You are currently browsing the category archive for the 'talk' category.
This week I was in Columbus, Ohio, attending a conference on equidistribution on manifolds. I talked about my recent paper with Ben Green on the quantitative behaviour of polynomial sequences in nilmanifolds, which I have blogged about previously. During my talk (and inspired by the immediately preceding talk of Vitaly Bergelson), I stated explicitly for the first time a generalisation of the van der Corput trick which morally underlies our paper, though it is somewhat buried there as we specialised it to our application at hand (and also had to deal with various quantitative issues that made the presentation more complicated). After the talk, several people asked me for a more precise statement of this trick, so I am presenting it here, and as an application reproving an old theorem of Leon Green that gives a necessary and sufficient condition as to whether a linear sequence 
This week I am at Rutgers University, giving the Lewis Memorial Lectures for this year, which are also concurrently part of a workshop in random matrices. I gave four lectures, three of which were on random matrices, and one of which was on the Szemerédi regularity lemma.
The titles, abstracts, and slides of these talks are as follows.
- Szemerédi’s lemma revisited. In this general-audience talk, I discuss the Szemerédi regularity lemma (which, roughly speaking, shows that an arbitrary large dense graph can always be viewed as the disjoint union of a bounded number of pseudorandom components), and how it has recently been reinterpreted in a more analytical (and infinitary) language using the theory of graph limits or of exchangeable measures. I also discuss arithmetic analogues of this lemma, including one which (implicitly) underlies my result with Ben Green that the primes contain arbitrarily long arithmetic progressions.
- Singularity and determinant of random matrices. Here, I present recent progress in understanding the question of how likely a random matrix (e.g. one whose entries are all +1 or -1 with equal probability) is to be invertible, as well as the related question of how large the determinant should be. The case of continuous matrix ensembles (such as the Gaussian ensemble) is well understood, but the discrete case contains some combinatorial difficulties and took longer to understand properly. In particular I present the results of Kahn-Komlós-Szemerédi and later authors showing that discrete random matrices are invertible with exponentially high probability, and also give some results for the distribution of the determinant.
- The least singular value of random matrices. A more quantitative version of the question “when is a matrix invertible?” is “what is the least singular value of that matrix”? I present here the recent results of Litvak-Pajor-Rudelson-Tomczak-Jaegermann, Rudelson, myself and Vu, and Rudelson-Vershynin on addressing this question in the discrete case. A central role is played by the inverse Littlewood-Offord theorems of additive combinatorics, which give reasonably sharp necessary conditions for a discrete random walk to concentrate in a small ball.
- The circular law. One interesting application of the above theory is to extend the circular law for the spectrum of random matrices from the continuous case to the discrete case. Previous arguments of Girko and Bai for the continuous case can be transplanted to the discrete case, but the key new ingredient needed is a least singular value bound for shifted matrices
in order to avoid the spectrum being overwhelmed by pseudospectrum. It turns out that the results of the preceding lecture are almost precisely what are needed to accomplish this.
[Update, Mar 31: first lecture slides corrected. Thanks to Yoshiyasu Ishigami for pointing out a slight inaccuracy in the text.]
Last month, at the joint AMS/MAA meeting in San Diego, I spoke at the AMS “Current Events” Bulletin on the topic “Why are solitons stable?“. This talk was supposed to be a survey of many of the developments on the rigorous stability theory of solitary waves in dispersive wave models (e.g. the Kortweg-de Vries equation and its generalisations, nonlinear Schrödinger equations, etc.), although my actual talk (which was the usual 50 minutes in length) only managed to cover about half of the material I had planned.
More recently, I completed the article that accompanies the talk, and which will be submitted to the Bulletin of the American Mathematical Society. In this paper I describe the key conflict in these wave models between dispersion (the tendency of waves of differing frequency to move at different speeds, thus causing any localised wave to disperse in space over time) and nonlinearity (which can cause any concentrated portion of the wave to self-amplify). Solitons seem to lie at the exact balancing point between these two forces, neither dispersing nor amplifying, but instead simply traveling at a constant velocity or oscillating in phase at a constant rate. In some cases, this balancing point is unstable; remove even a tiny amount of mass from the soliton and it eventually disperses completely into radiation, or one can add a tiny amount and cause the soliton to concentrate into a point and thence exhibit blowup in finite time. In other cases, the balancing point is stable; small perturbations to a soliton may end up changing the amplitude, position, and/or velocity of the soliton slightly, but the bulk of the solution still closely resembles a soliton in size, shape, and behaviour. Stability is sometimes enforced by linear properties, such as dispersive estimates or spectral properties of the linearised dynamics, but is also often enforced by nonlinear properties, such as nonlinear conservation laws, monotonicity formulae, and local propagation estimates for mass and energy (such as those provided by virial identities). The interplay between all these properties can be remarkably subtle, especially in the critical case when a key conserved quantity is scale-invariant (thus leading to an additional degeneracy in the soliton manifold). This is particularly evident in the remarkable series of papers by Martel and Merle establishing various stability and blowup properties near the ground state soliton of the critical generalised KdV equation, which I spend some time discussing (without going into too many of the (quite numerous) technical details). The focus in my paper is primarily on the non-integrable case, in which the techniques are primarily analytic rather than algebraic or geometric.
This week there is a conference here at IPAM on expanders in pure and applied mathematics. I was an invited speaker, but I don’t actually work in expanders per se (though I am certainly interested in them). So I spoke instead about the recent simplified proof by Kleiner of the celebrated theorem of Gromov on groups of polynomial growth. (This proof does not directly mention expanders, but the argument nevertheless hinges on the absence of expansion in the Cayley graph of a group of polynomial growth, which is exhibited through the smoothness properties of harmonic functions on such graphs.)
This Thursday I was at the University of Sydney, Australia, giving a public lecture on a favourite topic of mine, “Structure and randomness in the prime numbers“. My slides here are a merge between my slides for a Royal Society meeting and the slides I gave for the UCLA Science Colloquium; now that I figured out to use Powerpoint a little bit better, I was able to make the latter a bit more colourful (and the former less abridged).
This week I am in Australia, attending the ANZIAM annual meeting in Katoomba, New South Wales (in the picturesque Blue Mountains). I gave an overview talk on some recent developments in compressed sensing, particularly with regards to the basis pursuit approach to recovering sparse (or compressible) signals from incomplete measurements. The slides for my talk can be found here, with some accompanying pictures here. (There are of course by now many other presentations of compressed sensing on-line; see for instance this page at Rice.)
There was an interesting discussion after the talk. Some members of the audience asked the very good question as to whether any a priori information about a signal (e.g. some restriction about the support) could be incorporated to improve the performance of compressed sensing; a related question was whether one could perform an adaptive sequence of measurements to similarly improve performance. I don’t have good answers to these questions. Another pointed out that the restricted isometry property was like a local “well-conditioning” property for the matrix, which only applied when one viewed a few columns at a time.
This weekend I was (once again) in San Diego, this time for the Southern California Analysis and PDE (SCAPDE) meeting. I gave a talk on “The asymptotic behaviour of large data solutions to NLS”, which is based on two of my previous papers on what solutions to focusing nonlinear Schrödinger equations behave like as time goes to infinity. (Note that this is a specialist conference, and this talk will be a bit more technical than some of the general-audience talks that I have blogged about previously.)
Avi Wigderson’s final talk in his Distinguished Lecture Series on “Computational complexity” was entitled “Arithmetic computation“; the complexity theory of arithmetic circuits rather than boolean circuits.
On Thursday, Avi Wigderson continued his Distinguished Lecture Series here at UCLA on computational complexity with his second lecture “Expander Graphs - Constructions and Applications“. As in the previous lecture, he spent some additional time after the talk on an “encore”, which in this case was how lossless expanders could be used to obtain rapidly decodable error-correcting codes.
The talk was largely based on these slides. Avi also has a recent monograph with Hoory and Linial on these topics. (For a brief introduction to expanders, I can also recommend Peter Sarnak’s Notices article. I also mention expanders to some extent in my third Milliman lecture.)
The Distinguished Lecture Series at UCLA for this winter quarter is given by Avi Wigderson, who is lecturing on “some topics in computational complexity“. In his first lecture on Wednesday, Avi gave a wonderful talk (in his inimitably entertaining style) on “The power and weakness of randomness in computation“. The talk was based on these slides. He also gave a sort of “encore” on zero-knowledge proofs in more informal discussions after the main talk.
As always, any errors here are due to my transcription and interpretation.
This week I am in San Diego for the annual joint mathematics meeting of the American Mathematical Society and the Mathematical Association of America. I am giving two talks here. One is a lecture (for the AMS “Current Events” Bulletin) on recent developments (by Martel-Merle, Merle-Raphael, and others) on stability of solitons; I will post on that lecture at some point in the near future, once the survey paper associated to that lecture is finalised.
The other, which I am presenting here, is an address on “structure and randomness in the prime numbers“. Of course, I’ve talked about this general topic many times before, (e.g. at my Simons lecture at MIT, my Milliman lecture at U. Washington, and my Science Research Colloquium at UCLA), and I have given similar talks to the one here - which focuses on my original 2004 paper with Ben Green on long arithmetic progressions in the primes - about a dozen or so times. As such, this particular talk has probably run its course, and so I am “retiring” it by posting it here.
p.s. At this meeting, Endre Szemerédi was awarded the 2008 Steele prize for a seminal contribution to research, for his landmark paper establishing what is now known as Szemerédi’s theorem, which underlies the result I discuss in this talk. This prize is richly deserved - congratulations Endre! [The AMS and MAA also awarded prizes to several dozen other mathematicians, including many mentioned previously on this blog; rather than list them all here, let me just point you to their prize booklet.]
This is my final Milliman lecture, in which I talk about the sum-product phenomenon in arithmetic combinatorics, and some selected recent applications of this phenomenon to uniform distribution of exponentials, expander graphs, randomness extractors, and detecting (sieving) almost primes in group orbits, particularly as developed by Bourgain and his co-authors.
Read the rest of this entry »
This is my second Milliman lecture, in which I talk about recent applications of ideas from additive combinatorics (and in particular, from the inverse Littlewood-Offord problem) to the theory of discrete random matrices.
Read the rest of this entry »
This week I am visiting the University of Washington in Seattle, giving the Milliman Lecture Series for 2007-2008. My chosen theme here is “Recent developments in arithmetic combinatorics“. In my first lecture, I will speak (once again) on how methods in additive combinatorics have allowed us to detect additive patterns in the prime numbers, in particular discussing my joint work with Ben Green. In the second lecture I will discuss how additive combinatorics has made it possible to study the invertibility and spectral behaviour of random discrete matrices, in particular discussing my joint work with Van Vu; and in the third lecture I will discuss how sum-product estimates have recently led to progress in the theory of expanders relating to Lie groups, as well as to sieving over orbits of such groups, in particular presenting work of Jean Bourgain and his coauthors.
Today, Charlie wrapped up several loose ends in his lectures, including the connection with the classical Whitney extension theorem, the role of convex bodies and Whitney convexity, and a glimpse as to how one obtains the remarkably fast (almost linear time) algorithms in which one actually computes interpolation of functions from finite amounts of data.
On Thursday, Charlie Fefferman continued his lecture series on interpolation of functions. Here, he stated the main technical theorem about bundles that underlies all the results, answering the “cliffhanger” question from the last lecture, and broadly outlined the proof, except for a major technical wrinkle about “Whitney convexity” which he will discuss on Friday. Read the rest of this entry »
The first Distinguished Lecture Series at UCLA of this academic year is being given this week by my good friend and fellow Medalist Charlie Fefferman, who also happens to be my “older brother” (we were both students of Elias Stein). The theme of Charlie’s lectures is “Interpolation of functions on 
The general topic of extracting quantitative bounds from classical qualitative theorems is a subject that I am personally very fond of, and Charlie gave a wonderfully accessible presentation of the main results, though the actual details of the proofs were left to the next two lectures.
As usual, all errors and omissions here are my responsibility, and are not due to Charlie.
This month I have been at the Institute for Advanced Study, participating in their semester program on additive combinatorics. Today I gave a talk on my forthcoming paper with Tim Austin on the property testing of graphs and hypergraphs (I hope to make a preprint available here soon). There has been an immense amount of progress on these topics recently, based in large part on the graph and hypergraph regularity lemmas; but we have discovered some surprising subtleties regarding these results, namely a distinction between undirected and directed graphs, between graphs and hypergraphs, between partite hypergraphs and non-partite hypergraphs, and between monotone hypergraph properties and hereditary ones.
For simplicity let us first work with (uncoloured, undirected, loop-free) graphs G = (V,E). In the subject of graph property testing, one is given a property 
- G is planar.
- G is four-colourable.
- G has a number of edges equal to a power of two.
- G contains no triangles.
- G is bipartite.
- G is empty.
- G is a complete bipartite graph.
We assume that the labeling of the graph is irrelevant. More precisely, we assume that whenever two graphs G, G’ are isomorphic, that G satisfies
We shall think of G as being very large (so 
- (No false negatives) If G indeed satisfies
- (Few false positives in the
-far case) If G fails to satisfy
edges in G before
When a test with the above properties exists for each given
An example should illustrate this definition. Consider for instance property 6 above (the property that G is empty). To test whether a graph is empty, one can perform the following obvious algorithm: take k vertices in G at random and check whether they have any edges at all between them. If they do, then the test of course rejects G as being non-empty, while if they don’t, the test accepts G as being empty. Clearly there are no false negatives in this test, and if k is large enough depending on
On the other hand, it is intuitively obvious that property 3 (having an number of edges equal to a power of 2) is not testable with one-sided error.
So it is reasonable to ask: what types of graph properties are testable with one-sided error, and which ones are not?
I’ve just come back from the 48th Annual IEEE Symposium on the Foundations of Computer science, better known as FOCS; this year it was held at Providence, near Brown University. (This conference is also being officially reported on by the blog posts of Nicole Immorlica, Luca Trevisan, and Scott Aaronson.) I was there to give a tutorial on some of the tools used these days in additive combinatorics and graph theory to distinguish structure and randomness. In a previous blog post, I had already mentioned that my lecture notes for this were available on the arXiv; now the slides for my tutorial are available too (it covers much the same ground as the lecture notes, and also incorporates some material from my ICM slides, but in a slightly different format).
In the slides, I am tentatively announcing some very recent (and not yet fully written up) work of Ben Green and myself establishing the Gowers inverse conjecture in finite fields in the special case when the function f is a bounded degree polynomial (this is a case which already has some theoretical computer science applications). I hope to expand upon this in a future post. But I will describe here a neat trick I learned at the conference (from the FOCS submission of Bogdanov and Viola) which uses majority voting to enhance a large number of small independent correlations into a much stronger single correlation. This application of majority voting is widespread in computer science (and, of course, in real-world democracies), but I had not previously been aware of its utility to the type of structure/randomness problems I am interested in (in particular, it seems to significantly simplify some of the arguments in the proof of my result with Ben mentioned above); thanks to this conference, I now know to add majority voting to my “toolbox”.
Almost a year ago today, I was in Madrid attending the 2006 International Congress of Mathematicians (ICM). One of the many highlights of an ICM meeting are the plenary talks, which offer an excellent opportunity to hear about current developments in mathematics from leaders in various fields, aimed at a general mathematical audience. All the speakers sweat quite a lot over preparing these high-profile talks; for instance, I rewrote the slides for my own talk from scratch after the first version produced bemused reactions from those friends I had shown them to.
I didn’t write about these talks at the time, since my blog had not started then (and also, things were rather hectic for me in Madrid). During the congress, these talks were webcast live, but the video for these talks no longer seems to be available on-line.
A couple weeks ago, though, I received the first volume of the ICM proceedings, which is the one which among other things contains the articles contributed by the plenary speakers (the other two volumes were available at the congress itself). On reading through this volume, I discovered a pleasant surprise - the publishers had included a CD-ROM on the back page which had all the video and slides of the plenary talks, as well as the opening and closing ceremonies! This was a very nice bonus and I hope that the proceedings of future congresses also include something like this.
Of course, I won’t be able to put the data on that CD-ROM on-line, for both technical and legal reasons; but I thought I would discuss a particularly beautiful plenary lecture given by Étienne Ghys on “Knots and dynamics“. His talk was not only very clear and fascinating, but he also made a superb use of the computer, in particular using well-timed videos and images (developed in collaboration with Jos Leys) to illustrate key ideas and concepts very effectively. (The video on the CD-ROM unfortunately does not fully capture this, as it only has stills from his computer presentation rather than animations.) To give you some idea of how good the talk was, Étienne ended up running over time by about fifteen minutes or so; and yet, in an audience of over a thousand, only a handful of people actually left before the end.
The slides for Étienne’s talk can be found here, although, being in PDF format, they only have stills rather than full animations. Some of the animations though can be found on this page. (Étienne’s article for the proceedings can be found here, though like the contributions of most other plenary speakers, the print article is more detailed and technical than the talk.) I of course cannot replicate Étienne’s remarkable lecture style, but I can at least present the beautiful mathematics he discussed.
Read the rest of this entry »
Last week, as mentioned previously, I attended a very inspiring interdisciplinary meeting at the Royal Society. It would be impossible for me to describe all 47 talks in detail; but I thought that I would take the time to discuss one representative talk, presented by Rosemary Grant FRS, on her research (together with several collaborators, including her husband), on the ongoing evolution (and speciation) of Darwin’s finches in the Galápagos islands; this was a particularly striking talk for me, as I had not been aware that such dramatic microevolutionary changes could be seen in real-time (Grant and her co-authors have tracked multiple generations of these finches for 35 years!). They have several interesting results; the talk that I am reproducing here is largely based on this article in Evolution.
This week I was in London, attending the New Fellows Seminar at the Royal Society. This was a fairly low-key event preceding the formal admissions ceremony; for instance, it is not publicised on their web site. The format was very interesting: they had each of the new Fellows of the Society give a brief (15 minute) presentation of their work in quick succession, in a manner which would be accessible to a diverse audience in the physical and life sciences. The result was a wonderful two-day seminar on the state of the art in many areas of physics, chemistry, engineering, biology, medicine, and mathematics. For instance, I learnt
- How the solar neutrino problem was resolved by the discovery that the neutrino had mass, which did not commute with flavour and hence caused neutrino oscillations, which have since been detected experimentally;
- Why modern aircraft (such as the Dreamliner and A380) are now assembled using (incredibly tough and waterproofed) adhesives instead of bolts or welds, and how adhesion has been enhanced by nanoparticles;
- How the bacterium Helicobacter pylori was recently demonstrated (by two Aussies :-) ) to be a major cause of peptic ulcers (though the exact mechanism is not fully understood), but has also been proposed (somewhat paradoxically) to also have a preventative effect against esophageal cancer (cf. the hygiene hypothesis);
- How recent advances in machine learning and image segmentation (including graph cut methods!) now allow computers to identify and track many general classes of objects (e.g. people, cars, animals) simultaneously in real-world images and video, though not quite in real-time yet;
- How large-scale structure maps of the universe (such as the 2dF Galaxy Redshift Survey) combine with measurements of the cosmic background radiation (e.g. from WMAP) to demonstrate the existence of both dark matter and dark energy (they have different impacts on the evolution of the curvature of the universe and on the current distribution of visible matter);
- … and 42 other topics like this. (One strongly recurrent theme in the life science talks was just how much recent genomic technologies, such as the genome projects of various key species, have accelerated (by several orders of magnitude!) the ability to identify the genes, proteins, and mechanisms that underlie any given biological function or disease. To paraphrase one speaker, a modern genomics lab could now produce the equivalent of one 1970s PhD thesis in the subject every minute.)
This week I am in San Diego for the 39th ACM Symposium for the Theory of Computing (STOC). Today I presented my work with Van Vu on the condition number of randomly perturbed matrices, which was the subject of an earlier post on this blog. For this short talk (20 minutes), Van and I prepared some slides; of course, in such a short time frame one cannot hope to discuss many of the details of the result, but one can at least convey the statement of the result and a brief sketch of the main ideas in the proof.
One late update (which didn’t make it onto the slides): last week, an alternate proof of some cases of our main result (together with some further generalisations and other results, in particular a circular law for the eigenvalues of discrete random matrices) was obtained by Pan and Zhou, using earlier arguments by Rudelson and Vershynin.
[Update, June 16: It was pointed out to me that the Pan-Zhou result only recovers our result in the case when the unperturbed matrix has a spectral norm of 
I gave a non-technical talk today to the local chapter of the Pi Mu Epsilon society here at UCLA. I chose to talk on the cosmic distance ladder - the hierarchy of rather clever (yet surprisingly elementary) mathematical methods that astronomers use to indirectly measure very large distances, such as the distance to planets, nearby stars, or distant stars. This ladder was really started by the ancient Greeks, who used it to measure the size and relative locations of the Earth, Sun and Moon to reasonable accuracy, and then continued by Copernicus, Brahe and Kepler who then measured distances to the planets, and in the modern era to stars, galaxies, and (very recently) to the scale of the universe itself. It’s a great testament to the power of indirect measurement, and to the use of mathematics to cleverly augment observation.
For this (rather graphics-intensive) talk, I used Powerpoint for the first time; the slides (which are rather large - 3 megabytes) - can be downloaded here. [I gave an earlier version of this talk in Australia last year in a plainer PDF format, and had to get someone to convert it for me.]
[Update, May 31: In case the powerpoint file is too large or unreadable, I also have my older PDF version of the talk, which omits all the graphics.]
[Update, July 1 2008: John Hutchinson has made some computations to accompany these slides, which can be found at this page.]
On Friday, Yau concluded his lecture series by discussing the PDE approach to constructing geometric structures, particularly Einstein metrics, and their applications to many questions in low-dimensional topology (yes, this includes the Poincaré conjecture). Yau also discussed the situation in high-dimensional topology, which appears to be completely different (and much less well understood).
Yau’s slides for this talk are available here.
On Thursday, Yau continued his lecture series on geometric structures, focusing a bit more on the tools and philosophy that goes into actually building these structures. Much of the philosophy, in its full generality, is still rather vague and not properly formalised, but is nevertheless supported by a large number of rigorously worked out examples and results in special cases. A dominant theme in this talk was the interaction between geometry and physics, in particular general relativity and string theory.
As usual, there are likely to be some inaccuracies in my presentation of Yau’s talk (I am not really an expert in this subject), and corrections are welcome. Yau’s slides for this talk are available here.
Read the rest of this entry »
The final Distinguished Lecture Series for this academic year at UCLA was started on Tuesday by Shing-Tung Yau. (We’ve had a remarkably high-quality array of visitors this year; for instance, in addition to those already mentioned in this blog, mathematicians such as Peter Lax and Michael Freedman have come here and given lectures earlier this year.) Yau’s chosen topic is “Geometric Structures on Manifolds”, and the first talk was an introduction and overview of his later two, titled “What is a Geometric Structure.” Once again, I found this a great opportunity to learn about a field adjacent to my own areas of expertise, in this case geometric analysis (which is adjacent to nonlinear PDE).
As usual, all inaccuracies in these notes are due to myself and not to Yau, and I welcome corrections or comments. Yau’s slides for the talk are available here. Read the rest of this entry »
On Thursday Shou-wu Zhang concluded his lecture series by talking about the higher genus case 
This is perhaps the most technical of all the talks, but also the closest to recent developments, in particular the modern attacks on the abc conjecture. (Shou-wu made the point that one sometimes needs to move away from naive formulations of problems to obtain deeper formulations which are more difficult to understand, but can be easier to prove due to the availability of tools, structures, and intuition that were difficult to access in a naive setting, as well as the ability to precisely formulate and quantify what would otherwise be very fuzzy analogies.)
On Wednesday, Shou-wu Zhang continued his lecture series. Whereas the first lecture was a general overview of the rational points on curves problem, the second talk focused entirely on the genus 1 case - i.e. the problem of finding rational points on elliptic curves. This is already a very deep and important problem in number theory - for instance, this theory is decisive in Wiles’ proof of Fermat’s last theorem. It was also somewhat more technical than the previous talk, and I had more difficulty following all the details, but in any case here is my attempt to reconstruct the talk from my notes. Once again, the inevitable inaccuracies here are my fault and not Shou-wu’s, and corrections or comments are greatly appreciated.
NB: the talk here seems to be loosely based in part on Shou-wu’s “Current developments in Mathematics” article from 2001.
[This lecture is also doubling as this week's "open problem of the week", as it discusses the Birch and Swinnerton-Dyer conjecture and the effective Mordell conjecture.]
Like many other maths departments, UCLA has a distinguished lecture series for eminent mathematicians to present recent developments in a field of mathematics, both to a broad audience and to specialists. Unlike most departments, though, our lecture series goes by the descriptive (but unimaginative) name of “Distinguished Lecture Series“, supported by the Gill Foundation. This week the lecture series is given by Shou-wu Zhang from Columbia, and revolves around the topic of rational points on curves, a key subject of interest in arithmetic geometry and number theory. The first of three talks, which was on Tuesday, was a very accessible and enjoyable overview talk, which I am reproducing here (to use this opportunity to learn this stuff myself, and also to continue the diversification of subject matter here on this blog). As before, I do not vouch for 100% accuracy, and all errors are my responsibility rather than Shou-wu’s.
On Thursday, UCLA hosted a “Fields Medalist Symposium“, in which four of the six University of California-affiliated Fields Medalists (Vaughan Jones (1990), Efim Zelmanov (1994), Richard Borcherds (1998), and myself (2006)) gave talks of varying levels of technical sophistication. (The other two are Michael Freedman (1986) and Steven Smale (1966), who could not attend.) The slides for my own talks are available here.
The talks were in order of the year in which the medal was awarded: we began with Vaughan, who spoke on “Flatland: a great place to do algebra”, then Efim, who spoke on “Pro-finite groups”, Richard, who spoke on “What is a quantum field theory?”, and myself, on “Nilsequences and the primes.” The audience was quite mixed, ranging from mathematics faculty to undergraduates to alumni to curiosity seekers, and I severely doubt that every audience member understood every talk, but there was something for everyone, and for me personally it was fantastic to see some perspectives from first-class mathematicians on some wonderful areas of mathematics outside of my own fields of expertise.
Disclaimer: the summaries below are reconstructed from my notes and from some hasty web research; I don’t vouch for 100% accuracy of the mathematical content, and would welcome corrections.
For much of last week I was in Leiden, Holland, giving one of the Ostrowski prize lectures at the annual meeting of the Netherlands mathematical congress. My talk was not on the subject of the prize (arithmetic progressions in primes), as this was covered by a talk of Ben Green there, but rather on a certain “uniform uncertainty principle” in Fourier analysis, and its relation to compressed sensing; this is work which is joint with Emmanuel Candes and also partly with Justin Romberg.
[This lecture is also doubling as this week's "open problem of the week", as it (eventually) discusses the soliton resolution conjecture.]
In this third lecture, I will talk about how the dichotomy between structure and randomness pervades the study of two different types of partial differential equations (PDEs):
- Parabolic PDE, such as the heat equation
, which turn out to play an important role in the modern study of geometric topology; and
- Hamiltonian PDE, such as the Schrödinger equation
, which are heuristically related (via Liouville’s theorem) to measure-preserving actions of the real line (or time axis)
, somewhat in analogy to how combinatorial number theory and graph theory were related to measure-preserving actions of
and
respectively, as discussed in the previous lecture.
(In physics, one would also insert some physical constants, such as Planck’s constant 