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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 ${\Bbb R}^n$“, in the spirit of the classical Whitney extension theorem, except that now one is considering much more quantitative and computational extension problems (in particular, viewing the problem from a theoretical computer science perspective). Today Charlie introduced the basic problems in this subject, and stated some of the results of his joint work with Bo’az Klartag; he will continue the lectures on Thursday and Friday.

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.

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 $g \geq 2$, and in particular focusing on some recent work of his which is related to the effective Mordell conjecture and the abc conjecture. The higher genus case is substantially more difficult than the genus 0 or genus 1 cases, and one often needs to use techniques from many different areas of mathematics (together with one or two unproven conjectures) to get somewhere.

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.