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This week I once again gave some public lectures on the cosmic distance ladder in astronomy, once at Stanford and once at UCLA.  The slides I used were similar to the “version 3.0” slides I used for the same talk last year in Australia and elsewhere, but the images have been updated (and the permissions for copyrighted images secured), and some additional data has also been placed on them.    I am placing these slides here on this blog, in Powerpoint format and also in PDF format.  (Video for the UCLA talk should also be available on the UCLA web site at some point; I’ll add a link when it becomes available.)

These slides have evolved over a period of almost five years, particularly with regards to the imagery, but this is likely to be close to the final version.  Here are some of the older iterations of the slides:

I have found that working on and polishing a single public lecture over a period of several years has been very rewarding and educational, especially given that I had very little public speaking experience at the beginning; there are several other mathematicians I know of who are also putting some effort into giving good talks that communicate mathematics and science to the general public, but I think there could potentially be many more such talks like this.

A note regarding copyright: I am happy to have the text or layout of these slides used as the basis for other presentations, so long as the source is acknowledged.  However, some of the images in these slides are copyrighted by others, and permission by the copyright holders was granted only for the display of the slides in their current format.  (The list of such images is given at the end of the slides.)  So if you wish to adapt the slides for your own purposes, you may need to use slightly different imagery.

(Update, October 20: Some photos from the UCLA talk are available here.)

(Update, October 25: Video from the talk is available on Youtube and on Itunes.)

This week I was in my home town of Adelaide, Australia, for the 2009 annual meeting of the Australian Mathematical Society. This was a fairly large meeting (almost 500 participants). One of the highlights of such a large meeting is the ability to listen to plenary lectures in fields adjacent to one’s own, in which speakers can give high-level overviews of a subject without getting too bogged down in the technical details. From the talks here I learned a number of basic things which were well known to experts in that field, but which I had not fully appreciated, and so I wanted to share them here.

The first instance of this was from a plenary lecture by Danny Calegari entitled “faces of the stable commutator length (scl) ball”. One thing I learned from this talk is that in homotopy theory, there is a very close relationship between topological spaces (such as manifolds) on one hand, and groups (and generalisations of groups) on the other, so that homotopy-theoretic questions about the former can often be converted to purely algebraic questions about the latter, and vice versa; indeed, it seems that homotopy theorists almost think of topological spaces and groups as being essentially the same concept, despite looking very different at first glance. To get from a space ${X}$ to a group, one looks at homotopy groups ${\pi_n(X)}$ of that space, and in particular the fundamental group ${\pi_1(X)}$; conversely, to get from a group ${G}$ back to a topological space one can use the Eilenberg-Maclane spaces ${K(G,n)}$ associated to that group (and more generally, a Postnikov tower associated to a sequence of such groups, together with additional data). In Danny’s talk, he gave the following specific example: the problem of finding the least complicated embedded surface with prescribed (and homologically trivial) boundary in a space ${X}$, where “least complicated” is measured by genus (or more precisely, the negative component of Euler characteristic), is essentially equivalent to computing the commutator length of the element in the fundamental group ${\pi(X)}$ corresponding to that boundary (i.e. the least number of commutators one is required to multiply together to express the element); and the stable version of this problem (where one allows the surface to wrap around the boundary ${n}$ times for some large ${n}$, and one computes the asymptotic ratio between the Euler characteristic and ${n}$) is similarly equivalent to computing the stable commutator length of that group element. (Incidentally, there is a simple combinatorial open problem regarding commutator length in the free group, which I have placed on the polymath wiki.)

This theme was reinforced by another plenary lecture by Ezra Getzler entitled “${n}$-groups”, in which he showed how sequences of groups (such as the first ${n}$ homotopy groups ${\pi_1(X),\ldots,\pi_n(X)}$) can be enhanced into a more powerful structure known as an ${n}$-group, which is more complicated to define, requiring the machinery of simplicial complexes, sheaves, and nerves. Nevertheless, this gives a very topological and geometric interpretation of the concept of a group and its generalisations, which are of use in topological quantum field theory, among other things.

Mohammed Abuzaid gave a plenary lecture entitled “Functoriality in homological mirror symmetry”. One thing I learned from this talk was that the (partially conjectural) phenomenon of (homological) mirror symmetry is one of several types of duality, in which the behaviour of maps into one mathematical object ${X}$ (e.g. immersed or embedded curves, surfaces, etc.) are closely tied to the behaviour of maps out of a dual mathematical object ${\hat X}$ (e.g. functionals, vector fields, forms, sections, bundles, etc.). A familiar example of this is in linear algebra: by taking adjoints, a linear map into a vector space ${X}$ can be related to an adjoint linear map mapping out of the dual space ${X^*}$. Here, the behaviour of curves in a two-dimensional symplectic manifold (or more generally, Lagrangian submanifolds in a higher-dimensional symplectic manifold), is tied to the behaviour of holomorphic sections on bundles over a dual algebraic variety, where the precise definition of “behaviour” is category-theoretic, involving some rather complicated gadgets such as the Fukaya category of a symplectic manifold. As with many other applications of category theory, it is not just the individual pairings between an object and its dual which are of interest, but also the relationships between these pairings, as formalised by various functors between categories (and natural transformations between functors). (One approach to mirror symmetry was discussed by Shing-Tung Yau at a distinguished lecture at UCLA, as transcribed in this previous post.)

There was a related theme in a talk by Dennis Gaitsgory entitled “The geometric Langlands program”. From my (very superficial) understanding of the Langlands program, the behaviour of specific maps into a reductive Lie group ${G}$, such as representations in ${G}$ of a fundamental group, étale fundamental group, class group, or Galois group of a global field, is conjecturally tied to specific maps out of a dual reductive Lie group ${\hat G}$, such as irreducible automorphic representations of ${\hat G}$, or of various structures (such as derived categories) attached to vector bundles on ${\hat G}$. There are apparently some tentatively conjectured links (due to Witten?) between Langlands duality and mirror symmetry, but they seem at present to be fairly distinct phenomena (one is topological and geometric, the other is more algebraic and arithmetic). For abelian groups, Langlands duality is closely connected to the much more classical Pontryagin duality in Fourier analysis. (There is an analogue of Fourier analysis for nonabelian groups, namely representation theory, but the link from this to the Langlands program is somewhat murky, at least to me.)

Related also to this was a plenary talk by Akshay Venkatesh, entitled “The Cohen-Lenstra heuristics over global fields”. Here, the question concerned the conjectural behaviour of class groups of quadratic fields, and in particular to explain the numerically observed phenomenon that about ${75.4\%}$ of all quadratic fields ${{\Bbb Q}[\sqrt{d}]}$ (with $d$ prime) enjoy unique factorisation (i.e. have trivial class group). (Class groups, as I learned in these two talks, are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue of the full fundamental group.) One thing I learned here was that there was a canonical way to randomly generate a (profinite) abelian group, by taking the product of randomly generated finite abelian ${p}$-groups for each prime ${p}$. The way to canonically randomly generate a finite abelian ${p}$-group is to take large integers ${n, D}$, and look at the cokernel of a random homomorphism from ${({\mathbb Z}/p^n{\mathbb Z})^d}$ to ${({\mathbb Z}/p^n{\mathbb Z})^d}$. In the limit ${n,d \rightarrow \infty}$ (or by replacing ${{\mathbb Z}/p^n{\mathbb Z}}$ with the ${p}$-adics and just sending ${d \rightarrow \infty}$), this stabilises and generates any given ${p}$-group ${G}$ with probability

$\displaystyle \frac{1}{|\hbox{Aut}(G)|} \prod_{j=1}^\infty (1 - \frac{1}{p^j}), \ \ \ \ \ (1)$

where ${\hbox{Aut}(G)}$ is the group of automorphisms of ${G}$. In particular this leads to the strange identity

$\displaystyle \sum_G \frac{1}{|\hbox{Aut}(G)|} = \prod_{j=1}^\infty (1 - \frac{1}{p^j})^{-1} \ \ \ \ \ (2)$

where ${G}$ ranges over all ${p}$-groups; I do not know how to prove this identity other than via the above probability computation, the proof of which I give below the fold.

Based on the heuristic that the class group should behave “randomly” subject to some “obvious” constraints, it is expected that a randomly chosen real quadratic field ${{\Bbb Q}[\sqrt{d}]}$ has unique factorisation (i.e. the class group has trivial ${p}$-group component for every ${p}$) with probability

$\displaystyle \prod_{p \hbox{ odd}} \prod_{j=2}^\infty (1 - \frac{1}{p^j}) \approx 0.754,$

whereas a randomly chosen imaginary quadratic field ${{\Bbb Q}[\sqrt{-d}]}$ has unique factorisation with probability

$\displaystyle \prod_{p \hbox{ odd}} \prod_{j=1}^\infty (1 - \frac{1}{p^j}) = 0.$

The former claim is conjectural, whereas the latter claim follows from (for instance) Siegel’s theorem on the size of the class group, as discussed in this previous post. Ellenberg, Venkatesh, and Westerland have recently established some partial results towards the function field analogues of these heuristics.

Next month, I am scheduled to give a short speech (three to five minutes in length) at the annual induction ceremony of the American Academy of Arts and Sciences in Boston.  This is a bit different from the usual scientific talks that I am used to giving; there are no projectors, blackboards, or other visual aids available, and the audience of Academy members is split evenly between the humanities and the sciences (as well as people in industry and politics), so this will be an interesting new experience for me.  (The last time I gave a speech was in 1985.)

My chosen topic is on the future impact of internet-based technologies on academia (somewhat similar in theme to my recent talk on this topic).  I have a draft text below the fold, though it is currently too long and my actual speech is likely to be a significantly abridged version of the one below [Update, Oct 12: The abridged speech is now at the bottom of the post.]  In the spirit of the theme of the talk, I would of course welcome any comments and suggestions.

For comparison, the talks from last year’s ceremony, by Jim Simons, Peter Kim, Susan Athey, Earl Lewis, and Indra Nooyi, can be found here.  Jim’s chosen topic, incidentally, was what mathematics is, and why mathematicians do it.

[Update, Nov 3: Video of the various talks by myself and the other speakers (Emmylou Harris, James Earl Jones, Elizabeth Nabel, Ronald Marc George, and Edward Villela) is now available on the Academy web site here.]

I am posting the last two talks in my Clay-Mahler lecture series here:

[Update, Sep 14: Poincaré conjecture slides revised.]

[Update, Sep 18: Prime slides revised also.]

I am uploading another of my Clay-Mahler lectures here, namely my public talk on the cosmic distance ladder (4.3MB, PDF).  These slides are based on my previous talks of the same name, but I have updated and reorganised the graphics significantly as I was not fully satisfied with the previous arrangement.

[Update, Sep 4: slides updated.  The Powerpoint version of the slides (8MB) are available here.]

[Update, Oct 26: slides updated again.]

I’ll be in Australia for the next month or so, giving my share of the Clay-Mahler lectures at various institutions in the country.  My first lecture is next Monday at Melbourne University, entitled “Mathematical research and the internet“.  This public lecture discusses how various internet technologies (such as blogging) are beginning to transform the way mathematicians do research.

In the spirit of that article, I have decided to upload an advance copy of the talk here, and would welcome any comments or feedback (I still have a little bit of time to revise the article).   [NB: the PDF file is about 5MB in size; the original Powerpoint presentation was 10MB!]

[Update, August 31: the talk has been updated in view of feedback from this blog and elsewhere.  For comparison, the older version of the talk can be found here.]

[Update, Sep 4: Video of the talk and other information is available here.]

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

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

Below the fold is a version of my talk “Recent progress on the Kakeya conjecture” that I gave at the Fefferman conference.

I am currently at Princeton for the conference “The power of Analysis” honouring Charlie Fefferman‘s 60th birthday. I myself gave a talk at this conference entitled “Recent progress on the Kakeya conjecture”; I plan to post a version of this talk on this blog shortly.

But one nice thing about attending these sorts of conferences is that one can also learn some neat mathematical facts, and I wanted to show two such small gems here; neither is particularly deep, but I found both of them cute. The first one, which I learned from my former student Soonsik Kwon, is a unified way to view the mean, median, and mode of a probability distribution ${\mu}$ on the real line. If one assumes that this is a continuous distribution ${\mu = f(x)\ dx}$ for some smooth, rapidly decreasing function ${f: {\mathbb R} \rightarrow {\mathbb R}^+}$ with ${\int_{\mathbb R} f(x)\ dx = 1}$, then the mean is the value of ${x_0}$ that minimises the second moment

$\displaystyle \int_{\mathbb R} |x-x_0|^2 f(x)\ dx,$

the median is the value of ${x_0}$ that minimises the first moment

$\displaystyle \int_{\mathbb R} |x-x_0| f(x)\ dx,$

and the mode is the value of ${x_0}$ that maximises the “pseudo-negative first moment”

$\displaystyle \int_{\mathbb R} \delta(x-x_0) f(x)\ dx.$

(Note that the Dirac delta function ${\delta(x-x_0)}$ has the same scaling as ${|x-x_0|^{-1}}$, hence my terminology “pseudo-negative first moment”.)

The other fact, which I learned from my former classmate Diego Córdoba (and used in a joint paper of Diego with Antonio Córdoba), is a pointwise inequality

$\displaystyle |\nabla|^\alpha ( f^2 )(x) \leq 2 f(x) |\nabla|^\alpha f(x)$

for the fractional differentiation operators ${|\nabla|^\alpha}$ applied to a sufficiently nice real-valued function ${f: {\mathbb R}^d \rightarrow {\mathbb R}}$ (e.g. Schwartz class will do), in any dimension ${d}$ and for any ${0 \leq \alpha \leq 1}$; this should be compared with the product rule ${\nabla (f^2 ) = 2 f \nabla f}$.

The proof is as follows. By a limiting argument we may assume that ${0 < \alpha < 1}$. In this case, there is a formula

$\displaystyle |\nabla|^\alpha f(x) = c(\alpha) \int_{{\mathbb R}^d} \frac{f(x)-f(y)}{|x-y|^{d+\alpha}}\ dy$

for some explicit constant ${c(\alpha) > 0}$ (this can be seen by computations similar to those in my recent lecture notes on distributions, or by analytically continuing such computations; see also Stein’s “Singular integrals and differentiability properties of functions”). Using this formula, one soon sees that

$\displaystyle 2 f(x) |\nabla|^\alpha f(x) - |\nabla|^\alpha ( f^2 )(x) = c(\alpha) \int_{{\mathbb R}^d} \frac{|f(x)-f(y)|^2}{|x-y|^{d+\alpha}}\ dy$

and the claim follows.

I was recently at an international airport, trying to get from one end of a very long terminal to another.  It inspired in me the following simple maths puzzle, which I thought I would share here:

Suppose you are trying to get from one end A of a terminal to the other end B.  (For simplicity, assume the terminal is a one-dimensional line segment.)  Some portions of the terminal have moving walkways (in both directions); other portions do not.  Your walking speed is a constant $v$, but while on a walkway, it is boosted by the speed $u$ of the walkway for a net speed of $v+u$.  (Obviously, given a choice, one would only take those walkways that are going in the direction one wishes to travel in.)  Your objective is to get from A to B in the shortest time possible.

1. Suppose you need to pause for some period of time, say to tie your shoe.  Is it more efficient to do so while on a walkway, or off the walkway?  Assume the period of time required is the same in both cases.
2. Suppose you have a limited amount of energy available to run and increase your speed to a higher quantity $v'$ (or $v'+u$, if you are on a walkway).  Is it more efficient to run while on a walkway, or off the walkway?  Assume that the energy expenditure is the same in both cases.
3. Do the answers to the above questions change if one takes into account the various effects of special relativity?  (This is of course an academic question rather than a practical one.  But presumably it should be the time in the airport frame that one wants to minimise, not time in one’s personal frame.)

It is not too difficult to answer these questions on both a rigorous mathematical level and a physically intuitive level, but ideally one should be able to come up with a satisfying mathematical explanation that also corresponds well with one’s intuition.

[Update, Dec 11: Hints deleted, as they were based on an incorrect calculation of mine.]