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I’ve just uploaded to the arXiv my paper “On the universality of the incompressible Euler equation on compact manifolds“, submitted to Discrete and Continuous Dynamical Systems. This is a variant of my recent paper on the universality of potential well dynamics, but instead of trying to embed dynamical systems into a potential well , here we try to embed dynamical systems into the incompressible Euler equations

on a Riemannian manifold . (One is particularly interested in the case of flat manifolds , particularly or , but for the main result of this paper it is essential that one is permitted to consider curved manifolds.) This system, first studied by Ebin and Marsden, is the natural generalisation of the usual incompressible Euler equations to curved space; it can be viewed as the formal geodesic flow equation on the infinite-dimensional manifold of volume-preserving diffeomorphisms on (see this previous post for a discussion of this in the flat space case).

The Euler equations can be viewed as a nonlinear equation in which the nonlinearity is a quadratic function of the velocity field . It is thus natural to compare the Euler equations with quadratic ODE of the form

where is the unknown solution, and is a bilinear map, which we may assume without loss of generality to be symmetric. One can ask whether such an ODE may be linearly embedded into the Euler equations on some Riemannian manifold , which means that there is an injective linear map from to smooth vector fields on , as well as a bilinear map to smooth scalar fields on , such that the map takes solutions to (2) to solutions to (1), or equivalently that

for all .

For simplicity let us restrict to be compact. There is an obvious necessary condition for this embeddability to occur, which comes from energy conservation law for the Euler equations; unpacking everything, this implies that the bilinear form in (2) has to obey a cancellation condition

for some positive definite inner product on . The main result of the paper is the converse to this statement: if is a symmetric bilinear form obeying a cancellation condition (3), then it is possible to embed the equations (2) into the Euler equations (1) on some Riemannian manifold ; the catch is that this manifold will depend on the form and on the dimension (in fact in the construction I have, is given explicitly as , with a funny metric on it that depends on ).

As a consequence, any finite dimensional portion of the usual “dyadic shell models” used as simplified toy models of the Euler equation, can actually be embedded into a genuine Euler equation, albeit on a high-dimensional and curved manifold. This includes portions of the self-similar “machine” I used in a previous paper to establish finite time blowup for an averaged version of the Navier-Stokes (or Euler) equations. Unfortunately, the result in this paper does not apply to infinite-dimensional ODE, so I cannot yet establish finite time blowup for the Euler equations on a (well-chosen) manifold. It does not seem so far beyond the realm of possibility, though, that this could be done in the relatively near future. In particular, the result here suggests that one could construct something resembling a universal Turing machine within an Euler flow on a manifold, which was one ingredient I would need to engineer such a finite time blowup.

The proof of the main theorem proceeds by an “elimination of variables” strategy that was used in some of my previous papers in this area, though in this particular case the Nash embedding theorem (or variants thereof) are not required. The first step is to lessen the dependence on the metric by partially reformulating the Euler equations (1) in terms of the covelocity (which is a -form) instead of the velocity . Using the freedom to modify the dimension of the underlying manifold , one can also decouple the metric from the volume form that is used to obtain the divergence-free condition. At this point the metric can be eliminated, with a certain positive definiteness condition between the velocity and covelocity taking its place. After a substantial amount of trial and error (motivated by some “two-and-a-half-dimensional” reductions of the three-dimensional Euler equations, and also by playing around with a number of variants of the classic “separation of variables” strategy), I eventually found an ansatz for the velocity and covelocity that automatically solved most of the components of the Euler equations (as well as most of the positive definiteness requirements), as long as one could find a number of scalar fields that obeyed a certain nonlinear system of transport equations, and also obeyed a positive definiteness condition. Here I was stuck for a bit because the system I ended up with was overdetermined – more equations than unknowns. After trying a number of special cases I eventually found a solution to the transport system on the sphere, except that the scalar functions sometimes degenerated and so the positive definiteness property I wanted was only obeyed with positive semi-definiteness. I tried for some time to perturb this example into a strictly positive definite solution before eventually working out that this was not possible. Finally I had the brainwave to lift the solution from the sphere to an even more symmetric space, and this quickly led to the final solution of the problem, using the special orthogonal group rather than the sphere as the underlying domain. The solution ended up being rather simple in form, but it is still somewhat miraculous to me that it exists at all; in retrospect, given the overdetermined nature of the problem, relying on a large amount of symmetry to cut down the number of equations was basically the only hope.

I’ve just uploaded to the arXiv my paper “On the universality of potential well dynamics“, submitted to Dynamics of PDE. This is a spinoff from my previous paper on blowup of nonlinear wave equations, inspired by some conversations with Sungjin Oh. Here we focus mainly on the zero-dimensional case of such equations, namely the potential well equation

for a particle trapped in a potential well with potential , with as . This ODE always admits global solutions from arbitrary initial positions and initial velocities , thanks to conservation of the Hamiltonian . As this Hamiltonian is coercive (in that its level sets are compact), solutions to this equation are always almost periodic. On the other hand, as can already be seen using the harmonic oscillator (and direct sums of this system), this equation can generate periodic solutions, as well as quasiperiodic solutions.

All quasiperiodic motions are almost periodic. However, there are many examples of dynamical systems that admit solutions that are almost periodic but not quasiperiodic. So one can pose the question: are the dynamics of potential wells *universal* in the sense that they can capture all almost periodic solutions?

A precise question can be phrased as follows. Let be a compact manifold, and let be a smooth vector field on ; to avoid degeneracies, let us take to be *non-singular* in the sense that it is everywhere non-vanishing. Then the trajectories of the first-order ODE

for are always global and almost periodic. Can we then find a (coercive) potential for some , as well as a smooth embedding , such that every solution to (2) pushes forward under to a solution to (1)? (Actually, for technical reasons it is preferable to map into the phase space , rather than position space , but let us ignore this detail for this discussion.)

It turns out that the answer is no; there is a very specific obstruction. Given a pair as above, define a *strongly adapted -form* to be a -form on such that is pointwise positive, and the Lie derivative is an exact -form. We then have

Theorem 1A smooth compact non-singular dynamics can be embedded smoothly in a potential well system if and only if it admits a strongly adapted -form.

For the “only if” direction, the key point is that potential wells (viewed as a Hamiltonian flow on the phase space ) admit a strongly adapted -form, namely the canonical -form , whose Lie derivative is the derivative of the Lagrangian and is thus exact. The converse “if” direction is mainly a consequence of the Nash embedding theorem, and follows the arguments used in my previous paper.

Interestingly, the same obstruction also works for potential wells in a more general Riemannian manifold than , or for nonlinear wave equations with a potential; combining the two, the obstruction is also present for wave maps with a potential.

It is then natural to ask whether this obstruction is non-trivial, in the sense that there are at least some examples of dynamics that do not support strongly adapted -forms (and hence cannot be modeled smoothly by the dynamics of a potential well, nonlinear wave equation, or wave maps). I posed this question on MathOverflow, and Robert Bryant provided a very nice construction, showing that the vector field on the -torus had no strongly adapted -forms, and hence the dynamics of this vector field cannot be smoothly reproduced by a potential well, nonlinear wave equation, or wave map:

On the other hand, the suspension of any diffeomorphism does support a strongly adapted -form (the derivative of the time coordinate), and using this and the previous theorem I was able to embed a universal Turing machine into a potential well. In particular, there are flows for an explicitly describable potential well whose trajectories have behavior that is undecidable using the usual ZFC axioms of set theory! So potential well dynamics are “effectively” universal, despite the presence of the aforementioned obstruction.

In my previous work on blowup for Navier-Stokes like equations, I speculated that if one could somehow replicate a universal Turing machine within the Euler equations, one could use this machine to create a “von Neumann machine” that replicated smaller versions of itself, which on iteration would lead to a finite time blowup. Now that such a mechanism is present in nonlinear wave equations, it is tempting to try to make this scheme work in that setting. Of course, in my previous paper I had already demonstrated finite time blowup, at least in a three-dimensional setting, but that was a relatively simple discretely self-similar blowup in which no computation occurred. This more complicated blowup scheme would be significantly more effort to set up, but would be proof-of-concept that the same scheme would in principle be possible for the Navier-Stokes equations, assuming somehow that one can embed a universal Turing machine into the Euler equations. (But I’m still hopelessly stuck on how to accomplish this latter task…)

Kaisa Matomaki, Maksym Radziwill, and I have uploaded to the arXiv our paper “Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges“, submitted to Proceedings of the London Mathematical Society. This paper is concerned with the estimation of correlations such as

for medium-sized and large , where is the von Mangoldt function; we also consider variants of this sum in which one of the von Mangoldt functions is replaced with a (higher order) divisor function, but for sake of discussion let us focus just on the sum (1). Understanding this sum is very closely related to the problem of finding pairs of primes that differ by ; for instance, if one could establish a lower bound

then this would easily imply the twin prime conjecture.

The (first) Hardy-Littlewood conjecture asserts an asymptotic

as for any fixed positive , where the *singular series* is an arithmetic factor arising from the irregularity of distribution of at small moduli, defined explicitly by

when is even, and when is odd, where

is (half of) the twin prime constant. See for instance this previous blog post for a a heuristic explanation of this conjecture. From the previous discussion we see that (2) for would imply the twin prime conjecture. Sieve theoretic methods are only able to provide an upper bound of the form .

Needless to say, apart from the trivial case of odd , there are no values of for which the Hardy-Littlewood conjecture is known. However there are some results that say that this conjecture holds “on the average”: in particular, if is a quantity depending on that is somewhat large, there are results that show that (2) holds for most (i.e. for ) of the betwen and . Ideally one would like to get as small as possible, in particular one can view the full Hardy-Littlewood conjecture as the endpoint case when is bounded.

The first results in this direction were by van der Corput and by Lavrik, who established such a result with (with a subsequent refinement by Balog); Wolke lowered to , and Mikawa lowered further to . The main result of this paper is a further lowering of to . In fact (as in the preceding works) we get a better error term than , namely an error of the shape for any .

Our arguments initially proceed along standard lines. One can use the Hardy-Littlewood circle method to express the correlation in (2) as an integral involving exponential sums . The contribution of “major arc” is known by a standard computation to recover the main term plus acceptable errors, so it is a matter of controlling the “minor arcs”. After averaging in and using the Plancherel identity, one is basically faced with establishing a bound of the form

for any “minor arc” . If is somewhat close to a low height rational (specifically, if it is within of such a rational with ), then this type of estimate is roughly of comparable strength (by another application of Plancherel) to the best available prime number theorem in short intervals on the average, namely that the prime number theorem holds for most intervals of the form , and we can handle this case using standard mean value theorems for Dirichlet series. So we can restrict attention to the “strongly minor arc” case where is far from such rationals.

The next step (following some ideas we found in a paper of Zhan) is to rewrite this estimate not in terms of the exponential sums , but rather in terms of the Dirichlet polynomial . After a certain amount of computation (including some oscillatory integral estimates arising from stationary phase), one is eventually reduced to the task of establishing an estimate of the form

for any (with sufficiently large depending on ).

The next step, which is again standard, is the use of the Heath-Brown identity (as discussed for instance in this previous blog post) to split up into a number of components that have a Dirichlet convolution structure. Because the exponent we are shooting for is less than , we end up with five types of components that arise, which we call “Type “, “Type “, “Type “, “Type “, and “Type II”. The “Type II” sums are Dirichlet convolutions involving a factor supported on a range and is quite easy to deal with; the “Type ” terms are Dirichlet convolutions that resemble (non-degenerate portions of) the divisor function, formed from convolving together portions of . The “Type ” and “Type ” terms can be estimated satisfactorily by standard moment estimates for Dirichlet polynomials; this already recovers the result of Mikawa (and our argument is in fact slightly more elementary in that no Kloosterman sum estimates are required). It is the treatment of the “Type ” and “Type ” sums that require some new analysis, with the Type terms turning to be the most delicate. After using an existing moment estimate of Jutila for Dirichlet L-functions, matters reduce to obtaining a family of estimates, a typical one of which (relating to the more difficult Type sums) is of the form

for “typical” ordinates of size , where is the Dirichlet polynomial (a fragment of the Riemann zeta function). The precise definition of “typical” is a little technical (because of the complicated nature of Jutila’s estimate) and will not be detailed here. Such a claim would follow easily from the Lindelof hypothesis (which would imply that ) but of course we would like to have an unconditional result.

At this point, having exhausted all the Dirichlet polynomial estimates that are usefully available, we return to “physical space”. Using some further Fourier-analytic and oscillatory integral computations, we can estimate the left-hand side of (3) by an expression that is roughly of the shape

The phase can be Taylor expanded as the sum of and a lower order term , plus negligible errors. If we could discard the lower order term then we would get quite a good bound using the exponential sum estimates of Robert and Sargos, which control averages of exponential sums with purely monomial phases, with the averaging allowing us to exploit the hypothesis that is “typical”. Figuring out how to get rid of this lower order term caused some inefficiency in our arguments; the best we could do (after much experimentation) was to use Fourier analysis to shorten the sums, estimate a one-parameter average exponential sum with a binomial phase by a two-parameter average with a monomial phase, and then use the van der Corput process followed by the estimates of Robert and Sargos. This rather complicated procedure works up to it may be possible that some alternate way to proceed here could improve the exponent somewhat.

In a sequel to this paper, we will use a somewhat different method to reduce to a much smaller value of , but only if we replace the correlations by either or , and also we now only save a in the error term rather than .

In July I will be spending a week at Park City, being one of the mini-course lecturers in the Graduate Summer School component of the Park City Summer Session on random matrices. I have chosen to give some lectures on least singular values of random matrices, the circular law, and the Lindeberg exchange method in random matrix theory; this is a slightly different set of topics than I had initially advertised (which was instead about the Lindeberg exchange method and the local relaxation flow method), but after consulting with the other mini-course lecturers I felt that this would be a more complementary set of topics. I have uploaded an draft of my lecture notes (some portion of which is derived from my monograph on the subject); as always, comments and corrections are welcome.

<I>[Update, June 23: notes revised and reformatted to PCMI format. -T.]</I>

Suppose is a continuous (but nonlinear) map from one normed vector space to another . The continuity means, roughly speaking, that if are such that is small, then is also small (though the precise notion of “smallness” may depend on or , particularly if is not known to be uniformly continuous). If is known to be differentiable (in, say, the Fréchet sense), then we in fact have a linear bound of the form

for some depending on , if is small enough; one can of course make independent of (and drop the smallness condition) if is known instead to be Lipschitz continuous.

In many applications in analysis, one would like more explicit and quantitative bounds that estimate quantities like in terms of quantities like . There are a number of ways to do this. First of all, there is of course the trivial estimate arising from the triangle inequality:

This estimate is usually not very good when and are close together. However, when and are far apart, this estimate can be more or less sharp. For instance, if the magnitude of varies so much from to that is more than (say) twice that of , or vice versa, then (1) is sharp up to a multiplicative constant. Also, if is oscillatory in nature, and the distance between and exceeds the “wavelength” of the oscillation of at (or at ), then one also typically expects (1) to be close to sharp. Conversely, if does not vary much in magnitude from to , and the distance between and is less than the wavelength of any oscillation present in , one expects to be able to improve upon (1).

When is relatively simple in form, one can sometimes proceed simply by substituting . For instance, if is the squaring function in a commutative ring , one has

and thus

or in terms of the original variables one has

If the ring is not commutative, one has to modify this to

Thus, for instance, if are matrices and denotes the operator norm, one sees from the triangle inequality and the sub-multiplicativity of operator norm that

If involves (or various components of ) in several places, one can sometimes get a good estimate by “swapping” with at each of the places in turn, using a telescoping series. For instance, if we again use the squaring function in a non-commutative ring, we have

which for instance leads to a slight improvement of (2):

More generally, for any natural number , one has the identity

in a commutative ring, while in a non-commutative ring one must modify this to

and for matrices one has

Exercise 1If and are unitary matrices, show that the commutator obeys the inequality(

Hint:first control .)

Now suppose (for simplicity) that is a map between Euclidean spaces. If is continuously differentiable, then one can use the fundamental theorem of calculus to write

where is any continuously differentiable path from to . For instance, if one uses the straight line path , one has

In the one-dimensional case , this simplifies to

Among other things, this immediately implies the factor theorem for functions: if is a function for some that vanishes at some point , then factors as the product of and some function . Another basic consequence is that if is uniformly bounded in magnitude by some constant , then is Lipschitz continuous with the same constant .

Applying (4) to the power function , we obtain the identity

which can be compared with (3). Indeed, for and close to , one can use logarithms and Taylor expansion to arrive at the approximation , so (3) behaves a little like a Riemann sum approximation to (5).

Exercise 2For each , let and be random variables taking values in a measurable space , and let be a bounded measurable function.

- (i) (Lindeberg exchange identity) Show that
- (ii) (Knowles-Yin exchange identity) Show that
where is a mixture of and , with uniformly drawn from independently of each other and of the .

- (iii) Discuss the relationship between the identities in parts (i), (ii) with the identities (3), (5).
(The identity in (i) is the starting point for the

Lindeberg exchange methodin probability theory, discussed for instance in this previous post. The identity in (ii) can also be used in the Lindeberg exchange method; the terms in the right-hand side are slightly more symmetric in the indices , which can be a technical advantage in some applications; see this paper of Knowles and Yin for an instance of this.)

Exercise 3If is continuously times differentiable, establish Taylor’s theorem with remainderIf is bounded, conclude that

For real scalar functions , the average value of the continuous real-valued function must be attained at some point in the interval . We thus conclude the mean-value theorem

for some (that can depend on , , and ). This can for instance give a second proof of fact that continuously differentiable functions with bounded derivative are Lipschitz continuous. However it is worth stressing that the mean-value theorem is only available for *real scalar* functions; it is false for instance for complex scalar functions. A basic counterexample is given by the function ; there is no for which . On the other hand, as has magnitude , we still know from (4) that is Lipschitz of constant , and when combined with (1) we obtain the basic bounds

which are already very useful for many applications.

Exercise 4Let be matrices, and let be a non-negative real.

- (i) Establish the Duhamel formula
where denotes the matrix exponential of . (

Hint:Differentiate or in .)- (ii) Establish the
iterated Duhamel formulafor any .

- (iii) Establish the infinitely iterated Duhamel formula
- (iv) If is an matrix depending in a continuously differentiable fashion on , establish the variation formula
where is the adjoint representation applied to , and is the function

(thus for non-zero ), with defined using functional calculus.

We remark that further manipulation of (iv) of the above exercise using the fundamental theorem of calculus eventually leads to the Baker-Campbell-Hausdorff-Dynkin formula, as discussed in this previous blog post.

Exercise 5Let be positive definite matrices, and let be an matrix. Show that there is a unique solution to the Sylvester equationwhich is given by the formula

In the above examples we had applied the fundamental theorem of calculus along linear curves . However, it is sometimes better to use other curves. For instance, the circular arc can be useful, particularly if and are “orthogonal” or “independent” in some sense; a good example of this is the proof by Maurey and Pisier of the gaussian concentration inequality, given in Theorem 8 of this previous blog post. In a similar vein, if one wishes to compare a scalar random variable of mean zero and variance one with a Gaussian random variable of mean zero and variance one, it can be useful to introduce the intermediate random variables (where and are independent); note that these variables have mean zero and variance one, and after coupling them together appropriately they evolve by the Ornstein-Uhlenbeck process, which has many useful properties. For instance, one can use these ideas to establish monotonicity formulae for entropy; see e.g. this paper of Courtade for an example of this and further references. More generally, one can exploit curves that flow according to some geometrically natural ODE or PDE; several examples of this occur famously in Perelman’s proof of the Poincaré conjecture via Ricci flow, discussed for instance in this previous set of lecture notes.

In some cases, it is difficult to compute or the derivative directly, but one can instead proceed by implicit differentiation, or some variant thereof. Consider for instance the matrix inversion map (defined on the open dense subset of matrices consisting of invertible matrices). If one wants to compute for close to , one can write temporarily write , thus

Multiplying both sides on the left by to eliminate the term, and on the right by to eliminate the term, one obtains

and thus on reversing these steps we arrive at the basic identity

For instance, if are matrices, and we consider the resolvents

then we have the *resolvent identity*

as long as does not lie in the spectrum of or (for instance, if , are self-adjoint then one can take to be any strictly complex number). One can iterate this identity to obtain

for any natural number ; in particular, if has operator norm less than one, one has the Neumann series

Similarly, if is a family of invertible matrices that depends in a continuously differentiable fashion on a time variable , then by implicitly differentiating the identity

in using the product rule, we obtain

and hence

(this identity may also be easily derived from (6)). One can then use the fundamental theorem of calculus to obtain variants of (6), for instance by using the curve we arrive at

assuming that the curve stays entirely within the set of invertible matrices. While this identity may seem more complicated than (6), it is more symmetric, which conveys some advantages. For instance, using this identity it is easy to see that if are positive definite with in the sense of positive definite matrices (that is, is positive definite), then . (Try to prove this using (6) instead!)

Exercise 6If is an invertible matrix and are vectors, establish the Sherman-Morrison formulawhenever is a scalar such that is non-zero. (See also this previous blog post for more discussion of these sorts of identities.)

One can use the Cauchy integral formula to extend these identities to other functions of matrices. For instance, if is an entire function, and is a counterclockwise contour that goes around the spectrum of both and , then we have

and similarly

and hence by (7) one has

similarly, if depends on in a continuously differentiable fashion, then

as long as goes around the spectrum of .

Exercise 7If is an matrix depending continuously differentiably on , and is an entire function, establish the tracial chain rule

In a similar vein, given that the logarithm function is the antiderivative of the reciprocal, one can express the matrix logarithm of a positive definite matrix by the fundamental theorem of calculus identity

(with the constant term needed to prevent a logarithmic divergence in the integral). Differentiating, we see that if is a family of positive definite matrices depending continuously on , that

This can be used for instance to show that is a monotone increasing function, in the sense that whenever in the sense of positive definite matrices. One can of course integrate this formula to obtain some formulae for the difference of the logarithm of two positive definite matrices .

To compare the square root of two positive definite matrices is trickier; there are multiple ways to proceed. One approach is to use contour integration as before (but one has to take some care to avoid branch cuts of the square root). Another to express the square root in terms of exponentials via the formula

where is the gamma function; this formula can be verified by first diagonalising to reduce to the scalar case and using the definition of the Gamma function. Then one has

and one can use some of the previous identities to control . This is pretty messy though. A third way to proceed is via implicit differentiation. If for instance is a family of positive definite matrices depending continuously differentiably on , we can differentiate the identity

to obtain

This can for instance be solved using Exercise 5 to obtain

and this can in turn be integrated to obtain a formula for . This is again a rather messy formula, but it does at least demonstrate that the square root is a monotone increasing function on positive definite matrices: implies .

Several of the above identities for matrices can be (carefully) extended to operators on Hilbert spaces provided that they are sufficiently well behaved (in particular, if they have a good functional calculus, and if various spectral hypotheses are obeyed). We will not attempt to do so here, however.

Suppose one has a bounded sequence of real numbers. What kinds of limits can one form from this sequence?

Of course, we have the usual notion of limit , which in this post I will refer to as the *classical limit* to distinguish from the other limits discussed in this post. The classical limit, if it exists, is the unique real number such that for every , one has for all sufficiently large . We say that a sequence is (classically) convergent if its classical limit exists. The classical limit obeys many useful *limit laws* when applied to classically convergent sequences. Firstly, it is linear: if and are classically convergent sequences, then is also classically convergent with

and similarly for any scalar , is classically convergent with

It is also an algebra homomorphism: is also classically convergent with

We also have shift invariance: if is classically convergent, then so is with

and more generally in fact for any injection , is classically convergent with

The classical limit of a sequence is unchanged if one modifies any finite number of elements of the sequence. Finally, we have boundedness: for any classically convergent sequence , one has

One can in fact show without much difficulty that these laws uniquely determine the classical limit functional on convergent sequences.

One would like to extend the classical limit notion to more general bounded sequences; however, when doing so one must give up one or more of the desirable limit laws that were listed above. Consider for instance the sequence . On the one hand, one has for all , so if one wishes to retain the homomorphism property (3), any “limit” of this sequence would have to necessarily square to , that is to say it must equal or . On the other hand, if one wished to retain the shift invariance property (4) as well as the homogeneity property (2), any “limit” of this sequence would have to equal its own negation and thus be zero.

Nevertheless there are a number of useful generalisations and variants of the classical limit concept for non-convergent sequences that obey a significant portion of the above limit laws. For instance, we have the limit superior

and limit inferior

which are well-defined real numbers for any bounded sequence ; they agree with the classical limit when the sequence is convergent, but disagree otherwise. They enjoy the shift-invariance property (4), and the boundedness property (6), but do not in general obey the homomorphism property (3) or the linearity property (1); indeed, we only have the subadditivity property

for the limit superior, and the superadditivity property

for the limit inferior. The homogeneity property (2) is only obeyed by the limits superior and inferior for non-negative ; for negative , one must have the limit inferior on one side of (2) and the limit superior on the other, thus for instance

The limit superior and limit inferior are examples of limit points of the sequence, which can for instance be defined as points that are limits of at least one subsequence of the original sequence. Indeed, the limit superior is always the largest limit point of the sequence, and the limit inferior is always the smallest limit point. However, limit points can be highly non-unique (indeed they are unique if and only if the sequence is classically convergent), and so it is difficult to sensibly interpret most of the usual limit laws in this setting, with the exception of the homogeneity property (2) and the boundedness property (6) that are easy to state for limit points.

Another notion of limit are the Césaro limits

if this limit exists, we say that the sequence is Césaro convergent. If the sequence already has a classical limit, then it also has a Césaro limit that agrees with the classical limit; but there are additional sequences that have a Césaro limit but not a classical one. For instance, the non-classically convergent sequence discussed above is Césaro convergent, with a Césaro limit of . However, there are still bounded sequences that do not have Césaro limit, such as (exercise!). The Césaro limit is linear, bounded, and shift invariant, but not an algebra homomorphism and also does not obey the rearrangement property (5).

Using the Hahn-Banach theorem, one can extend the classical limit functional to *generalised limit functionals* , defined to be bounded linear functionals from the space of bounded real sequences to the real numbers that extend the classical limit functional (defined on the space of convergent sequences) without any increase in the operator norm. (In some of my past writings I made the slight error of referring to these generalised limit functionals as Banach limits, though as discussed below, the latter actually refers to a subclass of generalised limit functionals.) It is not difficult to see that such generalised limit functionals will range between the limit inferior and limit superior. In fact, for any specific sequence and any number lying in the closed interval , there exists at least one generalised limit functional that takes the value when applied to ; for instance, for any number in , there exists a generalised limit functional that assigns that number as the “limit” of the sequence . This claim can be seen by first designing such a limit functional on the vector space spanned by the convergent sequences and by , and then appealing to the Hahn-Banach theorem to extend to all sequences. This observation also gives a necessary and sufficient criterion for a bounded sequence to classically converge to a limit , namely that all generalised limits of this sequence must equal .

Because of the reliance on the Hahn-Banach theorem, the existence of generalised limits requires the axiom of choice (or some weakened version thereof); there are presumably models of set theory without the axiom of choice in which no generalised limits exist, but I do not know of an explicit reference for this.

Generalised limits can obey the shift-invariance property (4) or the algebra homomorphism property (2), but as the above analysis of the sequence shows, they cannot do both. Generalised limits that obey the shift-invariance property (4) are known as Banach limits; one can for instance construct them by applying the Hahn-Banach theorem to the Césaro limit functional; alternatively, if is any generalised limit, then the Césaro-type functional will be a Banach limit. The existence of Banach limits can be viewed as a demonstration of the amenability of the natural numbers (or integers); see this previous blog post for further discussion.

Generalised limits that obey the algebra homomorphism property (2) are known as *ultrafilter limits*. If one is given a generalised limit functional that obeys (2), then for any subset of the natural numbers , the generalised limit must equal its own square (since ) and is thus either or . If one defines to be the collection of all subsets of for which , one can verify that obeys the axioms of a non-principal ultrafilter. Conversely, if is a non-principal ultrafilter, one can define the associated generalised limit of any bounded sequence to be the unique real number such that the sets lie in for all ; one can check that this does indeed give a well-defined generalised limit that obeys (2). Non-principal ultrafilters can be constructed using Zorn’s lemma. In fact, they do not quite need the full strength of the axiom of choice; see the Wikipedia article on the ultrafilter lemma for examples.

We have previously noted that generalised limits of a sequence can converge to any point between the limit inferior and limit superior. The same is not true if one restricts to Banach limits or ultrafilter limits. For instance, by the arguments already given, the only possible Banach limit for the sequence is zero. Meanwhile, an ultrafilter limit must converge to a limit point of the original sequence, but conversely every limit point can be attained by at least one ultrafilter limit; we leave these assertions as an exercise to the interested reader. In particular, a bounded sequence converges classically to a limit if and only if all ultrafilter limits converge to .

There is no generalisation of the classical limit functional to any space that includes non-classically convergent sequences that obeys the subsequence property (5), since any non-classically-convergent sequence will have one subsequence that converges to the limit superior, and another subsequence that converges to the limit inferior, and one of these will have to violate (5) since the limit superior and limit inferior are distinct. So the above limit notions come close to the best generalisations of limit that one can use in practice.

We summarise the above discussion in the following table:

Limit | Always defined | Linear | Shift-invariant | Homomorphism | Constructive |

Classical | No | Yes | Yes | Yes | Yes |

Superior | Yes | No | Yes | No | Yes |

Inferior | Yes | No | Yes | No | Yes |

Césaro | No | Yes | Yes | No | Yes |

Generalised | Yes | Yes | Depends | Depends | No |

Banach | Yes | Yes | Yes | No | No |

Ultrafilter | Yes | Yes | No | Yes | No |

Ben Green and I have (finally!) uploaded to the arXiv our paper “New bounds for Szemerédi’s theorem, III: A polylogarithmic bound for “, submitted to Mathematika. This is the sequel to two previous papers (and an erratum to the former paper), concerning quantitative versions of Szemerédi’s theorem in the case of length four progressions. This sequel has been delayed for over a decade for a number of reasons, but we have finally managed to write the arguments up to our satisfaction and submit it (to a special issue of Mathematika honouring the work of Klaus Roth).

For any natural number , define to be the largest cardinality of a subset of which does not contain any non-trivial arithmetic progressions of length four (where “non-trivial” means that is non-zero). Trivially we have . In 1969, Szemerédi showed that . However, the decay rate that could be theoretically extracted from this argument (and from several subsequent proofs of this bound, including one by Roth) were quite poor. The first significant quantitative bound on this quantity was by Gowers, who showed that for some absolute constant . In the second paper in the above-mentioned series, we managed to improve this bound to . In this paper, we improve the bound further to , which seems to be the limit of the methods. (We remark that if we could take to be larger than one, this would imply the length four case of a well known conjecture of Erdös that any set of natural numbers whose sum of reciprocals diverges would contain arbitrarily long arithmetic progressions. Thanks to the work of Sanders and of Bloom, the corresponding case of the conjecture for length three conjectures is nearly settled, as it is known that for the analogous bound on one can take any less than one.)

Most of the previous work on bounding relied in some form or another on the *density increment argument* introduced by Roth back in 1953; roughly speaking, the idea is to show that if a dense subset of fails to contain arithmetic progressions of length four, one seeks to then locate a long subprogression of in which has increased density. This was the basic method for instance underlying our previous bound , as well as a finite field analogue of the bound ; however we encountered significant technical difficulties for several years in extending this argument to obtain the result of the current paper. Our method is instead based on “energy increment arguments”, and more specifically on establishing quantitative version of a Khintchine-type recurrence theorem, similar to the qualitative recurrence theorems established (in the ergodic theory context) by Bergelson-Host-Kra, and (in the current combinatorial context) by Ben Green and myself.

One way to phrase the latter recurrence theorem is as follows. Suppose that has density . Then one would expect a “randomly” selected arithmetic progression in (using the convention that random variables will be in boldface) to be contained in with probability about . This is not true in general, however it was shown by Ben and myself that for any , there was a set of shifts of cardinality , such that for any such one had

if was chosen uniformly at random from . This easily implies that , but does not give a particularly good bound on the decay rate, because the implied constant in the cardinality lower bound is quite poor (in fact of tower-exponential type, due to the use of regularity lemmas!), and so one has to take to be extremely large compared to to avoid the possibility that the set of shifts in the above theorem consists only of the trivial shift .

We do not know how to improve the lower bound on the set of shifts to the point where it can give bounds that are competitive with those in this paper. However, we can obtain better quantitative results if we permit ourselves to *couple* together the two parameters and of the length four progression. Namely, with , , as above, we are able to show that there exist random variables , not necessarily independent, such that

and such that we have the non-degeneracy bound

This then easily implies the main theorem.

The energy increment method is then deployed to locate a good pair of random variables that will obey the above bounds. One can get some intuition on how to proceed here by considering some model cases. Firstly one can consider a “globally quadratically structured” case in which the indicator function “behaves like” a globally quadratic function such as , for some irrational and some smooth periodic function of mean . If one then takes to be uniformly distributed in and respectively for some small , with no coupling between the two variables, then the left-hand side of (1) is approximately of the form

where the integral is with respect to the probability Haar measure, and the constraint ultimately arises from the algebraic constraint

However, an application of the Cauchy-Schwarz inequality and Fubini’s theorem shows that the integral in (2) is at least , which (morally at least) gives (1) in this case.

Due to the nature of the energy increment argument, it also becomes necessary to consider “locally quadratically structured” cases, in which is partitioned into some number of structured pieces (think of these as arithmetic progressions, or as “Bohr sets), and on each piece , behaves like a locally quadratic function such as , where now varies with , and the mean of will be approximately on the average after averaging in (weighted by the size of the pieces ). Now one should select and in the following coupled manner: first one chooses uniformly from , then one defines to be the label such that , and then selects uniformly from a set which is related to in much the same way that is related to . If one does this correctly, the analogue of (2) becomes

and one can again use Cauchy-Schwarz and Fubini’s theorem to conclude.

The general case proceeds, very roughly, by an iterative argument. At each stage of the iteration, one has some sort of quadratic model of which involves a decomposition of into structured pieces , and a quadratic approximation to on each piece. If this approximation is accurate enough (or more precisely, if a certain (averaged) local Gowers uniformity norm of the error is small enough) to model the count in (1) (for random variables determined by the above partition of into pieces ), and if the frequencies (such as ) involved in the quadratic approximation are “high rank” or “linearly independent over the rationals” in a suitably quantitative sense, then some version of the above arguments can be made to work. If there are some unwanted linear dependencies in the frequencies, we can do some linear algebra to eliminate one of the frequencies (using some geometry of numbers to keep the quantitative bounds under control) and continue the iteration. If instead the approximation is too inaccurate, then the error will be large in a certain averaged local Gowers uniformity norm . A significant fraction of the paper is then devoted to establishing a quantitative *inverse theorem* for that norm that concludes (with good bounds) that the error must then locally correlate with locally quadratic phases, which can be used to refine the quadratic approximation to in a manner that significantly increases its “energy” (basically an norm). Such energy increments cannot continue indefinitely, and when they terminate we obtain the desired claim.

There are existing inverse theorems for type norms in the literature, going back to the pioneering work of Gowers mentioned previously, and relying on arithmetic combinatorics tools such as Freiman’s theorem and the Balog-Szemerédi-Gowers lemma, which are good for analysing the “-structured homomorphisms” that arise in Gowers’ argument. However, when we applied these methods to the local Gowers norms we obtained inferior quantitative results that were not strong enough for our application. Instead, we use arguments from a different paper of Gowers in which he tackled Szemerédi’s theorem for arbitrary length progressions. This method produces “-structured homomorphisms” associated to any function with large Gowers uniformity norm; however the catch is that such homomorphisms are initially supported only on a sparse unstructured set, rather than a structured set such as a Bohr set. To proceed further, one first has to locate inside the sparse unstructured set a sparse *pseudorandom* subset of a Bohr set, and then use “error-correction” type methods (such as “majority-vote” based algorithms) to locally upgrade this -structured homomorphism on pseudorandom subsets of Bohr sets to a -structured homomorphism on the entirety of a Bohr set. It is then possible to use some “approximate cohomology” tools to “integrate” these homomorphisms (and discern a key “local symmetry” property of these homomorphisms) to locate the desired local quadratic structure (in much the same fashion that a -form on that varies linearly with the coordinates can be integrated to be the derivative of a quadratic function if we know that the -form is closed). These portions of the paper are unfortunately rather technical, but broadly follow the methods already used in previous literature.

A sequence of complex numbers is said to be quasiperiodic if it is of the form

for some real numbers and continuous function . For instance, linear phases such as (where ) are examples of quasiperiodic sequences; the top order coefficient (modulo ) can be viewed as a “frequency” of the integers, and an element of the Pontryagin dual of the integers. Any periodic sequence is also quasiperiodic (taking and to be the reciprocal of the period). A sequence is said to be almost periodic if it is the uniform limit of quasiperiodic sequences. For instance any Fourier series of the form

with real numbers and an absolutely summable sequence of complex coefficients, will be almost periodic.

These sequences arise in various “complexity one” problems in arithmetic combinatorics and ergodic theory. For instance, if is a measure-preserving system – a probability space equipped with a measure-preserving shift, and are bounded measurable functions, then the correlation sequence

can be shown to be an almost periodic sequence, plus an error term which is “null” in the sense that it has vanishing uniform density:

This can be established in a number of ways, for instance by writing as the Fourier coefficients of the spectral measure of the shift with respect to the functions , and then decomposing that measure into pure point and continuous components.

In the last two decades or so, it has become clear that there are natural higher order versions of these concepts, in which linear polynomials such as are replaced with higher degree counterparts. The most obvious candidates for these counterparts would be the polynomials , but this turns out to not be a complete set of higher degree objects needed for the theory. Instead, the higher order versions of quasiperiodic and almost periodic sequences are now known as *basic nilsequences* and *nilsequences* respectively, while the higher order version of a linear phase is a *nilcharacter*; each nilcharacter then has a *symbol* that is a higher order generalisation of the concept of a frequency (and the collection of all symbols forms a group that can be viewed as a higher order version of the Pontryagin dual of ). The theory of these objects is spread out in the literature across a number of papers; in particular, the theory of nilcharacters is mostly developed in Appendix E of this 116-page paper of Ben Green, Tamar Ziegler, and myself, and is furthermore written using nonstandard analysis and treating the more general setting of higher dimensional sequences. I therefore decided to rewrite some of that material in this blog post, in the simpler context of the qualitative asymptotic theory of one-dimensional nilsequences and nilcharacters rather than the quantitative single-scale theory that is needed for combinatorial applications (and which necessitated the use of nonstandard analysis in the previous paper).

For technical reasons (having to do with the non-trivial topological structure on nilmanifolds), it will be convenient to work with vector-valued sequences, that take values in a finite-dimensional complex vector space rather than in . By doing so, the space of sequences is now, technically, no longer a ring, as the operations of addition and multiplication on vector-valued sequences become ill-defined. However, we can still take complex conjugates of a sequence, and add sequences taking values in the same vector space , and for sequences taking values in different vector spaces , , we may utilise the tensor product , which we will normalise by defining

This product is associative and bilinear, and also commutative up to permutation of the indices. It also interacts well with the Hermitian norm

since we have .

The traditional definition of a basic nilsequence (as defined for instance by Bergelson, Host, and Kra) is as follows:

Definition 1 (Basic nilsequence, first definition)Anilmanifold of step at mostis a quotient , where is a connected, simply connected nilpotent Lie group of step at most (thus, all -fold commutators vanish) and is a discrete cocompact lattice in . Abasic nilsequence of degree at mostis a sequence of the form , where , , and is a continuous function.

For instance, it is not difficult using this definition to show that a sequence is a basic nilsequence of degree at most if and only if it is quasiperiodic. The requirement that be simply connected can be easily removed if desired by passing to a universal cover, but it is technically convenient to assume it (among other things, it allows for a well-defined logarithm map that obeys the Baker-Campbell-Hausdorff formula). When one wishes to perform a more quantitative analysis of nilsequences (particularly when working on a “single scale”. sich as on a single long interval ), it is common to impose additional regularity conditions on the function , such as Lipschitz continuity or smoothness, but ordinary continuity will suffice for the qualitative discussion in this blog post.

Nowadays, and particularly when one needs to understand the “single-scale” equidistribution properties of nilsequences, it is more convenient (as is for instance done in this ICM paper of Green) to use an alternate definition of a nilsequence as follows.

Definition 2Let . Afiltered group of degree at mostis a group together with a sequence of subgroups with and for . Apolynomial sequenceinto a filtered group is a function such that for all and , where is the difference operator. Afiltered nilmanifold of degree at mostis a quotient , where is a filtered group of degree at most such that and all of the subgroups are connected, simply connected nilpotent filtered Lie group, and is a discrete cocompact subgroup of such that is a discrete cocompact subgroup of . Abasic nilsequence of degree at mostis a sequence of the form , where is a polynomial sequence, is a filtered nilmanifold of degree at most , and is a continuous function which is -automorphic, in the sense that for all and .

One can easily identify a -automorphic function on with a function on , but there are some (very minor) advantages to working on the group instead of the quotient , as it becomes slightly easier to modify the automorphy group when needed. (But because the action of on is free, one can pass from -automorphic functions on to functions on with very little difficulty.) The main reason to work with polynomial sequences rather than geometric progressions is that they form a group, a fact essentially established by by Lazard and Leibman; see Corollary B.4 of this paper of Green, Ziegler, and myself for a proof in the filtered group setting.

It is easy to see that any sequence that is a basic nilsequence of degree at most in the sense of the first definition, is also a basic nilsequence of degree at most in the second definition, since a nilmanifold of degree at most can be filtered using the lower central series, and any linear sequence will be a polynomial sequence with respect to that filtration. The converse implication is a little trickier, but still not too hard to show: see Appendix C of this paper of Ben Green, Tamar Ziegler, and myself. There are two key examples of basic nilsequences to keep in mind. The first are the polynomially quasiperiodic sequences

where are polynomials of degree at most , and is a -automorphic (i.e., -periodic) continuous function. The map defined by is a polynomial map of degree at most , if one filters by defining to equal when , and for . The torus then becomes a filtered nilmanifold of degree at most , and is thus a basic nilsequence of degree at most as per the second definition. It is also possible explicitly describe as a basic nilsequence of degree at most as per the first definition, for instance (in the case) by taking to be the space of upper triangular unipotent real matrices, and the subgroup with integer coefficients; we leave the details to the interested reader.

The other key example is a sequence of the form

where are real numbers, denotes the fractional part of , and and is a -automorphic continuous function that vanishes in a neighbourhood of . To describe this as a nilsequence, we use the nilpotent connected, simply connected degree , Heisenberg group

with the lower central series filtration , , and for , to be the discrete compact subgroup

to be the polynomial sequence

and to be the -automorphic function

one easily verifies that this function is indeed -automorphic, and it is continuous thanks to the vanishing properties of . Also we have , so is a basic nilsequence of degree at most . One can concoct similar examples with replaced by other “bracket polynomials” of ; for instance

will be a basic nilsequence if now vanishes in a neighbourhood of rather than . See this paper of Bergelson and Leibman for more discussion of bracket polynomials (also known as generalised polynomials) and their relationship to nilsequences.

A *nilsequence of degree at most * is defined to be a sequence that is the uniform limit of basic nilsequences of degree at most . Thus for instance a sequence is a nilsequence of degree at most if and only if it is almost periodic, while a sequence is a nilsequence of degree at most if and only if it is constant. Such objects arise in higher order recurrence: for instance, if are integers, is a measure-preserving system, and , then it was shown by Leibman that the sequence

is equal to a nilsequence of degree at most , plus a null sequence. (The special case when the measure-preserving system was ergodic and for was previously established by Bergelson, Host, and Kra.) Nilsequences also arise in the inverse theory of the Gowers uniformity norms, as discussed for instance in this previous post.

It is easy to see that a sequence is a basic nilsequence of degree at most if and only if each of its components are. The scalar basic nilsequences of degree are easily seen to form a -algebra (that is to say, they are a complex vector space closed under pointwise multiplication and complex conjugation), which implies similarly that vector-valued basic nilsequences of degree at most form a complex vector space closed under complex conjugation for each , and that the tensor product of any two basic nilsequences of degree at most is another basic nilsequence of degree at most . Similarly with “basic nilsequence” replaced by “nilsequence” throughout.

Now we turn to the notion of a nilcharacter, as defined in this paper of Ben Green, Tamar Ziegler, and myself:

Definition 3 (Nilcharacters)Let . Asub-nilcharacter of degreeis a basic nilsequence of degree at most , such that obeys the additional modulation propertyfor all and , where is a continuous homomorphism . (Note from (1) and -automorphy that unless vanishes identically, must map to , thus without loss of generality one can view as an element of the Pontryagial dual of the torus .) If in addition one has for all , we call a

nilcharacterof degree .

In the degree one case , the only sub-nilcharacters are of the form for some vector and , and this is a nilcharacter if is a unit vector. Similarly, in higher degree, any sequence of the form , where is a vector and is a polynomial of degree at most , is a sub-nilcharacter of degree , and a character if is a unit vector. A nilsequence of degree at most is automatically a sub-nilcharacter of degree , and a nilcharacter if it is of magnitude . A further example of a nilcharacter is provided by the two-dimensional sequence defined by

where are continuous, -automorphic functions that vanish on a neighbourhood of and respectively, and which form a partition of unity in the sense that

for all . Note that one needs both and to be not identically zero in order for all these conditions to be satisfied; it turns out (for topological reasons) that there is no scalar nilcharacter that is “equivalent” to this nilcharacter in a sense to be defined shortly. In some literature, one works exclusively with sub-nilcharacters rather than nilcharacters, however the former space contains zero-divisors, which is a little annoying technically. Nevertheless, both nilcharacters and sub-nilcharacters generate the same set of “symbols” as we shall see later.

We claim that every degree sub-nilcharacter can be expressed in the form , where is a degree nilcharacter, and is a linear transformation. Indeed, by scaling we may assume where uniformly. Using partitions of unity, one can find further functions also obeying (1) for the same character such that is non-zero; by dividing out the by the square root of this quantity, and then multiplying by , we may assume that

and then

becomes a degree nilcharacter that contains amongst its components, giving the claim.

As we shall show below, nilsequences can be approximated uniformly by linear combinations of nilcharacters, in much the same way that quasiperiodic or almost periodic sequences can be approximated uniformly by linear combinations of linear phases. In particular, nilcharacters can be used as “obstructions to uniformity” in the sense of the inverse theory of the Gowers uniformity norms.

The space of degree nilcharacters forms a semigroup under tensor product, with the constant sequence as the identity. One can upgrade this semigroup to an abelian group by quotienting nilcharacters out by equivalence:

Definition 4Let . We say that two degree nilcharacters , areequivalentif is equal (as a sequence) to a basic nilsequence of degree at most . (We will later show that this is indeed an equivalence relation.) The equivalence class of such a nilcharacter will be called thesymbolof that nilcharacter (in analogy to the symbol of a differential or pseudodifferential operator), and the collection of such symbols will be denoted .

As we shall see below the fold, has the structure of an abelian group, and enjoys some nice “symbol calculus” properties; also, one can view symbols as precisely describing the obstruction to equidistribution for nilsequences. For , the group is isomorphic to the Ponytragin dual of the integers, and for should be viewed as higher order generalisations of this Pontryagin dual. In principle, this group can be explicitly described for all , but the theory rapidly gets complicated as increases (much as the classification of nilpotent Lie groups or Lie algebras of step rapidly gets complicated even for medium-sized such as or ). We will give an explicit description of the case here. There is however one nice (and non-trivial) feature of for – it is not just an abelian group, but is in fact a vector space over the rationals !

How many groups of order four are there? Technically, there are an enormous number, so much so, in fact, that the class of groups of order four is not even a set, but merely a proper class. This is because *any* four objects can be turned into a group by designating one of the four objects, say , to be the group identity, and imposing a suitable multiplication table (and inversion law) on the four elements in a manner that obeys the usual group axioms. Since all sets are themselves objects, the class of four-element groups is thus at least as large as the class of all sets, which by Russell’s paradox is known not to itself be a set (assuming the usual ZFC axioms of set theory).

A much better question is to ask how many groups of order four there are *up to isomorphism*, counting each isomorphism class of groups exactly once. Now, as one learns in undergraduate group theory classes, the answer is just “two”: the cyclic group and the Klein four-group .

More generally, given a class of objects and some equivalence relation on (which one should interpret as describing the property of two objects in being “isomorphic”), one can consider the number of objects in “up to isomorphism”, which is simply the cardinality of the collection of equivalence classes of . In the case where is finite, one can express this cardinality by the formula

thus one counts elements in , weighted by the reciprocal of the number of objects they are isomorphic to.

Example 1Let be the five-element set of integers between and . Let us say that two elements of are isomorphic if they have the same magnitude: . Then the quotient space consists of just three equivalence classes: , , and . Thus there are three objects in up to isomorphism, and the identity (1) is then justThus, to count elements in up to equivalence, the elements are given a weight of because they are each isomorphic to two elements in , while the element is given a weight of because it is isomorphic to just one element in (namely, itself).

Given a finite probability set , there is also a natural probability distribution on , namely the *uniform distribution*, according to which a random variable is set equal to any given element of with probability :

Given a notion of isomorphism on , one can then define the random equivalence class that the random element belongs to. But if the isomorphism classes are unequal in size, we now encounter a biasing effect: even if was drawn uniformly from , the equivalence class need not be uniformly distributed in . For instance, in the above example, if was drawn uniformly from , then the equivalence class will not be uniformly distributed in the three-element space , because it has a probability of being equal to the class or to the class , and only a probability of being equal to the class .

However, it is possible to remove this bias by changing the weighting in (1), and thus changing the notion of what cardinality means. To do this, we generalise the previous situation. Instead of considering sets with an equivalence relation to capture the notion of isomorphism, we instead consider groupoids, which are sets in which there are allowed to be *multiple* isomorphisms between elements in (and in particular, there are allowed to be multiple *automorphisms* from an element to itself). More precisely:

Definition 2A groupoid is a set (or proper class) , together with a (possibly empty) collection of “isomorphisms” between any pair of elements of , and a composition map from isomorphisms , to isomorphisms in for every , obeying the following group-like axioms:

- (Identity) For every , there is an identity isomorphism , such that for all and .
- (Associativity) If , , and for some , then .
- (Inverse) If for some , then there exists an inverse isomorphism such that and .
We say that two elements of a groupoid are

isomorphic, and write , if there is at least one isomorphism from to .

Example 3Any category gives a groupoid by taking to be the set (or class) of objects, and to be the collection of invertible morphisms from to . For instance, in the category of sets, would be the collection of bijections from to ; in the category of linear vector spaces over some given base field , would be the collection of invertible linear transformations from to ; and so forth.

Every set equipped with an equivalence relation can be turned into a groupoid by assigning precisely one isomorphism from to for any pair with , and no isomorphisms from to when , with the groupoid operations of identity, composition, and inverse defined in the only way possible consistent with the axioms. We will call this the *simply connected groupoid* associated with this equivalence relation. For instance, with as above, if we turn into a simply connected groupoid, there will be precisely one isomorphism from to , and also precisely one isomorphism from to , but no isomorphisms from to , , or .

However, one can also form multiply-connected groupoids in which there can be multiple isomorphisms from one element of to another. For instance, one can view as a space that is acted on by multiplication by the two-element group . This gives rise to two types of isomorphisms, an identity isomorphism from to for each , and a negation isomorphism from to for each ; in particular, there are *two* automorphisms of (i.e., isomorphisms from to itself), namely and , whereas the other four elements of only have a single automorphism (the identity isomorphism). One defines composition, identity, and inverse in this groupoid in the obvious fashion (using the group law of the two-element group ); for instance, we have .

For a finite multiply-connected groupoid, it turns out that the natural notion of “cardinality” (or as I prefer to call it, “cardinality up to isomorphism”) is given by the variant

of (1). That is to say, in the multiply connected case, the denominator is no longer the number of objects isomorphic to , but rather the number of *isomorphisms* from to other objects . Grouping together all summands coming from a single equivalence class in , we can also write this expression as

where is the automorphism group of , that is to say the group of isomorphisms from to itself. (Note that if belong to the same equivalence class , then the two groups and will be isomorphic and thus have the same cardinality, and so the above expression is well-defined.

For instance, if we take to be the simply connected groupoid on , then the number of elements of up to isomorphism is

exactly as before. If however we take the multiply connected groupoid on , in which has two automorphisms, the number of elements of up to isomorphism is now the smaller quantity

the equivalence class is now counted with weight rather than due to the two automorphisms on . Geometrically, one can think of this groupoid as being formed by taking the five-element set , and “folding it in half” around the fixed point , giving rise to two “full” quotient points and one “half” point . More generally, given a finite group acting on a finite set , and forming the associated multiply connected groupoid, the cardinality up to isomorphism of this groupoid will be , since each element of will have isomorphisms on it (whether they be to the same element , or to other elements of ).

The definition (2) can also make sense for some infinite groupoids; to my knowledge this was first explicitly done in this paper of Baez and Dolan. Consider for instance the category of finite sets, with isomorphisms given by bijections as in Example 3. Every finite set is isomorphic to for some natural number , so the equivalence classes of may be indexed by the natural numbers. The automorphism group of has order , so the cardinality of up to isomorphism is

(This fact is sometimes loosely stated as “the number of finite sets is “, but I view this statement as somewhat misleading if the qualifier “up to isomorphism” is not added.) Similarly, when one allows for multiple isomorphisms from a group to itself, the number of groups of order four up to isomorphism is now

because the cyclic group has two automorphisms, whereas the Klein four-group has six.

In the case that the cardinality of a groupoid up to isomorphism is finite and non-empty, one can now define the notion of a random isomorphism class in drawn “uniformly up to isomorphism”, by requiring the probability of attaining any given isomorphism class to be

thus the probability of being isomorphic to a given element will be inversely proportional to the number of automorphisms that has. For instance, if we take to be the set with the simply connected groupoid, will be drawn uniformly from the three available equivalence classes , with a probability of attaining each; but if instead one uses the multiply connected groupoid coming from the action of , and draws uniformly up to isomorphism, then and will now be selected with probability each, and will be selected with probability . Thus this distribution has accounted for the bias mentioned previously: if a finite group acts on a finite space , and is drawn uniformly from , then now still be drawn uniformly up to isomorphism from , if we use the multiply connected groupoid coming from the action, rather than the simply connected groupoid coming from just the -orbit structure on .

Using the groupoid of finite sets, we see that a finite set chosen uniformly up to isomorphism will have a cardinality that is distributed according to the Poisson distribution of parameter , that is to say it will be of cardinality with probability .

One important source of groupoids are the fundamental groupoids of a manifold (one can also consider more general topological spaces than manifolds, but for simplicity we will restrict this discussion to the manifold case), in which the underlying space is simply , and the isomorphisms from to are the equivalence classes of paths from to up to homotopy; in particular, the automorphism group of any given point is just the fundamental group of at that base point. The equivalence class of a point in is then the connected component of in . The cardinality up to isomorphism of the fundamental groupoid is then

where is the collection of connected components of , and is the order of the fundamental group of . Thus, simply connected components of count for a full unit of cardinality, whereas multiply connected components (which can be viewed as quotients of their simply connected cover by their fundamental group) will count for a fractional unit of cardinality, inversely to the order of their fundamental group.

This notion of cardinality up to isomorphism of a groupoid behaves well with respect to various basic notions. For instance, suppose one has an -fold covering map of one finite groupoid by another . This means that is a functor that is surjective, with all preimages of cardinality , with the property that given any pair in the base space and any in the preimage of , every isomorphism has a unique lift from the given initial point (and some in the preimage of ). Then one can check that the cardinality up to isomorphism of is times the cardinality up to isomorphism of , which fits well with the geometric picture of as the -fold cover of . (For instance, if one covers a manifold with finite fundamental group by its universal cover, this is a -fold cover, the base has cardinality up to isomorphism, and the universal cover has cardinality one up to isomorphism.) Related to this, if one draws an equivalence class of uniformly up to isomorphism, then will be an equivalence class of drawn uniformly up to isomorphism also.

Indeed, one can show that this notion of cardinality up to isomorphism for groupoids is uniquely determined by a small number of axioms such as these (similar to the axioms that determine Euler characteristic); see this blog post of Qiaochu Yuan for details.

The probability distributions on isomorphism classes described by the above recipe seem to arise naturally in many applications. For instance, if one draws a profinite abelian group up to isomorphism at random in this fashion (so that each isomorphism class of a profinite abelian group occurs with probability inversely proportional to the number of automorphisms of this group), then the resulting distribution is known as the *Cohen-Lenstra distribution*, and seems to emerge as the natural asymptotic distribution of many randomly generated profinite abelian groups in number theory and combinatorics, such as the class groups of random quadratic fields; see this previous blog post for more discussion. For a simple combinatorial example, the set of fixed points of a random permutation on elements will have a cardinality that converges in distribution to the Poisson distribution of rate (as discussed in this previous post), thus we see that the fixed points of a large random permutation asymptotically are distributed uniformly up to isomorphism. I’ve been told that this notion of cardinality up to isomorphism is also particularly compatible with stacks (which are a good framework to describe such objects as moduli spaces of algebraic varieties up to isomorphism), though I am not sufficiently acquainted with this theory to say much more than this.

Just a short post to note that Norwegian Academy of Science and Letters has just announced that the 2017 Abel prize has been awarded to Yves Meyer, “for his pivotal role in the development of the mathematical theory of wavelets”. The actual prize ceremony will be at Oslo in May.

I am actually in Oslo myself currently, having just presented Meyer’s work at the announcement ceremony (and also having written a brief description of some of his work). The Abel prize has a somewhat unintuitive (and occasionally misunderstood) arrangement in which the presenter of the work of the prize is selected independently of the winner of the prize (I think in part so that the choice of presenter gives no clues as to the identity of the laureate). In particular, like other presenters before me (which in recent years have included Timothy Gowers, Jordan Ellenberg, and Alex Bellos), I agreed to present the laureate’s work before knowing who the laureate was! But in this case the task was very easy, because Meyer’s areas of (both pure and applied) harmonic analysis and PDE fell rather squarely within my own area of expertise. (I had previously written about some other work of Meyer in this blog post.) Indeed I had learned about Meyer’s wavelet constructions as a graduate student while taking a course from Ingrid Daubechies. Daubechies also made extremely important contributions to the theory of wavelets, but due to a conflict of interest (as per the guidelines for the prize committee) arising from Daubechies’ presidency of the International Mathematical Union (which nominates the majority of the members of the Abel prize committee, who then serve for two years) from 2011 to 2014 (and her continuing service *ex officio* on the IMU executive committee from 2015 to 2018), she will not be eligible for the prize until 2021 at the earliest, and so I do not think this prize should be necessarily construed as a judgement on the relative contributions of Meyer and Daubechies to this field. (In any case I fully agree with the Abel prize committee’s citation of Meyer’s pivotal role in the development of the theory of wavelets.)

*[Update, Mar 28: link to prize committee guidelines and clarification of the extent of Daubechies’ conflict of interest added. -T]*

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