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For the past few months, Cambridge University Press (in consultation with a number of mathematicians, including Tim Gowers and myself) has been preparing to launch a new open access journal (or more precisely, a complex of journals – see below) in mathematics, under the title “Forum of Mathematics“, as an experiment in moving away from the traditional library subscription based model of mathematical academic publishing.   (The initial planning for this journal happened to precede the Cost of Knowledge boycott, but the philosophy behind the journal is certainly aligned with that of the boycott, which I believe is further evidence that the time has come for mathematical journal reform.)  The journal will formally begin accepting submissions on October 1st, but it has already been officially announced by Cambridge University Press, with an editorial board (with Rob Kirby as managing editor, thirteen other editors including Tim and myself, and a board of associate editors that is still in the process of being assembled) and FAQ already in place.

In many respects, Forum of Mathematics functions as a regular mathematics journal, in that papers are submitted by the authors, sent out to referees by the editors, and (if accepted) published by the Forum.  There are however a couple of important features that distinguish the Forum from traditional mathematics journals.  The first is the open access, publication-charge based publishing model (sometimes known as “gold open access”).    Namely, all articles will be freely available without subscription charges, but authors, upon their paper being accepted, be asked to pay a publication charge to cover costs.  The publication charges will be set at zero for the first three years, and then raised to somewhere around £500 GBP or \$750 USD after the initial three-year period (with fee waivers available for authors from developing countries).  (One reason that the publication charges are not entirely fixed at this point is that there is the possibility of obtaining additional funding sources for this journal, for instance from philanthropic organisations, which may allow for fee reductions or additional waivers.)    This is of course a non-trivial sum of money, but it is significantly lower than the charges for most other gold open access journals.   Also, editorial decisions will not be influenced by the author’s ability to pay for the charges, which only come into effect in the event that the paper is accepted for publication.

(One way in which Cambridge University Press is keeping costs low, by the way, is to keep the journal purely electronic, with physical issues available on a print-on-demand basis only.  A side benefit of this choice is that there is no hard constraint on how many or how few pages will be published each year, so that acceptance decisions will not be influenced by artificial constraints such as the size of the journal backlog.)

Another distinctive feature of Forum of Mathematics lies in its scope and structure.  It is not exactly a single journal, but is instead a complex consisting of a generalist flagship journal (officially known as Forum of Mathematics, Pi) and a specialist journal (Forum of Mathematics, Sigma) which is in turn loosely organised for editorial purposes into “clusters” for each of the major subfields of mathematics (analysis, topology, algebra, discrete mathematics, etc.).   As a first approximation, Pi is intended as a top-tier journal (on the level of, say, the Journal of the American Mathematical Society, Inventiones, or Annals of Mathematics) that only accepts significant papers of interest to a wide audience of mathematicians, while Sigma resembles a collection of specialist journals, one for each major subfield of mathematics.  However, the journals will be using the same editorial interface, and so it will be possible to easily transfer a submission between journals or clusters (while retaining all the referee reports and other editorial data).  This is meant to help address a common issue in traditional mathematical journals, namely that if the editorial board decides that a submission falls too far outside the scope of the journal, or is not quite at the desired level of quality, then the authors have to start all over again with a new journal (and new referee reports).    Of course, it is still possible (and perhaps even fairly common) that a submission to Forum of Mathematics will be deemed unsuitable for either Pi or Sigma, and thus rejected entirely; but the structure of the Forum should give some additional flexibility, to reduce the frequency that papers are rejected for artificial reasons such as being out of scope.  (Of course, we would still expect authors to aim their submission at the most appropriate location to begin with, in order to reduce the time and effort expended on processing the paper by everyone involved.)

Further discussion of this journal can be found at Tim Gowers’ blog.   It should be fully operational in a few months (barring last-minute hitches, we should be open for submissions on 1 October 2012).    Of course, a single journal is not going to resolve all the extant concerns about the need for journal publishing reform, such as those raised in the Cost of Knowledge boycott; but I feel that it is important to have some experimentation with different publishing models, to see what alternatives to the status quo are possible.

Two quick updates with regards to polymath projects.  Firstly, given the poll on starting the mini-polymath4 project, I will start the project at Thu July 12 2012 UTC 22:00.  As usual, the main research thread on this project will be held at the polymath blog, with the discussion thread hosted separately on this blog.

Second, the Polymath7 project, which seeks to establish the “hot spots conjecture” for acute-angled triangles, has made a fair amount of progress so far; for instance, the first part of the conjecture (asserting that the second Neumann eigenfunction of an acute non-equilateral triangle is simple) is now solved, and the second part (asserting that the “hot spots” (i.e. extrema) of that second eigenfunction lie on the boundary of the triangle) has been solved in a number of special cases (such as the isosceles case).  It’s been quite an active discussion in the last week or so, with almost 200 comments across two threads (and a third thread freshly opened up just now).  While the problem is still not completely solved, I feel optimistic that it should fall within the next few weeks (if nothing else, it seems that the problem is now at least amenable to a brute force numerical attack, though personally I would prefer to see a more conceptual solution).

Two polymath related items for this post. Firstly, there is a new polymath proposal over at the polymath blog, proposing to attack the “hot spots conjecture” (concerning a maximum principle for a heat equation) in the case when the domain is an acute-angled triangle (the case of the right and obtuse-angled triangles already being solved). Please feel free to comment on the proposal blog post if you are interested in participating.

Secondly, it is once again time to set up the annual “mini-polymath” project to collaboratively solve one of this year’s International Mathematical Olympiad problems. This year, the Olympiad is being held in Argentina, with the problems given out on July 10-11. As usual, there will be a wiki page, discussion thread, and research thread for the project. As in previous years, the first thing to resolve is the starting date and time, so I am setting up a poll here to fix a time (and also to get a preliminary indication of interest in the project).  (I am using 24-hour Coordinated Universal Time (UTC) for these times.  Here is a link that converts the first time given in the poll (Thu Jul 12 2012 UTC 6:00) into other time zones.) Given that the last three mini-polymaths were reasonably successful, I am not planning any changes to the format, but of course if there are any suggestions for changes, I’d be happy to hear them in the comments.

High school algebra marks a key transition point in one’s early mathematical education, and is a common point at which students feel that mathematics becomes really difficult. One of the reasons for this is that the problem solving process for a high school algebra question is significantly more free-form than the mechanical algorithms one is taught for elementary arithmetic, and a certain amount of planning and strategy now comes into play. For instance, if one wants to, say, write ${\frac{1,572,342}{4,124}}$ as a mixed fraction, there is a clear (albeit lengthy) algorithm to do this: one simply sets up the long division problem, extracts the quotient and remainder, and organises these numbers into the desired mixed fraction. After a suitable amount of drill, this is a task that can be accomplished by a large fraction of students at the middle school level. But if, for instance, one has to solve a system of equations such as

$\displaystyle a^2 + bc = 7$

$\displaystyle 2b - c = 1$

$\displaystyle a + 2 = c$

for ${a,b,c}$, there is no similarly mechanical procedure that can be easily taught to a high school student in order to solve such a problem “mindlessly”. (I doubt, for instance, that any attempt to teach Buchberger’s algorithm to such students will be all that successful.) Instead, one is taught the basic “moves” (e.g. multiplying both sides of an equation by some factor, subtracting one equation from another, substituting an expression for a variable into another equation, and so forth), and some basic principles (e.g. simplifying an expression whenever possible, for instance by gathering terms, or solving for one variable in terms of others in order to eliminate it from the system). It is then up to the student to find a suitable combination of moves that isolates each of the variables in turn, to reveal their identities.

Once one is sufficiently expert in algebraic manipulation, this is not all that difficult to do, but when one is just starting to learn algebra, this type of problem can be quite daunting, in part because of an “embarrasment of riches”; there are several possible “moves” one could try to apply to the equations given, and to the novice it is not always clear in advance which moves will simplify the problem and which ones will make it more complicated. Often, such a student may simply try moves at random, which can lead to a dishearteningly large amount of effort expended without getting any closer to a solution. What is worse, each move introduces the possibility of an arithmetic error (such as a sign error), the effect of which is usually not apparent until the student finally arrives at his or her solution and either checks it against the original equation, or submits the answer to be graded. (My own son can get quite frustrated after performing a lengthy series of computations to solve an algebra problem, only to be told that the answer was wrong due to an arithmetic error; I am sure this experience is common to many other schoolchildren.)

It occurred to me recently, though, that the set of problem-solving skills needed to solve algebra problems (and, to some extent, calculus problems also) is somewhat similar to the set of skills needed to solve puzzle-type computer games, in which a certain limited set of moves must be applied in a certain order to achieve a desired result. (There are countless games of this general type; a typical example is “Factory balls“.) Given that the computer game format is already quite familiar to many schoolchildren, one could then try to teach the strategy component of algebraic problem-solving via such a game, which could automate mechanical tasks such as gathering terms and performing arithmetic in order to reduce some of the more frustrating aspects of algebra. (Note that this is distinct from the type of maths games one often sees on educational web sites, which are usually based on asking the player to correctly answer some maths problem in order to advance within the game, making the game essentially a disguised version of a maths quiz. Here, the focus is not so much on being able to supply the correct answer, but on being able to select an effective problem-solving strategy.)

It is difficult to explain in words exactly what type of game I have in mind, so I decided to create a quick mockup of what a sample “level” would look like here (note: requires Java). I didn’t want to spend too much time to make this mockup, so I wrote it in Scratch, which is a somewhat limited programming language intended for use by children, but which has the benefit of being able to churn out simple but functional apps very quickly (the mockup took less than half an hour to write). (I would certainly not attempt to write a full game in this language, though.) In this mockup level, the objective is to solve a single linear equation in one variable, such as ${2x+7=11}$, with only two “moves” available: the ability to subtract ${1}$ from both sides of the equation, and the ability to divide both sides of the equation by ${2}$, which one performs by clicking on an appropriate icon.

It seems to me that one could actually teach a fair amount of algebra through a game such as this, with a progressively difficult sequence of levels that gradually introduce more and more types of “moves” that can handle increasingly difficult problems (e.g. simultaneous linear equations in several unknowns, quadratic equations in one or more variables, inequalities, etc.). Even within a single class of problem, such as solving linear equations, one could create additional challenge by placing some restriction on the available moves, for instance by limiting the number of available moves (as was done in the mockup), or requiring that each move cost some amount of game currency (which might possibly be waived if one is willing to perform the move “by hand”, i.e. by entering the transformed equation manually). And of course one could also make the graphics, sound, and gameplay fancier (e.g. by allowing for various competitive gameplay modes). One could also imagine some other types of high-school and lower-division undergraduate mathematics being amenable to this sort of gamification (calculus and ODE comes to mind, and maybe propositional logic), though I doubt that one could use it effectively for advanced undergraduate or graduate topics. (Though I have sometimes wished for an “integrate by parts” or “use Sobolev embedding” button when trying to control solutions to a PDE…)

This would however be a fair amount of work to actually implement, and is not something I could do by myself with the time I have available these days. But perhaps it may be possible to develop such a game (or platform for such a game) collaboratively, somewhat in the spirit of the polymath projects? I have almost no experience in modern software development (other than a summer programming job I had as a teenager, which hardly counts), so I would be curious to know how projects such as this actually get started in practice.

It has been a little over two weeks now since the protest site at thecostofknowledge.com was set up to register declarations of non-cooperation with Reed Elsevier in protest of their research publishing practices, inspired by this blog post of Tim Gowers.   Awareness of the protest has certainly grown in these two weeks; the number of signatories is now well over four thousand, across a broad array of academic disciplines, and the protest has been covered by many blogs and also the mainstream media (e.g. the Guardian, the Economist, Forbes, etc.), and even by Elsevier stock analysts.    (Elsevier itself released an open letter responding to the protest here.)  My interpretation of events is that there was a significant amount of latent or otherwise undisclosed dissatisfaction already with the publishing practices of Elsevier (and, to a lesser extent, some other commercial academic publishers), and a desire to support alternatives such as university or society publishers, and the more recent open access journals; and that this protest (and parallel protests, such as the movement to oppose the Research Works Act) served to drive these feelings out into the open.

The statement of the protest itself, though, is rather brief, reflecting the improvised manner in which the site was created.  A group of mathematicians including myself therefore decided to write and sign a more detailed explanation of why we supported this protest, giving more background and references to support our position.   The 34 signatories are Scott Aaronson, Douglas N. Arnold, Artur Avila, John Baez, Folkmar Bornemann, Danny Calegari, Henry Cohn, Ingrid Daubechies, Jordan Ellenberg, Matthew Emerton, Marie Farge, David Gabai, Timothy Gowers, Ben Green, Martin Grotschel, Michael Harris, Frederic Helein, Rob Kirby, Vincent Lafforgue, Gregory F. Lawler, Randall J. LeVeque, Laszlo Lovasz, Peter J. Olver, Olof Sisask, Richard Taylor, Bernard Teissier, Burt Totaro, Lloyd N. Trefethen, Takashi Tsuboi, Marie-France Vigneras, Wendelin Werner, Amie Wilkinson, Gunter M. Ziegler, and myself.  (Note that while Daubechies is current president of the International Mathematical Union, Lovasz is a past president, and Grotschel is the current secretary, they are signing this letter as individuals and not as representatives of the IMU. Similarly for Trefethen and Arnold (current and past president of SIAM).)

Of course, the 34 of us do not presume to speak for the remaining four thousand signatories to the protest, but I hope that our statement is somewhat representative of the position of many of its supporters.

Further discussion of this statement can be found at this blog post of Tim Gowers.

EDIT: I think it is appropriate to quote the following excerpt from our statement:

All mathematicians must decide for themselves whether, or to what extent, they wish to participate in the boycott. Senior mathematicians who have signed the boycott bear some responsibility towards junior colleagues who are forgoing the option of publishing in Elsevier journals, and should do their best to help minimize any negative career consequences.

Whether or not you decide to join the boycott, there are some simple actions that everyone can take, which seem to us to be uncontroversial:

1. Make sure that the final versions of all your papers, particularly new ones, are freely available online, ideally both on the arXiv and on your home page.
2.  If you are submitting a paper and there is a choice between an expensive journal and a cheap (or free) journal of the same standard, then always submit to the cheap one.

A few days ago, inspired by this recent post of Tim Gowers, a web page entitled “the cost of knowledge” has been set up as a location for mathematicians and other academics to declare a protest against the academic publishing practices of Reed Elsevier, in particular with regard to their exceptionally high journal prices, their policy of “bundling” journals together so that libraries are forced to purchase subscriptions to large numbers of low-quality journals in order to gain access to a handful of high-quality journals, and their opposition to the open access movement (as manifested, for instance, in their lobbying in support of legislation such as the Stop Online Piracy Act (SOPA) and the Research Works Act (RWA)).   [These practices have been documented in a number of places; this wiki page, which was set up in response to Tim's post, collects several relevant links for this purpose.  Some of the other commercial publishers have  exhibited similar behaviour, though usually not to the extent that Elsevier has, which is why this particular publisher is the focus of this protest.]  At the protest site, one can publicly declare a refusal to either publish at an Elsevier journal, referee for an Elsevier journal, or join the board of an Elsevier journal.

(In the past, the editorial boards of several Elsevier journals have resigned over the pricing policies of the journal, most famously the board of Topology in 2006, but also the Journal of Algorithms in 2003, and a number of journals in other sciences as well.  Several libraries, such as those of Harvard and Cornell, have also managed to negotiate an unbundling of Elsevier journals, but most libraries are still unable to subscribe to such journals individually.)

For a more thorough discussion as to why such a protest is warranted, please see Tim’s post on the matter (and the 100+ comments to that post).   Many of the issues regarding Elsevier were already known to some extent to many mathematicians (particularly those who have served on departmental library committees), several of whom had already privately made the decision to boycott Elsevier; but nevertheless it is important to bring these issues out into the open, to make them commonly known as opposed to merely mutually known.  (Amusingly, this distinction is also of crucial importance in my favorite logic puzzle, but that’s another story.)   One can also see Elsevier’s side of the story in this response to Tim’s post by David Clark (the Senior Vice President for Physical Sciences at Elsevier).

For my own part, though I have sent about 9% of my papers in the past to Elsevier journals (with one or two still in press), I have now elected not to submit any further papers to these journals, nor to serve on their editorial boards, though I will continue refereeing some papers from these journals.  As of this time of writing, over five hundred mathematicians and other academics have also signed on to the protest in the four days that the site has been active.

Admittedly, I am fortunate enough to be at a stage of career in which I am not pressured to publish in a very specific set of journals, and as such, I am not making a recommendation as to what anyone else should do or not do regarding this protest.  However, I do feel that it is worth spreading awareness, at least, of the fact that such protests exist (and some additional petitions on related issues can be found at the previously mentioned wiki page).

In the Winter quarter (starting on January 9), I will be teaching a graduate course on expansion in groups of Lie type.  This course will focus on constructions of expanding Cayley graphs on finite groups of Lie type (such as the special linear groups $SL_d({\bf F}_q)$, or their simple quotients $PSL_d({\bf F}_q)$, but also including more exotic “twisted” groups of Lie type, such as the Steinberg or Suzuki-Ree groups), including the “classical” constructions of Margulis and of Selberg, but also the more recent constructions of Bourgain-Gamburd and later authors (including some very recent work of Ben Green, Emmanuel Breuillard, Rob Guralnick, and myself which is nearing completion and which I plan to post about shortly).  As usual, I plan to start posting lecture notes on this blog before the course begins.

[Once again, some advertising on behalf of my department, following on a similar announcement in the previous two years.]

Two years ago, the UCLA mathematics department launched a scholarship opportunity for entering freshman students with exceptional background and promise in mathematics. We have offered one scholarship every year, but this year due to an additional source of funding, we will also be able to offer an additional scholarship for California residents.The UCLA Math Undergraduate Merit Scholarship provides for full tuition, and a room and board allowance for 4 years. In addition, scholarship recipients follow an individualized accelerated program of study, as determined after consultation with UCLA faculty.   The program of study leads to a Masters degree in Mathematics in four years.
More information and an application form for the scholarship can be found on the web at:
and
To be considered for Fall 2012, candidates must apply for the scholarship and also for admission to UCLA on or before November 30, 2011.

This fall (starting Monday, September 26), I will be teaching a graduate topics course which I have entitled “Hilbert’s fifth problem and related topics.” The course is going to focus on three related topics:

• Hilbert’s fifth problem on the topological description of Lie groups, as well as the closely related (local) classification of locally compact groups (the Gleason-Yamabe theorem).
• Approximate groups in nonabelian groups, and their classification via the Gleason-Yamabe theorem (this is very recent work of Emmanuel Breuillard, Ben Green, Tom Sanders, and myself, building upon earlier work of Hrushovski);
• Gromov’s theorem on groups of polynomial growth, as proven via the classification of approximate groups (as well as some consequences to fundamental groups of Riemannian manifolds).

I have already blogged about these topics repeatedly in the past (particularly with regard to Hilbert’s fifth problem), and I intend to recycle some of that material in the lecture notes for this course.

The above three families of results exemplify two broad principles (part of what I like to call “the dichotomy between structure and randomness“):

• (Rigidity) If a group-like object exhibits a weak amount of regularity, then it (or a large portion thereof) often automatically exhibits a strong amount of regularity as well;
• (Structure) This strong regularity manifests itself either as Lie type structure (in continuous settings) or nilpotent type structure (in discrete settings). (In some cases, “nilpotent” should be replaced by sister properties such as “abelian“, “solvable“, or “polycyclic“.)

Let me illustrate what I mean by these two principles with two simple examples, one in the continuous setting and one in the discrete setting. We begin with a continuous example. Given an ${n \times n}$ complex matrix ${A \in M_n({\bf C})}$, define the matrix exponential ${\exp(A)}$ of ${A}$ by the formula

$\displaystyle \exp(A) := \sum_{k=0}^\infty \frac{A^k}{k!} = 1 + A + \frac{1}{2!} A^2 + \frac{1}{3!} A^3 + \ldots$

which can easily be verified to be an absolutely convergent series.

Exercise 1 Show that the map ${A \mapsto \exp(A)}$ is a real analytic (and even complex analytic) map from ${M_n({\bf C})}$ to ${M_n({\bf C})}$, and obeys the restricted homomorphism property

$\displaystyle \exp(sA) \exp(tA) = \exp((s+t)A) \ \ \ \ \ (1)$

for all ${A \in M_n({\bf C})}$ and ${s,t \in {\bf C}}$.

Proposition 1 (Rigidity and structure of matrix homomorphisms) Let ${n}$ be a natural number. Let ${GL_n({\bf C})}$ be the group of invertible ${n \times n}$ complex matrices. Let ${\Phi: {\bf R} \rightarrow GL_n({\bf C})}$ be a map obeying two properties:

• (Group-like object) ${\Phi}$ is a homomorphism, thus ${\Phi(s) \Phi(t) = \Phi(s+t)}$ for all ${s,t \in {\bf R}}$.
• (Weak regularity) The map ${t \mapsto \Phi(t)}$ is continuous.

Then:

• (Strong regularity) The map ${t \mapsto \Phi(t)}$ is smooth (i.e. infinitely differentiable). In fact it is even real analytic.
• (Lie-type structure) There exists a (unique) complex ${n \times n}$ matrix ${A}$ such that ${\Phi(t) = \exp(tA)}$ for all ${t \in {\bf R}}$.

Proof: Let ${\Phi}$ be as above. Let ${\epsilon > 0}$ be a small number (depending only on ${n}$). By the homomorphism property, ${\Phi(0) = 1}$ (where we use ${1}$ here to denote the identity element of ${GL_n({\bf C})}$), and so by continuity we may find a small ${t_0>0}$ such that ${\Phi(t) = 1 + O(\epsilon)}$ for all ${t \in [-t_0,t_0]}$ (we use some arbitrary norm here on the space of ${n \times n}$ matrices, and allow implied constants in the ${O()}$ notation to depend on ${n}$).

The map ${A \mapsto \exp(A)}$ is real analytic and (by the inverse function theorem) is a diffeomorphism near ${0}$. Thus, by the inverse function theorem, we can (if ${\epsilon}$ is small enough) find a matrix ${B}$ of size ${B = O(\epsilon)}$ such that ${\Phi(t_0) = \exp(B)}$. By the homomorphism property and (1), we thus have

$\displaystyle \Phi(t_0/2)^2 = \Phi(t_0) = \exp(B) = \exp(B/2)^2.$

On the other hand, by another application of the inverse function theorem we see that the squaring map ${A \mapsto A^2}$ is a diffeomorphism near ${1}$ in ${GL_n({\bf C})}$, and thus (if ${\epsilon}$ is small enough)

$\displaystyle \Phi(t_0/2) = \exp(B/2).$

We may iterate this argument (for a fixed, but small, value of ${\epsilon}$) and conclude that

$\displaystyle \Phi(t_0/2^k) = \exp(B/2^k)$

for all ${k = 0,1,2,\ldots}$. By the homomorphism property and (1) we thus have

$\displaystyle \Phi(qt_0) = \exp(qB)$

whenever ${q}$ is a dyadic rational, i.e. a rational of the form ${a/2^k}$ for some integer ${a}$ and natural number ${k}$. By continuity we thus have

$\displaystyle \Phi(st_0) = \exp(sB)$

for all real ${s}$. Setting ${A := B/t_0}$ we conclude that

$\displaystyle \Phi(t) = \exp(tA)$

for all real ${t}$, which gives existence of the representation and also real analyticity and smoothness. Finally, uniqueness of the representation ${\Phi(t) = \exp(tA)}$ follows from the identity

$\displaystyle A = \frac{d}{dt} \exp(tA)|_{t=0}.$

$\Box$

Exercise 2 Generalise Proposition 1 by replacing the hypothesis that ${\Phi}$ is continuous with the hypothesis that ${\Phi}$ is Lebesgue measurable (Hint: use the Steinhaus theorem.). Show that the proposition fails (assuming the axiom of choice) if this hypothesis is omitted entirely.

Note how one needs both the group-like structure and the weak regularity in combination in order to ensure the strong regularity; neither is sufficient on its own. We will see variants of the above basic argument throughout the course. Here, the task of obtaining smooth (or real analytic structure) was relatively easy, because we could borrow the smooth (or real analytic) structure of the domain ${{\bf R}}$ and range ${M_n({\bf C})}$; but, somewhat remarkably, we shall see that one can still build such smooth or analytic structures even when none of the original objects have any such structure to begin with.

Now we turn to a second illustration of the above principles, namely Jordan’s theorem, which uses a discreteness hypothesis to upgrade Lie type structure to nilpotent (and in this case, abelian) structure. We shall formulate Jordan’s theorem in a slightly stilted fashion in order to emphasise the adherence to the above-mentioned principles.

Theorem 2 (Jordan’s theorem) Let ${G}$ be an object with the following properties:

• (Group-like object) ${G}$ is a group.
• (Discreteness) ${G}$ is finite.
• (Lie-type structure) ${G}$ is contained in ${U_n({\bf C})}$ (the group of unitary ${n \times n}$ matrices) for some ${n}$.

Then there is a subgroup ${G'}$ of ${G}$ such that

• (${G'}$ is close to ${G}$) The index ${|G/G'|}$ of ${G'}$ in ${G}$ is ${O_n(1)}$ (i.e. bounded by ${C_n}$ for some quantity ${C_n}$ depending only on ${n}$).
• (Nilpotent-type structure) ${G'}$ is abelian.

A key observation in the proof of Jordan’s theorem is that if two unitary elements ${g, h \in U_n({\bf C})}$ are close to the identity, then their commutator ${[g,h] = g^{-1}h^{-1}gh}$ is even closer to the identity (in, say, the operator norm ${\| \|_{op}}$). Indeed, since multiplication on the left or right by unitary elements does not affect the operator norm, we have

$\displaystyle \| [g,h] - 1 \|_{op} = \| gh - hg \|_{op}$

$\displaystyle = \| (g-1)(h-1) - (h-1)(g-1) \|_{op}$

and so by the triangle inequality

$\displaystyle \| [g,h] - 1 \|_{op} \leq 2 \|g-1\|_{op} \|h-1\|_{op}. \ \ \ \ \ (2)$

Now we can prove Jordan’s theorem.

Proof: We induct on ${n}$, the case ${n=1}$ being trivial. Suppose first that ${G}$ contains a central element ${g}$ which is not a multiple of the identity. Then, by definition, ${G}$ is contained in the centraliser ${Z(g)}$ of ${g}$, which by the spectral theorem is isomorphic to a product ${U_{n_1}({\bf C}) \times \ldots \times U_{n_k}({\bf C})}$ of smaller unitary groups. Projecting ${G}$ to each of these factor groups and applying the induction hypothesis, we obtain the claim.

Thus we may assume that ${G}$ contains no central elements other than multiples of the identity. Now pick a small ${\epsilon > 0}$ (one could take ${\epsilon=\frac{1}{10d}}$ in fact) and consider the subgroup ${G'}$ of ${G}$ generated by those elements of ${G}$ that are within ${\epsilon}$ of the identity (in the operator norm). By considering a maximal ${\epsilon}$-net of ${G}$ we see that ${G'}$ has index at most ${O_{n,\epsilon}(1)}$ in ${G}$. By arguing as before, we may assume that ${G'}$ has no central elements other than multiples of the identity.

If ${G'}$ consists only of multiples of the identity, then we are done. If not, take an element ${g}$ of ${G'}$ that is not a multiple of the identity, and which is as close as possible to the identity (here is where we crucially use that ${G}$ is finite). By (2), we see that if ${\epsilon}$ is sufficiently small depending on ${n}$, and if ${h}$ is one of the generators of ${G'}$, then ${[g,h]}$ lies in ${G'}$ and is closer to the identity than ${g}$, and is thus a multiple of the identity. On the other hand, ${[g,h]}$ has determinant ${1}$. Given that it is so close to the identity, it must therefore be the identity (if ${\epsilon}$ is small enough). In other words, ${g}$ is central in ${G'}$, and is thus a multiple of the identity. But this contradicts the hypothesis that there are no central elements other than multiples of the identity, and we are done. $\Box$

Commutator estimates such as (2) will play a fundamental role in many of the arguments we will see in this course; as we saw above, such estimates combine very well with a discreteness hypothesis, but will also be very useful in the continuous setting.

Exercise 3 Generalise Jordan’s theorem to the case when ${G}$ is a finite subgroup of ${GL_n({\bf C})}$ rather than of ${U_n({\bf C})}$. (Hint: The elements of ${G}$ are not necessarily unitary, and thus do not necessarily preserve the standard Hilbert inner product of ${{\bf C}^n}$. However, if one averages that inner product by the finite group ${G}$, one obtains a new inner product on ${{\bf C}^n}$ that is preserved by ${G}$, which allows one to conjugate ${G}$ to a subgroup of ${U_n({\bf C})}$. This averaging trick is (a small) part of Weyl’s unitary trick in representation theory.)

Exercise 4 (Inability to discretise nonabelian Lie groups) Show that if ${n \geq 3}$, then the orthogonal group ${O_n({\bf R})}$ cannot contain arbitrarily dense finite subgroups, in the sense that there exists an ${\epsilon = \epsilon_n > 0}$ depending only on ${n}$ such that for every finite subgroup ${G}$ of ${O_n({\bf R})}$, there exists a ball of radius ${\epsilon}$ in ${O_n({\bf R})}$ (with, say, the operator norm metric) that is disjoint from ${G}$. What happens in the ${n=2}$ case?

Remark 1 More precise classifications of the finite subgroups of ${U_n({\bf C})}$ are known, particularly in low dimensions. For instance, one can show that the only finite subgroups of ${SO_3({\bf R})}$ (which ${SU_2({\bf C})}$ is a double cover of) are isomorphic to either a cyclic group, a dihedral group, or the symmetry group of one of the Platonic solids.

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