I am totally stunned to learn that Maryam Mirzakhani died today yesterday, aged 40, after a severe recurrence of the cancer she had been fighting for several years.  I had planned to email her some wishes for a speedy recovery after learning about the relapse yesterday; I still can’t fully believe that she didn’t make it.

My first encounter with Maryam was in 2010, when I was giving some lectures at Stanford – one on Perelman’s proof of the Poincare conjecture, and another on random matrix theory.  I remember a young woman sitting in the front who asked perceptive questions at the end of both talks; it was only afterwards that I learned that it was Mirzakhani.  (I really wish I could remember exactly what the questions were, but I vaguely recall that she managed to put a nice dynamical systems interpretation on both of the topics of my talks.)

After she won the Fields medal in 2014 (as I posted about previously on this blog), we corresponded for a while.  The Fields medal is of course one of the highest honours one can receive in mathematics, and it clearly advances one’s career enormously; but it also comes with a huge initial burst of publicity, a marked increase in the number of responsibilities to the field one is requested to take on, and a strong expectation to serve as a public role model for mathematicians.  As the first female recipient of the medal, and also the first to come from Iran, Maryam was experiencing these pressures to a far greater extent than previous medallists, while also raising a small daughter and fighting off cancer.  I gave her what advice I could on these matters (mostly that it was acceptable – and in fact necessary – to say “no” to the vast majority of requests one receives).

Given all this, it is remarkable how productive she still was mathematically in the last few years.  Perhaps her greatest recent achievement has been her “magic wandtheorem with Alex Eskin, which is basically the analogue of the famous measure classification and orbit closure theorems of Marina Ratner, in the context of moduli spaces instead of unipotent flows on homogeneous spaces.  (I discussed Ratner’s theorems in this previous post.  By an unhappy coincidence, Ratner also passed away this month, aged 78.)  Ratner’s theorems are fundamentally important to any problem to which a homogeneous dynamical system can be associated (for instance, a special case of that theorem shows up in my work with Ben Green and Tamar Ziegler on the inverse conjecture for the Gowers norms, and on linear equations in primes), as it gives a good description of the equidistribution of any orbit of that system (if it is unipotently generated); and it seems the Eskin-Mirzakhani result will play a similar role in problems associated instead to moduli spaces.  The remarkable proof of this result – which now stands at over 200 pages, after three years of revision and updating – uses almost all of the latest techniques that had been developed for homogeneous dynamics, and ingeniously adapts them to the more difficult setting of moduli spaces, in a manner that had not been dreamed of being possible only a few years earlier.

Maryam was an amazing mathematician and also a wonderful and humble human being, who was at the peak of her powers.  Today was a huge loss for Maryam’s family and friends, as well as for mathematics.

[EDIT, Jul 16: New York times obituary here.]

[EDIT, Jul 18: New Yorker memorial here.]

 

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

\displaystyle  \partial_{tt} u = - (\nabla F)(u) \ \ \ \ \ (1)

for a particle {u: {\bf R} \rightarrow {\bf R}^m} trapped in a potential well with potential {F: {\bf R}^m \rightarrow {\bf R}}, with {F(z) \rightarrow +\infty} as {z \rightarrow \infty}. This ODE always admits global solutions from arbitrary initial positions {u(0)} and initial velocities {\partial_t u(0)}, thanks to conservation of the Hamiltonian {\frac{1}{2} |\partial_t u|^2 + F(u)}. 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 {\partial_{tt} u = - k^2 u} (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 {M} be a compact manifold, and let {X} be a smooth vector field on {M}; to avoid degeneracies, let us take {X} to be non-singular in the sense that it is everywhere non-vanishing. Then the trajectories of the first-order ODE

\displaystyle  \partial_t u = X(u) \ \ \ \ \ (2)

for {u: {\bf R} \rightarrow M} are always global and almost periodic. Can we then find a (coercive) potential {F: {\bf R}^m \rightarrow {\bf R}} for some {m}, as well as a smooth embedding {\phi: M \rightarrow {\bf R}^m}, such that every solution {u} to (2) pushes forward under {\phi} to a solution to (1)? (Actually, for technical reasons it is preferable to map into the phase space {{\bf R}^m \times {\bf R}^m}, rather than position space {{\bf R}^m}, 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 {(M,X)} as above, define a strongly adapted {1}-form to be a {1}-form {\phi} on {M} such that {\phi(X)} is pointwise positive, and the Lie derivative {{\mathcal L}_X \phi} is an exact {1}-form. We then have

Theorem 1 A smooth compact non-singular dynamics {(M,X)} can be embedded smoothly in a potential well system if and only if it admits a strongly adapted {1}-form.

For the “only if” direction, the key point is that potential wells (viewed as a Hamiltonian flow on the phase space {{\bf R}^m \times {\bf R}^m}) admit a strongly adapted {1}-form, namely the canonical {1}-form {p dq}, whose Lie derivative is the derivative {dL} of the Lagrangian {L := \frac{1}{2} |\partial_t u|^2 - F(u)} 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 {{\bf R}^m}, 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 {(M,X)} that do not support strongly adapted {1}-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 {(\sin(2\pi x), \cos(2\pi x))} on the {2}-torus {({\bf R}/{\bf Z})^2} had no strongly adapted {1}-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 {1}-form (the derivative {dt} 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…)

A few days ago, I was talking with Ed Dunne, who is currently the Executive Editor of Mathematical Reviews (and in particular with its online incarnation at MathSciNet).  At the time, I was mentioning how laborious it was for me to create a BibTeX file for dozens of references by using MathSciNet to locate each reference separately, and to export each one to BibTeX format.  He then informed me that underneath to every MathSciNet reference there was a little link to add the reference to a Clipboard, and then one could export the entire Clipboard at once to whatever format one wished.  In retrospect, this was a functionality of the site that had always been visible, but I had never bothered to explore it, and now I can populate a BibTeX file much more quickly.

This made me realise that perhaps there are many other useful features of popular mathematical tools out there that only a few users actually know about, so I wanted to create a blog post to encourage readers to post their own favorite tools, or features of tools, that are out there, often in plain sight, but not always widely known.  Here are a few that I was able to recall from my own workflow (though for some of them it took quite a while to consciously remember, since I have been so used to them for so long!):

  1. TeX for Gmail.  A Chrome plugin that lets one write TeX symbols in emails sent through Gmail (by writing the LaTeX code and pressing a hotkey, usually F8).
  2. Boomerang for Gmail.  Another Chrome plugin for Gmail, which does two main things.  Firstly, it can “boomerang” away an email from your inbox to return at some specified later date (e.g. one week from today).  I found this useful to declutter my inbox regarding mail that I needed to act on in the future, but was unable to deal with at present due to travel, or because I was waiting for some other piece of data to arrive first.   Secondly, it can send out email with some specified delay (e.g. by tomorrow morning), giving one time to cancel the email if necessary.  (Thanks to Julia Wolf for telling me about Boomerang!)
  3. Which just reminds me, the Undo Send feature on Gmail has saved me from embarrassment a few times (but one has to set it up first; it delays one’s emails by a short period, such as 30 seconds, during which time it is possible to undo the email).
  4. LaTeX rendering in Inkscape.  I used to use plain text to write mathematical formulae in my images, which always looked terrible.  It took me years to realise that Inkscape had the functionality to compile LaTeX within it.
  5. Bookmarks in TeXnicCenter.  I probably only use a tiny fraction of the functionality that TeXnicCenter offers, but one little feature I quite like is the ability to bookmark a portion of the TeX file (e.g. the bibliography at the end, or the place one is currently editing) with one hot-key (Ctrl-F2) and then one can cycle quickly between one bookmarked location and another with some further hot-keys (F2 and shift-F2).
  6. Actually, there are a number of Windows keyboard shortcuts that are worth experimenting with (and similarly for Mac or Linux systems of course).
  7. Detexify has been the quickest way for me to locate the TeX code for a symbol that I couldn’t quite remember (or when hunting for a new symbol that would roughly be shaped like something I had in mind).
  8. For writing LaTeX on my blog, I use Luca Trevisan’s LaTeX to WordPress Python script (together with a little batch file I wrote to actually run the python script).
  9. Using the camera on my phone to record a blackboard computation or a slide (or the wifi password at a conference centre, or any other piece of information that is written or displayed really).  If the phone is set up properly this can be far quicker than writing it down with pen and paper.  (I guess this particular trick is now quite widely used, but I still see people surprised when someone else uses a phone instead of a pen to record things.)
  10. Using my online calendar not only to record scheduled future appointments, but also to block out time to do specific tasks (e.g. reserve 2-3pm at Tuesday to read paper X, or do errand Y).  I have found I am able to get a much larger fraction of my “to do” list done on days in which I had previously blocked out such specific chunks of time, as opposed to days in which I had left several hours unscheduled (though sometimes those hours were also very useful for finding surprising new things to do that I had not anticipated).  (I learned of this little trick online somewhere, but I have long since lost the original reference.)

Anyway, I would very much like to hear what other little tools or features other readers have found useful in their work.

 

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

\displaystyle \sum_{n \leq X} \Lambda(n) \Lambda(n+h) \ \ \ \ \ (1)

for medium-sized {h} and large {X}, where {\Lambda} 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 {h}; for instance, if one could establish a lower bound

\displaystyle \sum_{n \leq X} \Lambda(n) \Lambda(n+2) \gg X

then this would easily imply the twin prime conjecture.

The (first) Hardy-Littlewood conjecture asserts an asymptotic

\displaystyle \sum_{n \leq X} \Lambda(n) \Lambda(n+h) = {\mathfrak S}(h) X + o(X) \ \ \ \ \ (2)

as {X \rightarrow \infty} for any fixed positive {h}, where the singular series {{\mathfrak S}(h)} is an arithmetic factor arising from the irregularity of distribution of {\Lambda} at small moduli, defined explicitly by

\displaystyle {\mathfrak S}(h) := 2 \Pi_2 \prod_{p|h; p>2} \frac{p-2}{p-1}

when {h} is even, and {{\mathfrak S}(h)=0} when {h} is odd, where

\displaystyle \Pi_2 := \prod_{p>2} (1-\frac{1}{(p-1)^2}) = 0.66016\dots

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 {h=2} would imply the twin prime conjecture. Sieve theoretic methods are only able to provide an upper bound of the form { \sum_{n \leq X} \Lambda(n) \Lambda(n+h) \ll {\mathfrak S}(h) X}.

Needless to say, apart from the trivial case of odd {h}, there are no values of {h} 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 {H} is a quantity depending on {X} that is somewhat large, there are results that show that (2) holds for most (i.e. for {1-o(1)}) of the {h} betwen {0} and {H}. Ideally one would like to get {H} as small as possible, in particular one can view the full Hardy-Littlewood conjecture as the endpoint case when {H} is bounded.

The first results in this direction were by van der Corput and by Lavrik, who established such a result with {H = X} (with a subsequent refinement by Balog); Wolke lowered {H} to {X^{5/8+\varepsilon}}, and Mikawa lowered {H} further to {X^{1/3+\varepsilon}}. The main result of this paper is a further lowering of {H} to {X^{8/33+\varepsilon}}. In fact (as in the preceding works) we get a better error term than {o(X)}, namely an error of the shape {O_A( X \log^{-A} X)} for any {A}.

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 {S(\alpha) := \sum_{n \leq X} \Lambda(n) e(\alpha n)}. The contribution of “major arc” {\alpha} is known by a standard computation to recover the main term {{\mathfrak S}(h) X} plus acceptable errors, so it is a matter of controlling the “minor arcs”. After averaging in {h} and using the Plancherel identity, one is basically faced with establishing a bound of the form

\displaystyle \int_{\beta-1/H}^{\beta+1/H} |S(\alpha)|^2\ d\alpha \ll_A X \log^{-A} X

for any “minor arc” {\beta}. If {\beta} is somewhat close to a low height rational {a/q} (specifically, if it is within {X^{-1/6-\varepsilon}} of such a rational with {q = O(\log^{O(1)} X)}), 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 {[x, x + x^{1/6+\varepsilon}]}, 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 {\beta} 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 {S(\alpha) := \sum_{n \leq X} \Lambda(n) e(\alpha n)}, but rather in terms of the Dirichlet polynomial {F(s) := \sum_{n \sim X} \frac{\Lambda(n)}{n^s}}. 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

\displaystyle \int_{t \sim \lambda X} (\sum_{t-\lambda H}^{t+\lambda H} |F(\frac{1}{2}+it')|\ dt')^2\ dt \ll_A \lambda^2 H^2 X \log^{-A} X

for any {X^{-1/6-\varepsilon} \ll \lambda \ll \log^{-B} X} (with {B} sufficiently large depending on {A}).

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 {\Lambda} into a number of components that have a Dirichlet convolution structure. Because the exponent {8/33} we are shooting for is less than {1/4}, we end up with five types of components that arise, which we call “Type {d_1}“, “Type {d_2}“, “Type {d_3}“, “Type {d_4}“, and “Type II”. The “Type II” sums are Dirichlet convolutions involving a factor supported on a range {[X^\varepsilon, X^{-\varepsilon} H]} and is quite easy to deal with; the “Type {d_j}” terms are Dirichlet convolutions that resemble (non-degenerate portions of) the {j^{th}} divisor function, formed from convolving together {j} portions of {1}. The “Type {d_1}” and “Type {d_2}” 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 {d_3}” and “Type {d_4}” sums that require some new analysis, with the Type {d_3} 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 {d_3} sums) is of the form

\displaystyle \int_{t - H}^{t+H} |M( \frac{1}{2} + it')|^2\ dt' \ll X^{\varepsilon^2} H \ \ \ \ \ (3)

for “typical” ordinates {t} of size {X}, where {M} is the Dirichlet polynomial {M(s) := \sum_{n \sim X^{1/3}} \frac{1}{n^s}} (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 {M(1/2 + it) \ll X^{o(1)}}) 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

\displaystyle \frac{H}{X^{1/3}} \sum_{\ell \sim X^{1/3}/H} |\sum_{m \sim X^{1/3}} e( \frac{t}{2\pi} \log \frac{m+\ell}{m-\ell} )|.

The phase {\frac{t}{2\pi} \log \frac{m+\ell}{m-\ell}} can be Taylor expanded as the sum of {\frac{t_j \ell}{\pi m}} and a lower order term {\frac{t_j \ell^3}{3\pi m^3}}, 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 {t} 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 {B} process followed by the estimates of Robert and Sargos. This rather complicated procedure works up to {H = X^{8/33+\varepsilon}} 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 {H} to a much smaller value of {\log^{O(1)} X}, but only if we replace the correlations {\sum_{n \leq X} \Lambda(n) \Lambda(n+h)} by either {\sum_{n \leq X} \Lambda(n) d_k(n+h)} or {\sum_{n \leq X} d_k(n) d_l(n+h)}, and also we now only save a {o(1)} in the error term rather than {O_A(\log^{-A} X)}.

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 {F: X \rightarrow Y} is a continuous (but nonlinear) map from one normed vector space {X} to another {Y}. The continuity means, roughly speaking, that if {x_0, x \in X} are such that {\|x-x_0\|_X} is small, then {\|F(x)-F(x_0)\|_Y} is also small (though the precise notion of “smallness” may depend on {x} or {x_0}, particularly if {F} is not known to be uniformly continuous). If {F} is known to be differentiable (in, say, the Fréchet sense), then we in fact have a linear bound of the form

\displaystyle  \|F(x)-F(x_0)\|_Y \leq C(x_0) \|x-x_0\|_X

for some {C(x_0)} depending on {x_0}, if {\|x-x_0\|_X} is small enough; one can of course make {C(x_0)} independent of {x_0} (and drop the smallness condition) if {F} is known instead to be Lipschitz continuous.

In many applications in analysis, one would like more explicit and quantitative bounds that estimate quantities like {\|F(x)-F(x_0)\|_Y} in terms of quantities like {\|x-x_0\|_X}. There are a number of ways to do this. First of all, there is of course the trivial estimate arising from the triangle inequality:

\displaystyle  \|F(x)-F(x_0)\|_Y \leq \|F(x)\|_Y + \|F(x_0)\|_Y. \ \ \ \ \ (1)

This estimate is usually not very good when {x} and {x_0} are close together. However, when {x} and {x_0} are far apart, this estimate can be more or less sharp. For instance, if the magnitude of {F} varies so much from {x_0} to {x} that {\|F(x)\|_Y} is more than (say) twice that of {\|F(x_0)\|_Y}, or vice versa, then (1) is sharp up to a multiplicative constant. Also, if {F} is oscillatory in nature, and the distance between {x} and {x_0} exceeds the “wavelength” of the oscillation of {F} at {x_0} (or at {x}), then one also typically expects (1) to be close to sharp. Conversely, if {F} does not vary much in magnitude from {x_0} to {x}, and the distance between {x} and {x_0} is less than the wavelength of any oscillation present in {F}, one expects to be able to improve upon (1).

When {F} is relatively simple in form, one can sometimes proceed simply by substituting {x = x_0 + h}. For instance, if {F: R \rightarrow R} is the squaring function {F(x) = x^2} in a commutative ring {R}, one has

\displaystyle  F(x_0+h) = (x_0+h)^2 = x_0^2 + 2x_0 h+ h^2

and thus

\displaystyle  F(x_0+h) - F(x_0) = 2x_0 h + h^2

or in terms of the original variables {x, x_0} one has

\displaystyle  F(x) - F(x_0) = 2 x_0 (x-x_0) + (x-x_0)^2.

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

\displaystyle  F(x) - F(x_0) = x_0 (x-x_0) + (x-x_0) x_0 + (x-x_0)^2.

Thus, for instance, if {A, B} are {n \times n} matrices and {\| \|_{op}} denotes the operator norm, one sees from the triangle inequality and the sub-multiplicativity {\| AB\|_{op} \leq \| A \|_{op} \|B\|_{op}} of operator norm that

\displaystyle  \| A^2 - B^2 \|_{op} \leq \| A - B \|_{op} ( 2 \|B\|_{op} + \|A - B \|_{op} ). \ \ \ \ \ (2)

If {F(x)} involves {x} (or various components of {x}) in several places, one can sometimes get a good estimate by “swapping” {x} with {x_0} at each of the places in turn, using a telescoping series. For instance, if we again use the squaring function {F(x) = x^2 = x x} in a non-commutative ring, we have

\displaystyle  F(x) - F(x_0) = x x - x_0 x_0

\displaystyle  = (x x - x_0 x) + (x_0 x - x_0 x_0)

\displaystyle  = (x-x_0) x + x_0 (x-x_0)

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

\displaystyle  \| A^2 - B^2 \|_{op} \leq \| A - B \|_{op} ( \| A\|_{op} + \|B\|_{op} ).

More generally, for any natural number {n}, one has the identity

\displaystyle  x^n - x_0^n = (x-x_0) (x^{n-1} + x^{n-2} x_0 + \dots + x x_0^{n-2} + x_0^{n-1}) \ \ \ \ \ (3)

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

\displaystyle  x^n - x_0^n = \sum_{i=0}^{n-1} x_0^i (x-x_0) x^{n-1-i},

and for matrices one has

\displaystyle  \| A^n - B^n \|_{op} \leq \| A-B\|_{op} ( \|A\|_{op}^{n-1} + \| A\|_{op}^{n-2} \| B\|_{op} + \dots + \|B\|_{op}^{n-1} ).

Exercise 1 If {U} and {V} are unitary {n \times n} matrices, show that the commutator {[U,V] := U V U^{-1} V^{-1}} obeys the inequality

\displaystyle  \| [U,V] - I \|_{op} \leq 2 \| U - I \|_{op} \| V - I \|_{op}.

(Hint: first control {\| UV - VU \|_{op}}.)

Now suppose (for simplicity) that {F: {\bf R}^d \rightarrow {\bf R}^{d'}} is a map between Euclidean spaces. If {F} is continuously differentiable, then one can use the fundamental theorem of calculus to write

\displaystyle  F(x) - F(x_0) = \int_0^1 \frac{d}{dt} F( \gamma(t) )\ dt

where {\gamma: [0,1] \rightarrow Y} is any continuously differentiable path from {x_0} to {x}. For instance, if one uses the straight line path {\gamma(t) := (1-t) x_0 + tx}, one has

\displaystyle  F(x) - F(x_0) = \int_0^1 ((x-x_0) \cdot \nabla F)( (1-t) x_0 + t x )\ dt.

In the one-dimensional case {d=1}, this simplifies to

\displaystyle  F(x) - F(x_0) = (x-x_0) \int_0^1 F'( (1-t) x_0 + t x )\ dt. \ \ \ \ \ (4)

Among other things, this immediately implies the factor theorem for {C^k} functions: if {F} is a {C^k({\bf R})} function for some {k \geq 1} that vanishes at some point {x_0}, then {F(x)} factors as the product of {x-x_0} and some {C^{k-1}} function {G}. Another basic consequence is that if {\nabla F} is uniformly bounded in magnitude by some constant {C}, then {F} is Lipschitz continuous with the same constant {C}.

Applying (4) to the power function {x \mapsto x^n}, we obtain the identity

\displaystyle  x^n - x_0^n = n (x-x_0) \int_0^1 ((1-t) x_0 + t x)^{n-1}\ dt \ \ \ \ \ (5)

which can be compared with (3). Indeed, for {x_0} and {x} close to {1}, one can use logarithms and Taylor expansion to arrive at the approximation {((1-t) x_0 + t x)^{n-1} \approx x_0^{(1-t) (n-1)} x^{t(n-1)}}, so (3) behaves a little like a Riemann sum approximation to (5).

Exercise 2 For each {i=1,\dots,n}, let {X^{(1)}_i} and {X^{(0)}_i} be random variables taking values in a measurable space {R_i}, and let {F: R_1 \times \dots \times R_n \rightarrow {\bf R}^m} be a bounded measurable function.

  • (i) (Lindeberg exchange identity) Show that

    \displaystyle  \mathop{\bf E} F(X^{(1)}_1,\dots,X^{(1)}_n) - \mathop{\bf E} F(X^{(0)}_1,\dots,X^{(0)}_n)

    \displaystyle = \sum_{i=1}^n \mathop{\bf E} F( X^{(1)}_1,\dots, X^{(1)}_{i-1}, X^{(1)}_i, X^{(0)}_{i+1}, \dots, X^{(0)}_n)

    \displaystyle - \mathop{\bf E} F( X^{(1)}_1,\dots, X^{(1)}_{i-1}, X^{(0)}_i, X^{(0)}_{i+1}, \dots, X^{(0)}_n).

  • (ii) (Knowles-Yin exchange identity) Show that

    \displaystyle  \mathop{\bf E} F(X^{(1)}_1,\dots,X^{(1)}_n) - \mathop{\bf E} F(X^{(0)}_1,\dots,X^{(0)}_n)

    \displaystyle = \int_0^1 \sum_{i=1}^n \mathop{\bf E} F( X^{(t)}_1,\dots, X^{(t)}_{i-1}, X^{(1)}_i, X^{(t)}_{i+1}, \dots, X^{(t)}_n)

    \displaystyle - \mathop{\bf E} F( X^{(t)}_1,\dots, X^{(t)}_{i-1}, X^{(0)}_i, X^{(t)}_{i+1}, \dots, X^{(t)}_n)\ dt,

    where {X^{(t)}_i = 1_{I_i \leq t} X^{(0)}_i + 1_{I_i > t} X^{(1)}_i} is a mixture of {X^{(0)}_i} and {X^{(1)}_i}, with {I_1,\dots,I_n} uniformly drawn from {[0,1]} independently of each other and of the {X^{(0)}_1,\dots,X^{(0)}_n, X^{(1)}_0,\dots,X^{(1)}_n}.

  • (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 method in 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 {1,\dots,n}, which can be a technical advantage in some applications; see this paper of Knowles and Yin for an instance of this.)

Exercise 3 If {F: {\bf R}^d \rightarrow {\bf R}^{d'}} is continuously {k} times differentiable, establish Taylor’s theorem with remainder

\displaystyle  F(x) = \sum_{j=0}^{k-1} \frac{1}{j!} (((x-x_0) \cdot \nabla)^j F)( x_0 )

\displaystyle + \int_0^1 \frac{(1-t)^{k-1}}{(k-1)!} (((x-x_0) \cdot \nabla)^k F)((1-t) x_0 + t x)\ dt.

If {\nabla^k F} is bounded, conclude that

\displaystyle  |F(x) - \sum_{j=0}^{k-1} \frac{1}{j!} (((x-x_0) \cdot \nabla)^j F)( x_0 )|

\displaystyle \leq \frac{|x-x_0|^k}{k!} \sup_{y \in {\bf R}^d} |\nabla^k F(y)|.

For real scalar functions {F: {\bf R}^d \rightarrow {\bf R}}, the average value of the continuous real-valued function {(x - x_0) \cdot \nabla F((1-t) x_0 + t x)} must be attained at some point {t} in the interval {[0,1]}. We thus conclude the mean-value theorem

\displaystyle  F(x) - F(x_0) = ((x - x_0) \cdot \nabla F)((1-t) x_0 + t x)

for some {t \in [0,1]} (that can depend on {x}, {x_0}, and {F}). This can for instance give a second proof of fact that continuously differentiable functions {F} 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 {e(x) := e^{2\pi i x}}; there is no {t \in [0,1]} for which {e(1) - e(0) = e'(t)}. On the other hand, as {e'} has magnitude {2\pi}, we still know from (4) that {e} is Lipschitz of constant {2\pi}, and when combined with (1) we obtain the basic bounds

\displaystyle  |e(x) - e(y)| \leq \min( 2, 2\pi |x-y| )

which are already very useful for many applications.

Exercise 4 Let {H_0, V} be {n \times n} matrices, and let {t} be a non-negative real.

  • (i) Establish the Duhamel formula

    \displaystyle  e^{t(H_0+V)} = e^{tH_0} + \int_0^t e^{(t-s) H_0} V e^{s (H_0+V)}\ ds

    \displaystyle  = e^{tH_0} + \int_0^t e^{(t-s) (H_0+V)} V e^{s H_0}\ ds

    where {e^A} denotes the matrix exponential of {A}. (Hint: Differentiate {e^{(t-s) H_0} e^{s (H_0+V)}} or {e^{(t-s) (H_0+V)} e^{s H_0}} in {s}.)

  • (ii) Establish the iterated Duhamel formula

    \displaystyle  e^{t(H_0+V)} = e^{tH_0} + \sum_{j=1}^k \int_{0 \leq t_1 \leq \dots \leq t_j \leq t}

    \displaystyle e^{(t-t_j) H_0} V e^{(t_j-t_{j-1}) H_0} V \dots e^{(t_2-t_1) H_0} V e^{t_1 H_0}\ dt_1 \dots dt_j

    \displaystyle  + \int_{0 \leq t_1 \leq \dots \leq t_{k+1} \leq t}

    \displaystyle  e^{(t-t_{k+1}) (H_0+V)} V e^{(t_{k+1}-t_k) H_0} V \dots e^{(t_2-t_1) H_0} V e^{t_1 H_0}\ dt_1 \dots dt_{k+1}

    for any {k \geq 0}.

  • (iii) Establish the infinitely iterated Duhamel formula

    \displaystyle  e^{t(H_0+V)} = e^{tH_0} + \sum_{j=1}^\infty \int_{0 \leq t_1 \leq \dots \leq t_j \leq t}

    \displaystyle e^{(t-t_j) H_0} V e^{(t_j-t_{j-1}) H_0} V \dots e^{(t_2-t_1) H_0} V e^{t_1 H_0}\ dt_1 \dots dt_j.

  • (iv) If {H(t)} is an {n \times n} matrix depending in a continuously differentiable fashion on {t}, establish the variation formula

    \displaystyle  \frac{d}{dt} e^{H(t)} = (F(\mathrm{ad}(H(t))) H'(t)) e^{H(t)}

    where {\mathrm{ad}(H)} is the adjoint representation {\mathrm{ad}(H)(V) = HV - VH} applied to {H}, and {F} is the function

    \displaystyle  F(z) := \int_0^1 e^{sz}\ ds

    (thus {F(z) = \frac{e^z-1}{z}} for non-zero {z}), with {F(\mathrm{ad}(H(t)))} 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 5 Let {A, B} be positive definite {n \times n} matrices, and let {Y} be an {n \times n} matrix. Show that there is a unique solution {X} to the Sylvester equation

\displaystyle  AX + X B = Y

which is given by the formula

\displaystyle  X = \int_0^\infty e^{-tA} Y e^{-tB}\ dt.

In the above examples we had applied the fundamental theorem of calculus along linear curves {\gamma(t) = (1-t) x_0 + t x}. However, it is sometimes better to use other curves. For instance, the circular arc {\gamma(t) = \cos(\pi t/2) x_0 + \sin(\pi t/2) x} can be useful, particularly if {x_0} and {x} 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 {X} of mean zero and variance one with a Gaussian random variable {G} of mean zero and variance one, it can be useful to introduce the intermediate random variables {\gamma(t) := (1-t)^{1/2} X + t^{1/2} G} (where {X} and {G} 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 {\gamma} 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 {F(x)-F(x_0)} or the derivative {\nabla F} directly, but one can instead proceed by implicit differentiation, or some variant thereof. Consider for instance the matrix inversion map {F(A) := A^{-1}} (defined on the open dense subset of {n \times n} matrices consisting of invertible matrices). If one wants to compute {F(B)-F(A)} for {B} close to {A}, one can write temporarily write {F(B) - F(A) = E}, thus

\displaystyle  B^{-1} - A^{-1} = E.

Multiplying both sides on the left by {B} to eliminate the {B^{-1}} term, and on the right by {A} to eliminate the {A^{-1}} term, one obtains

\displaystyle  A - B = B E A

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

\displaystyle  B^{-1} - A^{-1} = B^{-1} (A - B) A^{-1}. \ \ \ \ \ (6)

For instance, if {H_0, V} are {n \times n} matrices, and we consider the resolvents

\displaystyle  R_0(z) := (H_0 - z I)^{-1}; \quad R_V(z) := (H_0 + V - zI)^{-1}

then we have the resolvent identity

\displaystyle  R_V(z) - R_0(z) = - R_V(z) V R_0(z) \ \ \ \ \ (7)

as long as {z} does not lie in the spectrum of {H_0} or {H_0+V} (for instance, if {H_0}, {V} are self-adjoint then one can take {z} to be any strictly complex number). One can iterate this identity to obtain

\displaystyle  R_V(z) = \sum_{j=0}^k (-R_0(z) V)^j R_0(z) + (-R_V(z) V) (-R_0(z) V)^k R_0(z)

for any natural number {k}; in particular, if {R_0(z) V} has operator norm less than one, one has the Neumann series

\displaystyle  R_V(z) = \sum_{j=0}^\infty (-R_0(z) V)^j R_0(z).

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

\displaystyle  A(t) A(t)^{-1} = I

in {t} using the product rule, we obtain

\displaystyle  (\frac{d}{dt} A(t)) A(t)^{-1} + A(t) \frac{d}{dt} A(t)^{-1} = 0

and hence

\displaystyle  \frac{d}{dt} A(t)^{-1} = - A(t)^{-1} (\frac{d}{dt} A(t)) A(t)^{-1}

(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 {\gamma(t) = (1-t) A + tB} we arrive at

\displaystyle  B^{-1} - A^{-1} = \int_0^1 ((1-t)A + tB)^{-1} (A-B) ((1-t)A + tB)^{-1}\ dt

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 {A, B} are positive definite with {A>B} in the sense of positive definite matrices (that is, {A-B} is positive definite), then {B^{-1} > A^{-1}}. (Try to prove this using (6) instead!)

Exercise 6 If {A} is an invertible {n \times n} matrix and {u, v} are {n \times 1} vectors, establish the Sherman-Morrison formula

\displaystyle  (A + t uv^T)^{-1} = A^{-1} - \frac{t}{1 + t v^T A^{-1} u} A^{-1} uv^T A^{-1}

whenever {t} is a scalar such that {1 + t v^T A^{-1} u} 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 {F: {\bf C} \rightarrow {\bf C}} is an entire function, and {\gamma} is a counterclockwise contour that goes around the spectrum of both {H_0} and {H_0+V}, then we have

\displaystyle  F(H_0+V) = \frac{-1}{2\pi i} \int_\gamma F(z) R_V(z)\ dz

and similarly

\displaystyle  F(H_0) = \frac{-1}{2\pi i} \int_\gamma F(z) R_0(z)\ dz

and hence by (7) one has

\displaystyle  F(H_0+V) - F(H_0) = \frac{1}{2\pi i} \int_\gamma F(z) R_V(z) V F_0(z)\ dz;

similarly, if {H(t)} depends on {t} in a continuously differentiable fashion, then

\displaystyle  \frac{d}{dt} F(H(t)) = \frac{1}{2\pi i} \int_\gamma F(z) (H(t) - zI)^{-1} H'(t) (z) (H(t)-zI)^{-1}\ dz

as long as {\gamma} goes around the spectrum of {H(t)}.

Exercise 7 If {H(t)} is an {n \times n} matrix depending continuously differentiably on {t}, and {F: {\bf C} \rightarrow {\bf C}} is an entire function, establish the tracial chain rule

\displaystyle  \frac{d}{dt} \hbox{tr} F(H(t)) = \hbox{tr}(F'(H(t)) H'(t)).

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

\displaystyle  \log A = \int_0^\infty (I + sI)^{-1} - (A + sI)^{-1}\ ds

(with the constant term {(I+tI)^{-1}} needed to prevent a logarithmic divergence in the integral). Differentiating, we see that if {A(t)} is a family of positive definite matrices depending continuously on {t}, that

\displaystyle  \frac{d}{dt} \log A(t) = \int_0^\infty (A(t) + sI)^{-1} A'(t) (A(t)+sI)^{-1}\ dt.

This can be used for instance to show that {\log} is a monotone increasing function, in the sense that {\log A> \log B} whenever {A > B > 0} in the sense of positive definite matrices. One can of course integrate this formula to obtain some formulae for the difference {\log A - \log B} of the logarithm of two positive definite matrices {A,B}.

To compare the square root {A^{1/2} - B^{1/2}} of two positive definite matrices {A,B} 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

\displaystyle  A^{1/2} = \frac{1}{\Gamma(-1/2)} \int_0^\infty (e^{-tA} - I) t^{-1/2} \frac{dt}{t}

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

\displaystyle  A^{1/2} - B^{1/2} = \frac{1}{\Gamma(-1/2)} \int_0^\infty (e^{-tA} - e^{-tB}) t^{-1/2} \frac{dt}{t}

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

\displaystyle  A(t)^{1/2} A(t)^{1/2} = A(t)

to obtain

\displaystyle  A(t)^{1/2} \frac{d}{dt} A(t)^{1/2} + (\frac{d}{dt} A(t)^{1/2}) A(t)^{1/2} = \frac{d}{dt} A(t).

This can for instance be solved using Exercise 5 to obtain

\displaystyle  \frac{d}{dt} A(t)^{1/2} = \int_0^\infty e^{-sA(t)^{1/2}} A'(t) e^{-sA(t)^{1/2}}\ ds

and this can in turn be integrated to obtain a formula for {A^{1/2} - B^{1/2}}. 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: {A > B > 0} implies {A^{1/2} > B^{1/2} > 0}.

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 {(a_n)_{n=1}^\infty = (a_1, a_2, \dots)} of real numbers. What kinds of limits can one form from this sequence?

Of course, we have the usual notion of limit {\lim_{n \rightarrow \infty} a_n}, 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 {L} such that for every {\varepsilon>0}, one has {|a_n-L| \leq \varepsilon} for all sufficiently large {n}. 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 {(a_n)_{n=1}^\infty} and {(b_n)_{n=1}^\infty} are classically convergent sequences, then {(a_n+b_n)_{n=1}^\infty} is also classically convergent with

\displaystyle  \lim_{n \rightarrow \infty} (a_n + b_n) = (\lim_{n \rightarrow \infty} a_n) + (\lim_{n \rightarrow \infty} b_n) \ \ \ \ \ (1)

and similarly for any scalar {c}, {(ca_n)_{n=1}^\infty} is classically convergent with

\displaystyle  \lim_{n \rightarrow \infty} (ca_n) = c \lim_{n \rightarrow \infty} a_n. \ \ \ \ \ (2)

It is also an algebra homomorphism: {(a_n b_n)_{n=1}^\infty} is also classically convergent with

\displaystyle  \lim_{n \rightarrow \infty} (a_n b_n) = (\lim_{n \rightarrow \infty} a_n) (\lim_{n \rightarrow \infty} b_n). \ \ \ \ \ (3)

We also have shift invariance: if {(a_n)_{n=1}^\infty} is classically convergent, then so is {(a_{n+1})_{n=1}^\infty} with

\displaystyle  \lim_{n \rightarrow \infty} a_{n+1} = \lim_{n \rightarrow \infty} a_n \ \ \ \ \ (4)

and more generally in fact for any injection {\phi: {\bf N} \rightarrow {\bf N}}, {(a_{\phi(n)})_{n=1}^\infty} is classically convergent with

\displaystyle  \lim_{n \rightarrow \infty} a_{\phi(n)} = \lim_{n \rightarrow \infty} a_n. \ \ \ \ \ (5)

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 {(a_n)_{n=1}^\infty}, one has

\displaystyle  \inf_n a_n \leq \lim_{n \rightarrow \infty} a_n \leq \sup_n a_n. \ \ \ \ \ (6)

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 {a_n = (-1)^n}. On the one hand, one has {a_n^2 = 1} for all {n}, so if one wishes to retain the homomorphism property (3), any “limit” of this sequence {a_n} would have to necessarily square to {1}, that is to say it must equal {+1} or {-1}. 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

\displaystyle  \limsup_{n \rightarrow \infty} a_n := \inf_N \sup_{n \geq N} a_n

and limit inferior

\displaystyle  \liminf_{n \rightarrow \infty} a_n := \sup_N \inf_{n \geq N} a_n

which are well-defined real numbers for any bounded sequence {(a_n)_{n=1}^\infty}; 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

\displaystyle  \limsup_{n \rightarrow \infty} (a_n + b_n) \leq (\limsup_{n \rightarrow \infty} a_n) + (\limsup_{n \rightarrow \infty} b_n)

for the limit superior, and the superadditivity property

\displaystyle  \liminf_{n \rightarrow \infty} (a_n + b_n) \geq (\liminf_{n \rightarrow \infty} a_n) + (\liminf_{n \rightarrow \infty} b_n)

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

\displaystyle  \limsup_{n \rightarrow \infty} (-a_n) = - \liminf_{n \rightarrow \infty} a_n.

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

\displaystyle  \mathrm{C}\!-\!\lim_{n \rightarrow \infty} a_n := \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N a_n;

if this limit exists, we say that the sequence is Césaro convergent. If the sequence {(a_n)_{n=1}^\infty} 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 {a_n= (-1)^n} discussed above is Césaro convergent, with a Césaro limit of {0}. However, there are still bounded sequences that do not have Césaro limit, such as {a_n := \sin( \log n )} (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 {\mathop{\widetilde \lim}_{n \rightarrow \infty} a_n}, defined to be bounded linear functionals from the space {\ell^\infty({\bf N})} of bounded real sequences to the real numbers {{\bf R}} that extend the classical limit functional (defined on the space {c_0({\bf N}) + {\bf R}} 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 {(a_n)_{n=1}^\infty} and any number {L} lying in the closed interval {[\liminf_{n \rightarrow \infty} a_n, \limsup_{n \rightarrow \infty} a_n]}, there exists at least one generalised limit functional {\mathop{\widetilde \lim}_{n \rightarrow \infty}} that takes the value {L} when applied to {a_n}; for instance, for any number {\theta} in {[-1,1]}, there exists a generalised limit functional that assigns that number {\theta} as the “limit” of the sequence {a_n = (-1)^n}. This claim can be seen by first designing such a limit functional on the vector space spanned by the convergent sequences and by {(a_n)_{n=1}^\infty}, 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 {(a_n)_{n=1}^\infty} to classically converge to a limit {L}, namely that all generalised limits of this sequence must equal {L}.

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 {a_n = (-1)^n} 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 {\mathop{\widetilde \lim}} is any generalised limit, then the Césaro-type functional {(a_n)_{n=1}^\infty \mapsto \mathop{\widetilde \lim}_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N a_n} 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 {p\!-\!\lim_{n \rightarrow \infty}} that obeys (2), then for any subset {A} of the natural numbers {{\bf N}}, the generalised limit {p\!-\!\lim_{n \rightarrow \infty} 1_A(n)} must equal its own square (since {1_A(n)^2 = 1_A(n)}) and is thus either {0} or {1}. If one defines {p \subset 2^{2^{\bf N}}} to be the collection of all subsets {A} of {{\bf N}} for which {p\!-\!\lim_{n \rightarrow \infty} 1_A(n) = 1}, one can verify that {p} obeys the axioms of a non-principal ultrafilter. Conversely, if {p} is a non-principal ultrafilter, one can define the associated generalised limit {p\!-\!\lim_{n \rightarrow \infty} a_n} of any bounded sequence {(a_n)_{n=1}^\infty} to be the unique real number {L} such that the sets {\{ n \in {\bf N}: |a_n - L| \leq \varepsilon \}} lie in {p} for all {\varepsilon>0}; 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 {a_n = (-1)^n} 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 {L} if and only if all ultrafilter limits converge to {L}.

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 {r_4(N)}“, 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 {N}, define {r_4(N)} to be the largest cardinality of a subset {A} of {[N] = \{1,\dots,N\}} which does not contain any non-trivial arithmetic progressions {a, a+r, a+2r, a+3r} of length four (where “non-trivial” means that {r} is non-zero). Trivially we have {r_4(N) \leq N}. In 1969, Szemerédi showed that {r_4(N) = o(N)}. 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 {r_4(N) \ll N (\log \log N)^{-c}} for some absolute constant {c>0}. In the second paper in the above-mentioned series, we managed to improve this bound to {r_4(N) \ll N \exp( - c \sqrt{\log \log N})}. In this paper, we improve the bound further to {r_4(N) \ll N (\log N)^{-c}}, which seems to be the limit of the methods. (We remark that if we could take {c} 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 {r_3(N)} one can take any {c} less than one.)

Most of the previous work on bounding {r_4(N)} 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 {A} of {[N]} fails to contain arithmetic progressions of length four, one seeks to then locate a long subprogression of {[N]} in which {A} has increased density. This was the basic method for instance underlying our previous bound {r_4(N) \ll N \exp( - c \sqrt{\log \log N})}, as well as a finite field analogue of the bound {r_4(N) \ll N (\log N)^{-c}}; 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 {A \subset [N]} has density {\delta}. Then one would expect a “randomly” selected arithmetic progression {{\bf a}, {\bf a}+{\bf r}, {\bf a}+2{\bf r}, {\bf a}+3{\bf r}} in {[N]} (using the convention that random variables will be in boldface) to be contained in {A} with probability about {\delta^4}. This is not true in general, however it was shown by Ben and myself that for any {\eta>0}, there was a set of shifts {r \in [-N,N]} of cardinality {\gg_{\delta,\eta} N}, such that for any such {r} one had

\displaystyle {\bf P}( {\bf a}, {\bf a}+r, {\bf a}+2r, {\bf a}+3r \in A ) \geq \delta^4 - \eta

if {{\bf a}} was chosen uniformly at random from {[N]}. This easily implies that {r_4(N) = o(N)}, but does not give a particularly good bound on the decay rate, because the implied constant in the cardinality lower bound {\gg_{\delta,\eta} N} is quite poor (in fact of tower-exponential type, due to the use of regularity lemmas!), and so one has to take {N} to be extremely large compared to {\delta,\eta} to avoid the possibility that the set of shifts in the above theorem consists only of the trivial shift {r=0}.

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 {{\bf a}} and {{\bf r}} of the length four progression. Namely, with {A}, {\delta}, {\eta} as above, we are able to show that there exist random variables {{\bf a}, {\bf r}}, not necessarily independent, such that

\displaystyle {\bf P}( {\bf a}, {\bf a}+{\bf r}, {\bf a}+2{\bf r}, {\bf a}+3{\bf r} \in A ) \geq \delta^4 - \eta \ \ \ \ \ (1)

and such that we have the non-degeneracy bound

\displaystyle {\bf P}( {\bf r} = 0 ) \ll \exp( - \eta^{-O(1)} ) / N.

This then easily implies the main theorem.

The energy increment method is then deployed to locate a good pair {({\bf a}, {\bf r})} 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 {1_A} “behaves like” a globally quadratic function such as {F( \alpha n^2 )}, for some irrational {\alpha} and some smooth periodic function {F: {\bf R}/{\bf Z} \rightarrow {\bf R}} of mean {\delta}. If one then takes {{\bf a}, {\bf r}} to be uniformly distributed in {[N]} and {[-\varepsilon N, \varepsilon N]} respectively for some small {\varepsilon>0}, with no coupling between the two variables, then the left-hand side of (1) is approximately of the form

\displaystyle \int_{(x,y,z,w) \in ({\bf R}/{\bf Z})^4: x-3y+3z-w = 0} F(x) F(y) F(z) F(w) \ \ \ \ \ (2)

where the integral is with respect to the probability Haar measure, and the constraint {x-3y+3z-w=0} ultimately arises from the algebraic constraint

\displaystyle \alpha {\bf a}^2 - 3 \alpha ({\bf a}+{\bf r})^2 + 3 \alpha ({\bf a}+2{\bf r})^2 - \alpha ({\bf a}+3{\bf r})^2 = 0.

However, an application of the Cauchy-Schwarz inequality and Fubini’s theorem shows that the integral in (2) is at least {(\int_{{\bf R}/{\bf Z}} F)^4}, 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 {[N]} is partitioned into some number of structured pieces {B_c} (think of these as arithmetic progressions, or as “Bohr sets), and on each piece {B_c}, {1_A} behaves like a locally quadratic function such as {F_c( \alpha_c n^2 )}, where {\alpha_c} now varies with {c}, and the mean of {F_c} will be approximately {\delta} on the average after averaging in {c} (weighted by the size of the pieces {B_c}). Now one should select {{\bf a}} and {{\bf r}} in the following coupled manner: first one chooses {{\bf a}} uniformly from {[N]}, then one defines {{\bf c}} to be the label {c} such that {{\bf a} \in B_c}, and then selects {{\bf r}} uniformly from a set {B_{c,\varepsilon}} which is related to {B_c} in much the same way that {[-\varepsilon N, \varepsilon N]} is related to {[N]}. If one does this correctly, the analogue of (2) becomes

\displaystyle {\bf E} \int_{(x,y,z,w) \in ({\bf R}/{\bf Z})^4: x-3y+3z-w = 0} F_{\mathbf c}(x) F_{\mathbf c}(y) F_{\mathbf c}(z) F_{\mathbf c}(w),

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 {1_A} which involves a decomposition of {[N]} into structured pieces {B_c}, and a quadratic approximation to {1_A} on each piece. If this approximation is accurate enough (or more precisely, if a certain (averaged) local Gowers uniformity norm {U^3} of the error is small enough) to model the count in (1) (for random variables {{\bf a}, {\bf r}} determined by the above partition of {[N]} into pieces {B_c}), and if the frequencies (such as {\alpha_c}) 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 {U^3}. 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 {1_A} in a manner that significantly increases its “energy” (basically an {L^2} norm). Such energy increments cannot continue indefinitely, and when they terminate we obtain the desired claim.

There are existing inverse theorems for {U^3} 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 “{1\%}-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 “{99\%}-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 {99\%}-structured homomorphism on pseudorandom subsets of Bohr sets to a {100\%}-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 {1}-form on {{\bf R}^n} that varies linearly with the coordinates can be integrated to be the derivative of a quadratic function if we know that the {1}-form is closed). These portions of the paper are unfortunately rather technical, but broadly follow the methods already used in previous literature.

A sequence {a: {\bf Z} \rightarrow {\bf C}} of complex numbers is said to be quasiperiodic if it is of the form

\displaystyle a(n) = F( \alpha_1 n \hbox{ mod } 1, \dots, \alpha_k n \hbox{ mod } 1)

for some real numbers {\alpha_1,\dots,\alpha_k} and continuous function {F: ({\bf R}/{\bf Z})^k \rightarrow {\bf C}}. For instance, linear phases such as {n \mapsto e(\alpha n + \beta)} (where {e(\theta) := e^{2\pi i \theta}}) are examples of quasiperiodic sequences; the top order coefficient {\alpha} (modulo {1}) can be viewed as a “frequency” of the integers, and an element of the Pontryagin dual {\hat {\bf Z} \equiv {\bf R}/{\bf Z}} of the integers. Any periodic sequence is also quasiperiodic (taking {k=1} and {\alpha_1} 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

\displaystyle a(n) = \sum_{j=1}^\infty c_j e(\alpha_j n)

with {\alpha_1,\alpha_2,\dots} real numbers and {c_1,c_2,\dots} 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 {(X, \mu, T)} is a measure-preserving system – a probability space {(X,\mu)} equipped with a measure-preserving shift, and {f_1,f_2 \in L^\infty(X)} are bounded measurable functions, then the correlation sequence

\displaystyle a(n) := \int_X f_1(x) f_2(T^n x)\ d\mu(x)

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

\displaystyle \sup_N \frac{1}{M} \sum_{n=N+1}^{N+M} |b_n| \rightarrow 0 \hbox{ as } M \rightarrow \infty.

This can be established in a number of ways, for instance by writing {a(n)} as the Fourier coefficients of the spectral measure of the shift {T} with respect to the functions {f_1,f_2}, 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 {\alpha n + \beta} are replaced with higher degree counterparts. The most obvious candidates for these counterparts would be the polynomials {\alpha_d n^d + \dots + \alpha_0}, 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 {{\bf Z}}). 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 {{\bf C}^m} rather than in {{\bf C}}. 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 {{\bf C}^m}, and for sequences taking values in different vector spaces {{\bf C}^m}, {{\bf C}^{m'}}, we may utilise the tensor product {\otimes: {\bf C}^m \times {\bf C}^{m'} \rightarrow {\bf C}^{mm'}}, which we will normalise by defining

\displaystyle (z_1,\dots,z_m) \otimes (w_1,\dots,w_{m'}) = (z_1 w_1, \dots, z_1 w_{m'}, \dots, z_m w_1, \dots, z_m w_{m'} ).

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

\displaystyle \| (z_1,\dots,z_m) \| := \sqrt{|z_1|^2 + \dots + |z_m|^2}

since we have {\|z \otimes w\| = \|z\| \|w\|}.

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) A nilmanifold of step at most {d} is a quotient {G/\Gamma}, where {G} is a connected, simply connected nilpotent Lie group of step at most {d} (thus, all {d+1}-fold commutators vanish) and {\Gamma} is a discrete cocompact lattice in {G}. A basic nilsequence of degree at most {d} is a sequence of the form {n \mapsto F(g^n g_0 \Gamma)}, where {g_0 \Gamma \in G/\Gamma}, {g \in G}, and {F: G/\Gamma \rightarrow {\bf C}^m} 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 {1} if and only if it is quasiperiodic. The requirement that {G} 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 {\{ N+1, \dots, N+M\}}), it is common to impose additional regularity conditions on the function {F}, 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 2 Let {d \geq 0}. A filtered group of degree at most {d} is a group {G} together with a sequence {G_\bullet = (G_0,G_1,G_2,\dots)} of subgroups {G \geq G_0 \geq G_1 \geq \dots} with {G_{d+1}=\{\hbox{id}\}} and {[G_i,G_j] \subset G_{i+j}} for {i,j \geq 0}. A polynomial sequence {g: {\bf Z} \rightarrow G} into a filtered group is a function such that {\partial_{h_i} \dots \partial_{h_1} g(n) \in G_i} for all {i \geq 0} and {n,h_1,\dots,h_i \in{\bf Z}}, where {\partial_h g(n) := g(n+h) g(n)^{-1}} is the difference operator. A filtered nilmanifold of degree at most {s} is a quotient {G/\Gamma}, where {G} is a filtered group of degree at most {s} such that {G} and all of the subgroups {G_i} are connected, simply connected nilpotent filtered Lie group, and {\Gamma} is a discrete cocompact subgroup of {G} such that {\Gamma_i := \Gamma \cap G_i} is a discrete cocompact subgroup of {G_i}. A basic nilsequence of degree at most {d} is a sequence of the form {n \mapsto F(g(n))}, where {g: {\bf Z} \rightarrow G} is a polynomial sequence, {G/\Gamma} is a filtered nilmanifold of degree at most {d}, and {F: G \rightarrow {\bf C}^m} is a continuous function which is {\Gamma}-automorphic, in the sense that {F(g \gamma) = F(g)} for all {g \in G} and {\gamma \in \Gamma}.

One can easily identify a {\Gamma}-automorphic function on {G} with a function on {G/\Gamma}, but there are some (very minor) advantages to working on the group {G} instead of the quotient {G/\Gamma}, as it becomes slightly easier to modify the automorphy group {\Gamma} when needed. (But because the action of {\Gamma} on {G} is free, one can pass from {\Gamma}-automorphic functions on {G} to functions on {G/\Gamma} with very little difficulty.) The main reason to work with polynomial sequences {n \mapsto g(n)} rather than geometric progressions {n \mapsto g^n g_0 \Gamma} 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 {d} in the sense of the first definition, is also a basic nilsequence of degree at most {d} in the second definition, since a nilmanifold of degree at most {d} can be filtered using the lower central series, and any linear sequence {n \mapsto g^n g_0} 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

\displaystyle a(n) = F( P_1(n), \dots, P_k(n) ),

where {P_1,\dots,P_k: {\bf Z} \rightarrow {\bf R}} are polynomials of degree at most {d}, and {F: {\bf R}^k \rightarrow {\bf C}^m} is a {{\bf Z}^k}-automorphic (i.e., {{\bf Z}^k}-periodic) continuous function. The map {P: {\bf Z} \rightarrow {\bf R}^k} defined by {P(n) := (P_1(n),\dots,P_k(n))} is a polynomial map of degree at most {d}, if one filters {{\bf R}^k} by defining {({\bf R}^k)_i} to equal {{\bf R}^k} when {i \leq d}, and {\{0\}} for {i > d}. The torus {{\bf R}^k/{\bf Z}^k} then becomes a filtered nilmanifold of degree at most {d}, and {a(n)} is thus a basic nilsequence of degree at most {d} as per the second definition. It is also possible explicitly describe {a_n} as a basic nilsequence of degree at most {d} as per the first definition, for instance (in the {k=1} case) by taking {G} to be the space of upper triangular unipotent {d+1 \times d+1} real matrices, and {\Gamma} the subgroup with integer coefficients; we leave the details to the interested reader.

The other key example is a sequence of the form

\displaystyle a(n) = F( \alpha n, \{ \alpha n \} \beta n )

where {\alpha,\beta} are real numbers, {\{ \alpha n \} = \alpha n - \lfloor \alpha n \rfloor} denotes the fractional part of {\alpha n}, and and {F: {\bf R}^2 \rightarrow {\bf C}^m} is a {{\bf Z}^2}-automorphic continuous function that vanishes in a neighbourhood of {{\bf Z} \times {\bf R}}. To describe this as a nilsequence, we use the nilpotent connected, simply connected degree {2}, Heisenberg group

\displaystyle G := \begin{pmatrix} 1 & {\bf R} & {\bf R} \\ 0 & 1 & {\bf R} \\ 0 & 0 & 1 \end{pmatrix}

with the lower central series filtration {G_0=G_1=G}, {G_2= [G,G] = \begin{pmatrix} 1 &0 & {\bf R} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}}, and {G_i = \{ \mathrm{id} \}} for {i > 2}, {\Gamma} to be the discrete compact subgroup

\displaystyle \Gamma := \begin{pmatrix} 1 & {\bf Z} & {\bf Z} \\ 0 & 1 & {\bf Z} \\ 0 & 0 & 1 \end{pmatrix},

{g: {\bf Z} \rightarrow G} to be the polynomial sequence

\displaystyle g(n) := \begin{pmatrix} 1 & \beta n & \alpha \beta n^2 \\ 0 & 1 & \alpha n \\ 0 & 0 & 1 \end{pmatrix}

and {\tilde F: G \rightarrow {\bf C}^m} to be the {\Gamma}-automorphic function

\displaystyle \tilde F( \begin{pmatrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{pmatrix} ) = F( \{ z \}, y - \lfloor z \rfloor x );

one easily verifies that this function is indeed {\Gamma}-automorphic, and it is continuous thanks to the vanishing properties of {F}. Also we have {a(n) = \tilde F(g(n))}, so {a} is a basic nilsequence of degree at most {2}. One can concoct similar examples with {\{ \alpha n \} \beta n} replaced by other “bracket polynomials” of {n}; for instance

\displaystyle a(n) = F( \alpha n, \{ \alpha n - \frac{1}{2} \} \beta n )

will be a basic nilsequence if {F} now vanishes in a neighbourhood of {(\frac{1}{2}+{\bf Z}) \times {\bf R}} rather than {{\bf Z} \times {\bf R}}. 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 {d} is defined to be a sequence that is the uniform limit of basic nilsequences of degree at most {d}. Thus for instance a sequence is a nilsequence of degree at most {1} if and only if it is almost periodic, while a sequence is a nilsequence of degree at most {0} if and only if it is constant. Such objects arise in higher order recurrence: for instance, if {h_0,\dots,h_d} are integers, {(X,\mu,T)} is a measure-preserving system, and {f_0,\dots,f_d \in L^\infty(X)}, then it was shown by Leibman that the sequence

\displaystyle n \mapsto \int_X f_0(T^{h_0 n} x) \dots f_d(T^{h_d n} x)\ d\mu(x)

is equal to a nilsequence of degree at most {d}, plus a null sequence. (The special case when the measure-preserving system was ergodic and {h_i = i} for {i=0,\dots,d} 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 {a: {\bf Z} \rightarrow {\bf C}^m} is a basic nilsequence of degree at most {d} if and only if each of its {m} components are. The scalar basic nilsequences {a: {\bf Z} \rightarrow {\bf C}} of degree {d} 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 {a: {\bf Z} \rightarrow {\bf C}^m} of degree at most {d} form a complex vector space closed under complex conjugation for each {m}, and that the tensor product of any two basic nilsequences of degree at most {d} is another basic nilsequence of degree at most {d}. 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 {d \geq 1}. A sub-nilcharacter of degree {d} is a basic nilsequence {\chi: n \mapsto F(g(n))} of degree at most {d}, such that {F} obeys the additional modulation property

\displaystyle F( g_d g ) = e( \xi \cdot g_d ) F(g) \ \ \ \ \ (1)

for all {g \in G} and {g_d \in G_d}, where {\xi: G_d \rightarrow {\bf R}} is a continuous homomorphism {g_d \mapsto \xi \cdot g_d}. (Note from (1) and {\Gamma}-automorphy that unless {F} vanishes identically, {\xi} must map {\Gamma_d} to {{\bf Z}}, thus without loss of generality one can view {\xi} as an element of the Pontryagial dual of the torus {G_d / \Gamma_d}.) If in addition one has {\|F(g)\|=1} for all {g \in G}, we call {\chi} a nilcharacter of degree {d \geq 1}.

In the degree one case {d=1}, the only sub-nilcharacters are of the form {\chi(n) = e(\alpha n)} for some vector {c \in {\bf C}^m} and {\alpha \in {\bf R}}, and this is a nilcharacter if {c} is a unit vector. Similarly, in higher degree, any sequence of the form {\chi(n) = c e(P(n))}, where {c \in {\bf C}^m} is a vector and {P: {\bf Z} \rightarrow {\bf R}} is a polynomial of degree at most {d}, is a sub-nilcharacter of degree {d}, and a character if {c} is a unit vector. A nilsequence of degree at most {d-1} is automatically a sub-nilcharacter of degree {d}, and a nilcharacter if it is of magnitude {1}. A further example of a nilcharacter is provided by the two-dimensional sequence {\chi: {\bf Z} \rightarrow {\bf C}^2} defined by

\displaystyle \chi(n) := ( F_0( \alpha n ) e( \{ \alpha n \} \beta n ), F_{1/2}( \alpha n ) e( \{ \alpha n - \frac{1}{2} \} \beta n ) ) \ \ \ \ \ (2)

where {F_0, F_{1/2}: {\bf R} \rightarrow {\bf C}} are continuous, {{\bf Z}}-automorphic functions that vanish on a neighbourhood of {{\bf Z}} and {\frac{1}{2}+{\bf Z}} respectively, and which form a partition of unity in the sense that

\displaystyle |F_0(x)|^2 + |F_{1/2}(x)|^2 = 1

for all {x \in {\bf R}}. Note that one needs both {F_0} and {F_{1/2}} 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 {d} sub-nilcharacter {f: {\bf Z} \rightarrow {\bf C}^m} can be expressed in the form {f = c \chi}, where {\chi: {\bf Z} \rightarrow {\bf C}^{m'}} is a degree {d} nilcharacter, and {c: {\bf C}^{m'} \rightarrow {\bf C}^m} is a linear transformation. Indeed, by scaling we may assume {f(n) = F(g(n))} where {|F| < 1} uniformly. Using partitions of unity, one can find further functions {F_1,\dots,F_m} also obeying (1) for the same character {\xi} such that {|F_1|^2 + \dots + |F_m|^2} is non-zero; by dividing out the {F_1,\dots,F_m} by the square root of this quantity, and then multiplying by {\sqrt{1-|F|^2}}, we may assume that

\displaystyle |F|^2 + |F_1|^2 + \dots + |F_m|^2 = 1,

and then

\displaystyle \chi(n) := (F(g(n)), F_1(g(n)), \dots, F_m(g(n)))

becomes a degree {d} nilcharacter that contains {f(n)} 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 {d} nilcharacters forms a semigroup under tensor product, with the constant sequence {1} as the identity. One can upgrade this semigroup to an abelian group by quotienting nilcharacters out by equivalence:

Definition 4 Let {d \geq 1}. We say that two degree {d} nilcharacters {\chi: {\bf Z} \rightarrow {\bf C}^m}, {\chi': {\bf Z} \rightarrow {\bf C}^{m'}} are equivalent if {\chi \otimes \overline{\chi'}: {\bf Z} \rightarrow {\bf C}^{mm'}} is equal (as a sequence) to a basic nilsequence of degree at most {d-1}. (We will later show that this is indeed an equivalence relation.) The equivalence class {[\chi]_{\mathrm{Symb}^d({\bf Z})}} of such a nilcharacter will be called the symbol of that nilcharacter (in analogy to the symbol of a differential or pseudodifferential operator), and the collection of such symbols will be denoted {\mathrm{Symb}^d({\bf Z})}.

As we shall see below the fold, {\mathrm{Symb}^d({\bf Z})} 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 {d=1}, the group is isomorphic to the Ponytragin dual {\hat {\bf Z} = {\bf R}/{\bf Z}} of the integers, and {\mathrm{Symb}^d({\bf Z})} for {d > 1} should be viewed as higher order generalisations of this Pontryagin dual. In principle, this group can be explicitly described for all {d}, but the theory rapidly gets complicated as {d} increases (much as the classification of nilpotent Lie groups or Lie algebras of step {d} rapidly gets complicated even for medium-sized {d} such as {d=3} or {d=4}). We will give an explicit description of the {d=2} case here. There is however one nice (and non-trivial) feature of {\mathrm{Symb}^d({\bf Z})} for {d \geq 2} – it is not just an abelian group, but is in fact a vector space over the rationals {{\bf Q}}!

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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 {a,b,c,d} can be turned into a group {\{a,b,c,d\}} by designating one of the four objects, say {a}, 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 {C_4} and the Klein four-group {C_2 \times C_2}.

More generally, given a class of objects {X} and some equivalence relation {\sim} on {X} (which one should interpret as describing the property of two objects in {X} being “isomorphic”), one can consider the number {|X / \sim|} of objects in {X} “up to isomorphism”, which is simply the cardinality of the collection {X/\sim} of equivalence classes {[x]:=\{y\in X:x \sim {}y \}} of {X}. In the case where {X} is finite, one can express this cardinality by the formula

\displaystyle |X/\sim| = \sum_{x \in X} \frac{1}{|[x]|}, \ \ \ \ \ (1)

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

Example 1 Let {X} be the five-element set {\{-2,-1,0,1,2\}} of integers between {-2} and {2}. Let us say that two elements {x, y} of {X} are isomorphic if they have the same magnitude: {x \sim y \iff |x| = |y|}. Then the quotient space {X/\sim} consists of just three equivalence classes: {\{-2,2\} = [2] = [-2]}, {\{-1,1\} = [-1] = [1]}, and {\{0\} = [0]}. Thus there are three objects in {X} up to isomorphism, and the identity (1) is then just

\displaystyle 3 = \frac{1}{2} + \frac{1}{2} + 1 + \frac{1}{2} + \frac{1}{2}.

Thus, to count elements in {X} up to equivalence, the elements {-2,-1,1,2} are given a weight of {1/2} because they are each isomorphic to two elements in {X}, while the element {0} is given a weight of {1} because it is isomorphic to just one element in {X} (namely, itself).

Given a finite probability set {X}, there is also a natural probability distribution on {X}, namely the uniform distribution, according to which a random variable {\mathbf{x} \in X} is set equal to any given element {x} of {X} with probability {\frac{1}{|X|}}:

\displaystyle {\bf P}( \mathbf{x} = x ) = \frac{1}{|X|}.

Given a notion {\sim} of isomorphism on {X}, one can then define the random equivalence class {[\mathbf{x}] \in X/\sim} that the random element {\mathbf{x}} belongs to. But if the isomorphism classes are unequal in size, we now encounter a biasing effect: even if {\mathbf{x}} was drawn uniformly from {X}, the equivalence class {[\mathbf{x}]} need not be uniformly distributed in {X/\sim}. For instance, in the above example, if {\mathbf{x}} was drawn uniformly from {\{-2,-1,0,1,2\}}, then the equivalence class {[\mathbf{x}]} will not be uniformly distributed in the three-element space {X/\sim}, because it has a {2/5} probability of being equal to the class {\{-2,2\}} or to the class {\{-1,1\}}, and only a {1/5} probability of being equal to the class {\{0\}}.

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 {X} with an equivalence relation {\sim} to capture the notion of isomorphism, we instead consider groupoids, which are sets {X} in which there are allowed to be multiple isomorphisms between elements in {X} (and in particular, there are allowed to be multiple automorphisms from an element to itself). More precisely:

Definition 2 A groupoid is a set (or proper class) {X}, together with a (possibly empty) collection {\mathrm{Iso}(x \rightarrow y)} of “isomorphisms” between any pair {x,y} of elements of {X}, and a composition map {f, g \mapsto g \circ f} from isomorphisms {f \in \mathrm{Iso}(x \rightarrow y)}, {g \in \mathrm{Iso}(y \rightarrow z)} to isomorphisms in {\mathrm{Iso}(x \rightarrow z)} for every {x,y,z \in X}, obeying the following group-like axioms:

  • (Identity) For every {x \in X}, there is an identity isomorphism {\mathrm{id}_x \in \mathrm{Iso}(x \rightarrow x)}, such that {f \circ \mathrm{id}_x = \mathrm{id}_y \circ f = f} for all {f \in \mathrm{Iso}(x \rightarrow y)} and {x,y \in X}.
  • (Associativity) If {f \in \mathrm{Iso}(x \rightarrow y)}, {g \in \mathrm{Iso}(y \rightarrow z)}, and {h \in \mathrm{Iso}(z \rightarrow w)} for some {x,y,z,w \in X}, then {h \circ (g \circ f) = (h \circ g) \circ f}.
  • (Inverse) If {f \in \mathrm{Iso}(x \rightarrow y)} for some {x,y \in X}, then there exists an inverse isomorphism {f^{-1} \in \mathrm{Iso}(y \rightarrow x)} such that {f \circ f^{-1} = \mathrm{id}_y} and {f^{-1} \circ f = \mathrm{id}_x}.

We say that two elements {x,y} of a groupoid are isomorphic, and write {x \sim y}, if there is at least one isomorphism from {x} to {y}.

Example 3 Any category gives a groupoid by taking {X} to be the set (or class) of objects, and {\mathrm{Iso}(x \rightarrow y)} to be the collection of invertible morphisms from {x} to {y}. For instance, in the category {\mathbf{Set}} of sets, {\mathrm{Iso}(x \rightarrow y)} would be the collection of bijections from {x} to {y}; in the category {\mathbf{Vec}/k} of linear vector spaces over some given base field {k}, {\mathrm{Iso}(x \rightarrow y)} would be the collection of invertible linear transformations from {x} to {y}; and so forth.

Every set {X} equipped with an equivalence relation {\sim} can be turned into a groupoid by assigning precisely one isomorphism {\iota_{x \rightarrow y}} from {x} to {y} for any pair {x,y \in X} with {x \sim y}, and no isomorphisms from {x} to {y} when {x \not \sim y}, 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 {X = \{-2,-1,0,1,2\}} as above, if we turn {X} into a simply connected groupoid, there will be precisely one isomorphism from {2} to {-2}, and also precisely one isomorphism from {2} to {2}, but no isomorphisms from {2} to {-1}, {0}, or {1}.

However, one can also form multiply-connected groupoids in which there can be multiple isomorphisms from one element of {X} to another. For instance, one can view {X = \{-2,-1,0,1,2\}} as a space that is acted on by multiplication by the two-element group {\{-1,+1\}}. This gives rise to two types of isomorphisms, an identity isomorphism {(+1)_x} from {x} to {x} for each {x \in X}, and a negation isomorphism {(-1)_x} from {x} to {-x} for each {x \in X}; in particular, there are two automorphisms of {0} (i.e., isomorphisms from {0} to itself), namely {(+1)_0} and {(-1)_0}, whereas the other four elements of {X} 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 {\{-1,+1\}}); for instance, we have {(-1)_{-2} \circ (-1)_2 = (+1)_2}.

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

\displaystyle \sum_{x \in X} \frac{1}{|\{ f: f \in \mathrm{Iso}(x \rightarrow y) \hbox{ for some } y\}|}

of (1). That is to say, in the multiply connected case, the denominator is no longer the number of objects {y} isomorphic to {x}, but rather the number of isomorphisms from {x} to other objects {y}. Grouping together all summands coming from a single equivalence class {[x]} in {X/\sim}, we can also write this expression as

\displaystyle \sum_{[x] \in X/\sim} \frac{1}{|\mathrm{Aut}(x)|} \ \ \ \ \ (2)

where {\mathrm{Aut}(x) := \mathrm{Iso}(x \rightarrow x)} is the automorphism group of {x}, that is to say the group of isomorphisms from {x} to itself. (Note that if {x,x'} belong to the same equivalence class {[x]}, then the two groups {\mathrm{Aut}(x)} and {\mathrm{Aut}(x')} will be isomorphic and thus have the same cardinality, and so the above expression is well-defined.

For instance, if we take {X} to be the simply connected groupoid on {\{-2,-1,0,1,2\}}, then the number of elements of {X} up to isomorphism is

\displaystyle \frac{1}{2} + \frac{1}{2} + 1 + \frac{1}{2} + \frac{1}{2} = 1 + 1 + 1 = 3

exactly as before. If however we take the multiply connected groupoid on {\{-2,-1,0,1,2\}}, in which {0} has two automorphisms, the number of elements of {X} up to isomorphism is now the smaller quantity

\displaystyle \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = 1 + \frac{1}{2} + 1 = \frac{5}{2};

the equivalence class {[0]} is now counted with weight {1/2} rather than {1} due to the two automorphisms on {0}. Geometrically, one can think of this groupoid as being formed by taking the five-element set {\{-2,-1,0,1,2\}}, and “folding it in half” around the fixed point {0}, giving rise to two “full” quotient points {[1], [2]} and one “half” point {[0]}. More generally, given a finite group {G} acting on a finite set {X}, and forming the associated multiply connected groupoid, the cardinality up to isomorphism of this groupoid will be {|X|/|G|}, since each element {x} of {X} will have {|G|} isomorphisms on it (whether they be to the same element {x}, or to other elements of {X}).

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 {\mathbf{FinSet}} of finite sets, with isomorphisms given by bijections as in Example 3. Every finite set is isomorphic to {\{1,\dots,n\}} for some natural number {n}, so the equivalence classes of {\mathbf{FinSet}} may be indexed by the natural numbers. The automorphism group {S_n} of {\{1,\dots,n\}} has order {n!}, so the cardinality of {\mathbf{FinSet}} up to isomorphism is

\displaystyle \sum_{n=0}^\infty \frac{1}{n!} = e.

(This fact is sometimes loosely stated as “the number of finite sets is {e}“, 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

\displaystyle \frac{1}{2} + \frac{1}{6} = \frac{2}{3}

because the cyclic group {C_4} has two automorphisms, whereas the Klein four-group {C_2 \times C_2} has six.

In the case that the cardinality of a groupoid {X} up to isomorphism is finite and non-empty, one can now define the notion of a random isomorphism class {[\mathbf{x}]} in {X/\sim} drawn “uniformly up to isomorphism”, by requiring the probability of attaining any given isomorphism class {[x]} to be

\displaystyle {\mathbf P}([\mathbf{x}] = [x]) = \frac{1 / |\mathrm{Aut}(x)|}{\sum_{[y] \in X/\sim} 1/|\mathrm{Aut}(y)|},

thus the probability of being isomorphic to a given element {x} will be inversely proportional to the number of automorphisms that {x} has. For instance, if we take {X} to be the set {\{-2,-1,0,1,2\}} with the simply connected groupoid, {[\mathbf{x}]} will be drawn uniformly from the three available equivalence classes {[0], [1], [2]}, with a {1/3} probability of attaining each; but if instead one uses the multiply connected groupoid coming from the action of {\{-1,+1\}}, and draws {[\mathbf{x}]} uniformly up to isomorphism, then {[1]} and {[2]} will now be selected with probability {2/5} each, and {[0]} will be selected with probability {1/5}. Thus this distribution has accounted for the bias mentioned previously: if a finite group {G} acts on a finite space {X}, and {\mathbf{x}} is drawn uniformly from {X}, then {[\mathbf{x}]} now still be drawn uniformly up to isomorphism from {X/G}, if we use the multiply connected groupoid coming from the {G} action, rather than the simply connected groupoid coming from just the {G}-orbit structure on {X}.

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 {1}, that is to say it will be of cardinality {n} with probability {\frac{e^{-1}}{n!}}.

One important source of groupoids are the fundamental groupoids {\pi_1(M)} of a manifold {M} (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 {M}, and the isomorphisms from {x} to {y} are the equivalence classes of paths from {x} to {y} up to homotopy; in particular, the automorphism group of any given point is just the fundamental group of {M} at that base point. The equivalence class {[x]} of a point in {M} is then the connected component of {x} in {M}. The cardinality up to isomorphism of the fundamental groupoid is then

\displaystyle \sum_{M' \in \pi_0(M)} \frac{1}{|\pi_1(M')|}

where {\pi_0(M)} is the collection of connected components {M'} of {M}, and {|\pi_1(M')|} is the order of the fundamental group of {M'}. Thus, simply connected components of {M} 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 {n}-fold covering map {\pi: X \rightarrow Y} of one finite groupoid {Y} by another {X}. This means that {\pi} is a functor that is surjective, with all preimages of cardinality {n}, with the property that given any pair {y,y'} in the base space {Y} and any {x} in the preimage {\pi^{-1}(\{y\})} of {y}, every isomorphism {f \in \mathrm{Iso}(y \rightarrow y')} has a unique lift {\tilde f \in \mathrm{Iso}(x \rightarrow x')} from the given initial point {x} (and some {x'} in the preimage of {y'}). Then one can check that the cardinality up to isomorphism of {X} is {n} times the cardinality up to isomorphism of {Y}, which fits well with the geometric picture of {X} as the {n}-fold cover of {Y}. (For instance, if one covers a manifold {M} with finite fundamental group by its universal cover, this is a {|\pi_1(M)|}-fold cover, the base has cardinality {1/|\pi_1(M)|} up to isomorphism, and the universal cover has cardinality one up to isomorphism.) Related to this, if one draws an equivalence class {[\mathrm{x}]} of {X} uniformly up to isomorphism, then {\pi([\mathrm{x}])} will be an equivalence class of {Y} 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 {[G]} of a profinite abelian group {G} 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 {n} elements will have a cardinality that converges in distribution to the Poisson distribution of rate {1} (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.

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