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

- “Structure and randomness in the prime numbers“. This public lecture is slightly updated from a previous talk of the same name given last year, but is largely the same material.
- “Perelman’s proof of the Poincaré conjecture“. Here I try (perhaps ambitiously) to give an overview of Perelman’s proof of the Poincaré conjecture into an hour-length talk for a general mathematical audience. It is a little unpolished, as I have not given any version of this talk before, but I hope to update it a bit in the future.

[*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 am posting here four more of my Mahler lectures, each of which is based on earlier talks of mine:

- Compressed sensing. This is an updated and reformatted version of my ANZIAM talk on this topic.
- Discrete random matrices. This talk is a survey on recent developments on the universality phenomenon in random matrices, including work of myself and Van Vu. It covers similar material to my Netanyahu lecture, which has not previously appeared in electronic form.
- Recent progress in additive prime number theory. This is an updated and reformatted version of my AMS lecture on this topic.
- Recent progress on the Kakeya conjecture. This is an updated and reformatted version of my Fefferman conference lecture.

As always, comments, corrections, and other feedback are welcome.

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.

Seven videotaped lectures from 1964 by Richard Feynman given in Cornell, on “The Character of Physical Law“, have recently been put online (by Microsoft Research, through the purchase of these lectures from the Feynman estate by Bill Gates, as described in this interview with Gates), with a number of multimedia enhancements (external links, subtitles, etc.). These lectures, intended for a general audience, broadly cover the same type of material that is in his famous lectures on physics.

I have just finished the first lecture, describing the history and impact of the law of gravitation as a model example of a physical law; I had of course known of Feynman’s reputation as an outstandingly clear, passionate, and entertaining lecturer, but it is quite something else to see that lecturing style directly. The lectures are each about an hour long, but I recommend setting aside the time to view at least one of them, both for the substance of the lecture and for the presentation. His introduction to the first lecture is surprisingly poetic:

The artists of the Renaissance said that man’s main concern should be for man.

And yet, there are some other things of interest in this world: even the artist appreciates sunsets, and ocean waves, and the march of the stars across the heavens.

And there is some reason, then, to talk of other things sometimes.

As we look into these things, we get an aesthetic pleasure directly on observation, but there’s also a rhythm and pattern between the phenomena of nature, which isn’t apparent to the eye, but only to the eye of analysis.

And it’s these rhythms and patterns that we call physical laws.

What I want to talk about in this series of lectures is the general characteristics of these physical laws. …

The talk then shifts to the very concrete and specific topic of gravitation, though, as can be seen in this portion of the video:

Coincidentally, I covered some of the material in Feynman’s first lecture in my own talk on the cosmic distance ladder, though I was approaching the topic from a rather different angle, and with a less elegant presentation.

[*Via the New York Times*. Note that some browsers may need a Silverlight extension in order to view the videos. Youtube versions of three of the seven lectures are available here. Another, much more recent, series of videotaped lectures of Feynman is available here.]

[*Update*, July 15: Of particular interest to mathematicians is his second lecture “The relation of mathematics and physics”. He draws several important contrasts between the reasoning of physics and the axiomatic reasoning of formal, settled mathematics, of the type found in textbooks; but it is quite striking to me that the reasoning of *unsettled* mathematics – recent fields in which the precise axioms and theoretical framework has not yet been fully formalised and standardised – matches Feynman’s description of physical reasoning in many ways. I suspect that Feynman’s impressions of mathematics as performed by mathematicians in 1964 may differ a little from the way mathematics is performed today.]

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

In his final lecture, Prof. Margulis talked about some of the ideas around the theory of unipotent flows on homogeneous spaces, culminating in the orbit closure, equidsitribution, and measure classification theorems of Ratner in the subject. Margulis also discussed the application to metric theory of Diophantine approximation which was not covered in the preceding lecture.

Today, Prof. Margulis continued his lecture series, focusing on two specific examples of homogeneous dynamics applications to number theory, namely counting lattice points on algebraic varieties, and quantitative versions of the Oppenheim conjecture. (Due to lack of time, the third application mentioned in the previous lecture, namely metric theory of Diophantine approximation, was not covered.)

The final distinguished lecture series for the academic year here at UCLA is being given this week by Gregory Margulis, who is giving three lectures on “homogeneous dynamics and number theory”. In his first lecture, Prof. Margulis surveyed some classical problems in number theory that turn out, rather surprisingly, to have more or less equivalent counterparts in homogeneous dynamics – the theory of dynamical systems on homogeneous spaces .

As usual, any errors in this post are due to my transcription of the talk.

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