line of thought leads to the Palais-Smale condition, which is a substitute for compactness in some calculus of variations situations.

But in many situations, one cannot weaken the topology to the point where the domain E becomes compact, without destroying the continuity (or semi-continuity) of F, though one can often at least find an intermediate topology (or metric) in which F is continuous, but for which E is still not quite compact.  Thus one can find sequences in E which do not have any subsequences that converge to a constant element , even in this intermediate metric.  (As we shall see shortly, one major cause of this failure of compactness is the existence of a non-trivial action of a non-compact group G on E; such a group action can cause compensated compactness or the Palais-Smale condition to fail also.)  Because of this, it is a priori conceivable that a continuous function F need not attain its supremum or infimum.

Nevertheless, even though a sequence does not have any subsequences that converge to a constant x, it may have a subsequence (which we also call ) which converges to some non-constant sequence (in the sense that the distance between the subsequence and the new sequence in a this intermediate metric), where the approximating sequence is of a very structured form (e.g. “concentrating” to a point, or “travelling” off to infinity, or a superposition of several concentrating or travelling profiles of this form).  This weaker form of compactness, in which superpositions of a certain type of profile completely describe all the failures (or defects) of compactness, is known as concentration compactness, and the decomposition of the subsequence is known as the profile decomposition.  In many applications, it is a sufficiently good substitute for compactness that one can still do things like locate extremisers for functionals F -  though one often has to make some additional assumptions of F to compensate for the more complicated nature of the compactness.  This phenomenon was systematically studied by P.L. Lions in the 80s, and found great application in calculus of variations and nonlinear elliptic PDE.  More recently, concentration compactness has been a crucial and powerful tool in the non-perturbative analysis of nonlinear dispersive PDE, in particular being used to locate “minimal energy blowup solutions” or “minimal mass blowup solutions” for such a PDE (analogously to how one can use calculus of variations to find minimal energy solutions to a nonlinear elliptic equation); see for instance this recent survey by Killip and Visan.

In typical applications, the concentration compactness phenomenon is exploited in moderately sophisticated function spaces (such as Sobolev spaces or Strichartz spaces), with the failure of traditional compactness being connected to a moderately complicated group G of symmetries (e.g. the group generated by translations and dilations).  Because of this, concentration compactness can appear to be a rather complicated and technical concept when it is first encountered.  In this note, I would like to illustrate concentration compactness in a simple toy setting, namely in the space of absolutely summable sequences, with the uniform () metric playing the role of the intermediate metric, and the translation group playing the role of the symmetry group G.  This toy setting is significantly simpler than any model that one would actually use in practice [for instance, in most applications X is a Hilbert space], but hopefully it serves to illuminate this useful concept in a less technical fashion.

254A, Lecture 16: A Ratner-type theorem for nilmanifolds

The last two lectures of this course will be on Ratner’s theorems on equidistribution of orbits on homogeneous spaces. Due to lack of time, I will not be able to cover all the material here that I had originally planned; in particular, for an introduction to this family of results, and its connections with number theory, I will have to refer readers to my previous blog post on these theorems. In this course, I will discuss two special cases of Ratner-type theorems. In this lecture, I will talk about Ratner-type theorems for discrete actions (of the integers on nilmanifolds; this case is much simpler than the general case, because there is a simple criterion in the nilmanifold case to test whether any given orbit is equidistributed or not. Ben Green and I had need recently to develop quantitative versions of such theorems for a number-theoretic application. In the next and final lecture of this course, I will discuss Ratner-type theorems for actions of , which is simpler in a different way (due to the semisimplicity of , and lack of compact factors).

254A, Lecture 15: The Furstenberg-Zimmer structure theorem and the Furstenberg recurrence theorem

In this lecture – the final one on general measure-preserving dynamics – we put together the results from the past few lectures to establish the Furstenberg-Zimmer structure theorem for measure-preserving systems, and then use this to finish the proof of the Furstenberg recurrence theorem.

254A, Lecture 13: Compact extensions

In Lecture 11, we studied compact measure-preserving systems – those systems in which every function was almost periodic, which meant that their orbit was precompact in the topology. Among other things, we were able to easily establish the Furstenberg recurrence theorem (Theorem 1 from Lecture 11) for such systems.

In this lecture, we generalise these results to a “relative” or “conditional” setting, in which we study systems which are compact relative to some factor of . Such systems are to compact systems as isometric extensions are to isometric systems in topological dynamics. The main result we establish here is that the Furstenberg recurrence theorem holds for such compact extensions whenever the theorem holds for the base. The proof is essentially the same as in the compact case; the main new trick is to not to work in the Hilbert spaces over the complex numbers, but rather in the Hilbert module over the (commutative) von Neumann algebra . (Modules are to rings as vector spaces are to fields.) Because of the compact nature of the extension, it turns out that results from topological dynamics (and in particular, van der Waerden’s theorem) can be exploited to good effect in this argument.

[Note: this operator-algebraic approach is not the only way to understand these extensions; one can also proceed by disintegrating into fibre measures for almost every and working fibre by fibre. We will discuss the connection between the two approaches below.]

Structure and randomness in the prime numbers

This Thursday I was at the University of Sydney, Australia, giving a public lecture on a favourite topic of mine, “Structure and randomness in the prime numbers“. My slides here are a merge between my slides for a Royal Society meeting and the slides I gave for the UCLA Science Colloquium; now that I figured out to use Powerpoint a little bit better, I was able to make the latter a bit more colourful (and the former less abridged).

254A, Lecture 7: Structural theory of topological dynamical systems

In our final lecture on topological dynamics, we discuss a remarkable theorem of Furstenberg that classifies a major type of topological dynamical system – distal systems – in terms of highly structured (from an algebraic point of view) systems, namely towers of isometric extensions. This theorem is also a model for an important analogous result in ergodic theory, the Furstenberg-Zimmer structure theorem, which we will turn to in a few lectures. We will not be able to prove Furstenberg’s structure theorem for distal systems here in full, but we hope to illustrate some of the key points and ideas.

254A, Lecture 6: Isometric systems and isometric extensions

In this lecture, we move away from recurrence, and instead focus on the structure of topological dynamical systems. One remarkable feature of this subject is that starting from fairly “soft” notions of structure, such as topological structure, one can extract much more “hard” or “rigid” notions of structure, such as geometric or algebraic structure. The key concept needed to capture this structure is that of an isometric system, or more generally an isometric extension, which we shall discuss in this lecture. As an application of this theory we characterise the distribution of polynomial sequences in torii (a baby case of a variant of Ratner’s theorem due to (Leon) Green, which we will cover later in this course).

Milliman Lecture I: Additive combinatorics and the primes

This week I am visiting the University of Washington in Seattle, giving the Milliman Lecture Series for 2007-2008. My chosen theme here is “Recent developments in arithmetic combinatorics“. In my first lecture, I will speak (once again) on how methods in additive combinatorics have allowed us to detect additive patterns in the prime numbers, in particular discussing my joint work with Ben Green. In the second lecture I will discuss how additive combinatorics has made it possible to study the invertibility and spectral behaviour of random discrete matrices, in particular discussing my joint work with Van Vu; and in the third lecture I will discuss how sum-product estimates have recently led to progress in the theory of expanders relating to Lie groups, as well as to sieving over orbits of such groups, in particular presenting work of Jean Bourgain and his coauthors.