This is a (very much) belated update post from the beginning of March (2016).
I spent six weeks in February (2016) in New Haven. It was very nice to get back to Yale after more than two years. Here is a picture from the spectacular Yale’s art museum.
Micha Sageev’s construction. I got the gist of it (at last…)
A few years ago I wrote about amazing developments in low dimensional topology. Ian Agol proved the Thurston’s virtual Haken conjecture. The proof relies on earlier major works by Dani Wise and others (see this post) and an important ingredient was Micha Sageev work on cubical complexes arising from 3-manifolds. I remember thinking that as a mathematician in another field it is unrealistic for me to want to understand these developments in any detail (which is not an obstacle for writing about them), but that one day I want to understand Sageev’s construction. At Yale, Ian Agol gave three lectures on his proof, mentioned a combinatorial form of Sageev construction, and gave me some references. A few days after landing Dani Wise talked at our seminar and he explained to me at lunch how it goes in details. (BTW, a few days before landing Micha Sageev gave a colloquium at HUJI that I missed.) I have some notes, and it is not so difficult so stay tuned for some details in a future post!. (Update: I will need some refreshing for writing about it. But I can assure you that Micha Sageev’s construction is a beautiful combinatorial construction that we can understand and perhaps use.)
Unorthodox PNT (and even weak forms of RH).
Hee Oh talked at Yale about various analogs of the prime number theorem (PNT) and in some cases even of weak forms of the Riemann hypothesis for hyperbolic manifolds and for rational maps. This is based on a recent exciting paper by Oh and Dale Winter. Some results for hyperbolic manifolds were known before but moving to dynamics of rational maps is completely new and it follows a “dictionary” between hyperbolic manifolds and rational maps offered by Dennis Sullivan. It was nice to see in Oh’s lecture unexpected connections between the mathematical objects studied by three Yale mathematicians, Mandelbrot, Margulis, and Minsky. Are these results related to the real Riemann Hypothesis? I don’t know.
Weil conjectures (even for curves): from the very concrete to the very abstract,
Going well over my head I want to tell you about somethings I learned from two lectures. It is about Weil’s conjectures for curves including the “Riemann hypothesis for curves over finite fields.” One lecture is by Peter Sarnak and the other by Ravi Vakil. Both lectures were given twice (perhaps a little differently) in Jerusalem and at Yale few weeks apart. Peter Sarnak mentioned in a talk the very concrete proof by Stepanov to the Rieman hypothesis for curves over finite fields. The proof uses some sort of the polynomial method, and it is this concrete proof that is useful for a recent work on Markoff triples by Bourgain, Gamburd and Sarnak. Ravi Vakhil mentioned a very very abstract form of the conjecture or, more precisely, of the “rationality” part of it proved in 1960 by Bernard Dwork (yes, Cynthia’s father!). It started from an extremely abstract version offered by Kapranov in 2000 to which Larsen and Lunts found a counterexample. A certain weaker form of Kapranov’s conjecture that Ravi discussed might still be correct. (A related paper to Ravi’s talk is Discriminants in the Grothendieck ring by Ravi Vakil and Melanie Matchett Wood.) These very abstract forms of the conjectues are also extremely appealing and it is especially appealing to see the wide spectrum from the very concrete to the very abstract. It is also nice that both the polynomial method and the quest for rationality (of power series) are present also in combinatorics.
HD expanders, HD combinatorics, and more.
Alex Lubotzky and I gave a 6-weeks course on high dimensional expanders. This was the fourth time we gave a course on the subject and quite a lot have happened since we first taught a similar course some years ago so it was quite interesting to get back to the subject. I certainly plan to devote a few posts to HD-expanders and Ramanujan complexes at some point in the future.
Update: There will be a special semester on high-dimensional combinatorics and the Israeli Institute for Advanced Studies in Jerusalem in the academic year 2017/2018.
At the Simons Center in NYC Rafał Latała gave a beautiful lecture on the solution by Thomas Royen of the Gaussian correlation conjecture. Here is a review paper by Rafał Latała and Dariusz Matlak.
Update: Here is an article about this proof in Quanta magazine.
I also gave three lectures at Yale about polymath projects (at that time both Polymath10 and Polymath11 were active), and a special welcome lecture to fresh Ph. D. potential students POLYMATH and more – Mathematics over the Internet (click for the presentation) about polymath projects and mainly Polymath5, MathOverflow, mathematical aspects of Angry Birds and why they should all choose Yale.
An advantage of being 60.
One of repeated rather unpleasant dreams I had over the years (less so in the last decade) was that the Israeli army discovered that I still owe some months of service, and I find myself confusingly and inconveniently back in uniform. A few weeks (+ one year) ago I had a new variant of that dream appropriately scaled to my current age: MIT discovered that I still owe some months of my postdoctoral service. This was much more pleasant!