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The main purpose of this post is to roll over the discussion from the previous Polymath8 thread, which has become rather full with comments. We are still writing the paper, but it appears to have stabilised in a near-final form (source files available here); the main remaining tasks are proofreading, checking the mathematics, and polishing the exposition. We also have a tentative consensus to submit the paper to Algebra and Number Theory when the proofreading is all complete.
The paper is quite large now (164 pages!) but it is fortunately rather modular, and thus hopefully somewhat readable (particularly regarding the first half of the paper, which does not need any of the advanced exponential sum estimates). The size should not be a major issue for the journal, so I would not seek to artificially shorten the paper at the expense of readability or content.
Define a partition of to be a finite or infinite multiset of real numbers in the interval (that is, an unordered set of real numbers in , possibly with multiplicity) whose total sum is : . For instance, is a partition of . Such partitions arise naturally when trying to decompose a large object into smaller ones, for instance:
- (Prime factorisation) Given a natural number , one can decompose it into prime factors (counting multiplicity), and then the multiset
is a partition of .
- (Cycle decomposition) Given a permutation on labels , one can decompose into cycles , and then the multiset
is a partition of .
- (Normalisation) Given a multiset of positive real numbers whose sum is finite and non-zero, the multiset
is a partition of .
In the spirit of the universality phenomenon, one can ask what is the natural distribution for what a “typical” partition should look like; thus one seeks a natural probability distribution on the space of all partitions, analogous to (say) the gaussian distributions on the real line, or GUE distributions on point processes on the line, and so forth. It turns out that there is one natural such distribution which is related to all three examples above, known as the Poisson-Dirichlet distribution. To describe this distribution, we first have to deal with the problem that it is not immediately obvious how to cleanly parameterise the space of partitions, given that the cardinality of the partition can be finite or infinite, that multiplicity is allowed, and that we would like to identify two partitions that are permutations of each other
One way to proceed is to random partition as a type of point process on the interval , with the constraint that , in which case one can study statistics such as the counting functions
(where the cardinality here counts multiplicity). This can certainly be done, although in the case of the Poisson-Dirichlet process, the formulae for the joint distribution of such counting functions is moderately complicated. Another way to proceed is to order the elements of in decreasing order
with the convention that one pads the sequence by an infinite number of zeroes if is finite; this identifies the space of partitions with an infinite dimensional simplex
However, it turns out that the process of ordering the elements is not “smooth” (basically because functions such as and are not smooth) and the formulae for the joint distribution in the case of the Poisson-Dirichlet process is again complicated.
It turns out that there is a better (or at least “smoother”) way to enumerate the elements of a partition than the ordered method, although it is random rather than deterministic. This procedure (which I learned from this paper of Donnelly and Grimmett) works as follows.
- Given a partition , let be an element of chosen at random, with each element having a probability of being chosen as (so if occurs with multiplicity , the net probability that is chosen as is actually ). Note that this is well-defined since the elements of sum to .
- Now suppose is chosen. If is empty, we set all equal to zero and stop. Otherwise, let be an element of chosen at random, with each element having a probability of being chosen as . (For instance, if occurred with some multiplicity in , then can equal with probability .)
- Now suppose are both chosen. If is empty, we set all equal to zero and stop. Otherwise, let be an element of , with ech element having a probability of being chosen as .
- We continue this process indefinitely to create elements .
We denote the random sequence formed from a partition in the above manner as the random normalised enumeration of ; this is a random variable in the infinite unit cube , and can be defined recursively by the formula
with drawn randomly from , with each element chosen with probability , except when in which case we instead have
with the convention that one discards any zero elements on the right-hand side. Thus can be viewed as a (stochastic) parameterisation of the space of partitions by the unit cube , which is a simpler domain to work with than the infinite-dimensional simplex mentioned earlier.
Note that this random enumeration procedure can also be adapted to the three models described earlier:
- Given a natural number , one can randomly enumerate its prime factors by letting each prime factor of be equal to with probability , then once is chosen, let each remaining prime factor of be equal to with probability , and so forth.
- Given a permutation , one can randomly enumerate its cycles by letting each cycle in be equal to with probability , and once is chosen, let each remaining cycle be equalto with probability , and so forth. Alternatively, one traverse the elements of in random order, then let be the first cycle one encounters when performing this traversal, let be the next cycle (not equal to one encounters when performing this traversal, and so forth.
- Given a multiset of positive real numbers whose sum is finite, we can randomly enumerate the elements of this sequence by letting each have a probability of being set equal to , and then once is chosen, let each remaining have a probability of being set equal to , and so forth.
We then have the following result:
Proposition 1 (Existence of the Poisson-Dirichlet process) There exists a random partition whose random enumeration has the uniform distribution on , thus are independently and identically distributed copies of the uniform distribution on .
where are iid copies of the uniform distribution of , although it is not immediately obvious from this definition that is indeed uniformly distributed on . We prove this proposition below the fold.
where is a random element of with each having a probability of being equal to , is a uniform variable on that is independent of , and denotes equality of distribution. This can be viewed as a sort of stochastic self-similarity property of : if one randomly removes one element from and rescales, one gets a new copy of .
It turns out that each of the three ways to generate partitions listed above can lead to the Poisson-Dirichlet process, either directly or in a suitable limit. We begin with the third way, namely by normalising a Poisson process to have sum :
Proposition 2 (Poisson-Dirichlet processes via Poisson processes) Let , and let be a Poisson process on with intensity function . Then the sum is almost surely finite, and the normalisation is a Poisson-Dirichlet process.
Again, we prove this proposition below the fold. Now we turn to the second way (a topic, incidentally, that was briefly touched upon in this previous blog post):
Proposition 3 (Large cycles of a typical permutation) For each natural number , let be a permutation drawn uniformly at random from . Then the random partition converges in the limit to a Poisson-Dirichlet process in the following sense: given any fixed sequence of intervals (independent of ), the joint discrete random variable converges in distribution to .
Finally, we turn to the first way:
- (Uniform distribution) is drawn uniformly at random from the natural numbers in .
- (Shifted uniform distribution) is drawn uniformly at random from the natural numbers in .
- (Zeta distribution) Each natural number has a probability of being equal to , where and .
Then converges as to a Poisson-Dirichlet process in the same fashion as in Proposition 3.
The process was first studied by Billingsley (and also later by Knuth-Trabb Pardo and by Vershik, but the formulae were initially rather complicated; the proposition above is due to of Donnelly and Grimmett, although the third case of the proposition is substantially easier and appears in the earlier work of Lloyd. We prove the proposition below the fold.
The previous two propositions suggests an interesting analogy between large random integers and large random permutations; see this ICM article of Vershik and this non-technical article of Granville (which, incidentally, was once adapted into a play) for further discussion.
As a sample application, consider the problem of estimating the number of integers up to which are not divisible by any prime larger than (i.e. they are –smooth), where is a fixed real number. This is essentially (modulo some inessential technicalities concerning the distinction between the intervals and ) the probability that avoids , which by the above theorem converges to the probability that avoids . Below the fold we will show that this function is given by the Dickman function, defined by setting for and for , thus recovering the classical result of Dickman that .
I thank Andrew Granville and Anatoly Vershik for showing me the nice link between prime factors and the Poisson-Dirichlet process. The material here is standard, and (like many of the other notes on this blog) was primarily written for my own benefit, but it may be of interest to some readers. In preparing this article I found this exposition by Kingman to be helpful.
Note: this article will emphasise the computations rather than rigour, and in particular will rely on informal use of infinitesimals to avoid dealing with stochastic calculus or other technicalities. We adopt the convention that we will neglect higher order terms in infinitesimal calculations, e.g. if is infinitesimal then we will abbreviate simply as .
Emmanuel Breuillard, Ben Green, Bob Guralnick, and I have just uploaded to the arXiv our joint paper “Expansion in finite simple groups of Lie type“. This long-delayed paper (announced way back in 2010!) is a followup to our previous paper in which we showed that, with one possible exception, generic pairs of elements of a simple algebraic group (over an uncountable field) generated a free group which was strongly dense in the sense that any nonabelian subgroup of this group was Zariski dense. The main result of this paper is to establish the analogous result for finite simple groups of Lie type (as defined in the previous blog post) and bounded rank, namely that almost all pairs of elements of such a group generate a Cayley graph which is a (two-sided) expander, with expansion constant bounded below by a quantity depending on the rank of the group. (Informally, this means that the random walk generated by spreads out in logarithmic time to be essentially uniformly distributed across the group, as opposed for instance to being largely trapped in an algebraic subgroup. Thus if generic elements did not generate a strongly dense group, one would probably expect expansion to fail.)
There are also some related results established in the paper. Firstly, as we discovered after writing our first paper, there was one class of algebraic groups for which our demonstration of strongly dense subgroups broke down, namely the groups in characteristic three. In the current paper we provide in a pair of appendices a new argument that covers this case (or more generally, in odd characteristic), by first reducing to the case of affine groups (which can be found inside as a subgroup) and then using a ping-pong argument (in a p-adic metric) in the latter context.
Secondly, we show that the distinction between one-sided expansion and two-sided expansion (see this set of lecture notes of mine for definitions) is erased in the context of Cayley graphs of bounded degree, in the sense that such graphs are one-sided expanders if and only if they are two-sided expanders (perhaps with slightly different expansion constants). The argument turns out to be an elementary combinatorial one, based on the “pivot” argument discussed in these lecture notes of mine.
Now to the main result of the paper, namely the expansion of random Cayley graphs. This result had previously been established for by Bourgain and Gamburd, and Ben, Emmanuel and I had used the Bourgain-Gamburd method to achieve the same result for Suzuki groups. For the other finite simple groups of Lie type, expander graphs had been constructed by Kassabov, Lubotzky, and Nikolov, but they required more than two generators, which were placed deterministically rather than randomly. (Here, I am skipping over a large number of other results on expanding Cayley graphs; see this survey of Lubotzsky for a fairly recent summary of developments.) The current paper also uses the “Bourgain-Gamburd machine”, as discussed in these lecture notes of mine, to demonstrate expansion. This machine shows how expansion of a Cayley graph follows from three basic ingredients, which we state informally as follows:
- Non-concentration (A random walk in this graph does not concentrate in a proper subgroup);
- Product theorem (A medium-sized subset of this group which is not trapped in a proper subgroup will expand under multiplication); and
- Quasirandomness (The group has no small non-trivial linear representations).
Quasirandomness of arbitrary finite simple groups of Lie type was established many years ago (predating, in fact, the introduction of the term “quasirandomness” by Gowers for this property) by Landazuri-Seitz and Seitz-Zalesskii, and the product theorem was already established by Pyber-Szabo and independently by Breuillard, Green, and myself. So the main problem is to establish non-concentration: that for a random Cayley graph on a finite simple group of Lie type, random walks did not concentrate in proper subgroups.
The first step was to classify the proper subgroups of . Fortunately, these are all known; in particular, such groups are either contained in proper algebraic subgroups of the algebraic group containing (or a bounded cover thereof) with bounded complexity, or are else arising (up to conjugacy) from a version of the same group associated to a proper subfield of the field respectively; this follows for instance from the work of Larsen and Pink, but also can be deduced using the classification of finite simple groups, together with some work of Aschbacher, Liebeck-Seitz, and Nori. We refer to the two types of subgroups here as “structural subgroups” and “subfield subgroups”.
To preclude concentration in a structural subgroup, we use our previous result that generic elements of an algebraic group generate a strongly dense subgroup, and so do not concentrate in any algebraic subgroup. To translate this result from the algebraic group setting to the finite group setting, we need a Schwarz-Zippel lemma for finite simple groups of Lie type. This is straightforward for Chevalley groups, but turns out to be a bit trickier for the Steinberg and Suzuki-Ree groups, and we have to go back to the Chevalley-type parameterisation of such groups in terms of (twisted) one-parameter subgroups, that can be found for instance in the text of Carter; this “twisted Schwartz-Zippel lemma” may possibly have further application to analysis on twisted simple groups of Lie type. Unfortunately, the Schwartz-Zippel estimate becomes weaker in twisted settings, and particularly in the case of triality groups , which require a somewhat ad hoc additional treatment that relies on passing to a simpler subgroup present in a triality group, namely a central product of two different ‘s.
To rule out concentration in a conjugate of a subfield group, we repeat an argument we introduced in our Suzuki paper and pass to a matrix model and analyse the coefficients of the characteristic polynomial of words in this Cayley graph, to prevent them from concentrating in a subfield. (Note that these coefficients are conjugation-invariant.)
In this previous post I recorded some (very standard) material on the structural theory of finite-dimensional complex Lie algebras (or Lie algebras for short), with a particular focus on those Lie algebras which were semisimple or simple. Among other things, these notes discussed the Weyl complete reducibility theorem (asserting that semisimple Lie algebras are the direct sum of simple Lie algebras) and the classification of simple Lie algebras (with all such Lie algebras being (up to isomorphism) of the form , , , , , , , , or ).
Among other things, the structural theory of Lie algebras can then be used to build analogous structures in nearby areas of mathematics, such as Lie groups and Lie algebras over more general fields than the complex field (leading in particular to the notion of a Chevalley group), as well as finite simple groups of Lie type, which form the bulk of the classification of finite simple groups (with the exception of the alternating groups and a finite number of sporadic groups).
In the case of complex Lie groups, it turns out that every simple Lie algebra is associated with a finite number of connected complex Lie groups, ranging from a “minimal” Lie group (the adjoint form of the Lie group) to a “maximal” Lie group (the simply connected form of the Lie group) that finitely covers , and occasionally also a number of intermediate forms which finitely cover , but are in turn finitely covered by . For instance, is associated with the projective special linear group as its adjoint form and the special linear group as its simply connected form, and intermediate groups can be created by quotienting out by some subgroup of its centre (which is isomorphic to the roots of unity). The minimal form is simple in the group-theoretic sense of having no normal subgroups, but the other forms of the Lie group are merely quasisimple, although traditionally all of the forms of a Lie group associated to a simple Lie algebra are known as simple Lie groups.
Thanks to the work of Chevalley, a very similar story holds for algebraic groups over arbitrary fields ; given any Dynkin diagram, one can define a simple Lie algebra with that diagram over that field, and also one can find a finite number of connected algebraic groups over (known as Chevalley groups) with that Lie algebra, ranging from an adjoint form to a universal form , with every form having an isogeny (the analogue of a finite cover for algebraic groups) to the adjoint form, and in turn receiving an isogeny from the universal form. Thus, for instance, one could construct the universal form of the algebraic group over a finite field of finite order.
When one restricts the Chevalley group construction to adjoint forms over a finite field (e.g. ), one usually obtains a finite simple group (with a finite number of exceptions when the rank and the field are very small, and in some cases one also has to pass to a bounded index subgroup, such as the derived group, first). One could also use other forms than the adjoint form, but one then recovers the same finite simple group as before if one quotients out by the centre. This construction was then extended by Steinberg, Suzuki, and Ree by taking a Chevalley group over a finite field and then restricting to the fixed points of a certain automorphism of that group; after some additional minor modifications such as passing to a bounded index subgroup or quotienting out a bounded centre, this gives some additional finite simple groups of Lie type, including classical examples such as the projective special unitary groups , as well as some more exotic examples such as the Suzuki groups or the Ree groups.
While I learned most of the classical structural theory of Lie algebras back when I was an undergraduate, and have interacted with Lie groups in many ways in the past (most recently in connection with Hilbert’s fifth problem, as discussed in this previous series of lectures), I have only recently had the need to understand more precisely the concepts of a Chevalley group and of a finite simple group of Lie type, as well as better understand the structural theory of simple complex Lie groups. As such, I am recording some notes here regarding these concepts, mainly for my own benefit, but perhaps they will also be of use to some other readers. The material here is standard, and was drawn from a number of sources, but primarily from Carter, Gorenstein-Lyons-Solomon, and Fulton-Harris, as well as the lecture notes on Chevalley groups by my colleague Robert Steinberg. The arrangement of material also reflects my own personal preferences; in particular, I tend to favour complex-variable or Riemannian geometry methods over algebraic ones, and this influenced a number of choices I had to make regarding how to prove certain key facts. The notes below are far from a comprehensive or fully detailed discussion of these topics, and I would refer interested readers to the references above for a properly thorough treatment.
The main purpose of this post is to roll over the discussion from the previous Polymath8 thread, which has become rather full with comments. As with the previous thread, the main focus on the comments to this thread are concerned with writing up the results of the Polymath8 “bounded gaps between primes” project; the latest files on this writeup may be found at this directory, with the most recently compiled PDF file (clocking in at about 90 pages so far, with a few sections still to be written!) being found here. There is also still some active discussion on improving the numerical results, with a particular focus on improving the sieving step that converts distribution estimates such as into weak prime tuples results . (For a discussion of the terminology, and for a general overview of the proof strategy, see this previous progress report on the Polymath8 project.) This post can also contain any other discussion pertinent to any aspect of the polymath8 project, of course.
There are a few sections that still need to be written for the draft, mostly concerned with the Type I, Type II, and Type III estimates. However, the proofs of these estimates exist already on this blog, so I hope to transcribe them to the paper fairly shortly (say by the end of this week). Barring any unexpected surprises, or major reorganisation of the paper, it seems that the main remaining task in the writing process would be the proofreading and polishing, and turning from the technical mathematical details to expository issues. As always, feedback from casual participants, as well as those who have been closely involved with the project, would be very valuable in this regard. (One small comment, by the way, regarding corrections: as the draft keeps changing with time, referring to a specific line of the paper using page numbers and line numbers can become inaccurate, so if one could try to use section numbers, theorem numbers, or equation numbers as reference instead (e.g. “the third line after (5.35)” instead of “the twelfth line of page 54”) that would make it easier to track down specific portions of the paper.)
Also, we have set up a wiki page for listing the participants of the polymath8 project, their contact information, and grant information (if applicable). We have two lists of participants; one for those who have been making significant contributions to the project (comparable to that of a co-author of a traditional mathematical research paper), and another list for those who have made auxiliary contributions (e.g. typos, stylistic suggestions, or supplying references) that would typically merit inclusion in the Acknowledgments section of a traditional paper. It’s difficult to exactly draw the line between the two types of contributions, but we have relied in the past on self-reporting, which has worked pretty well so far. (By the time this project concludes, I may go through the comments to previous posts and see if any further names should be added to these lists that have not already been self-reported.)