### Souvenirs from San Diego

Every time that I fly to a conference, I think about the airport puzzle that I once read in Terry Tao’s blog. Suppose that you are trying to get quickly from point A to point B in an airport, and that part of the way has moving walkways, and part of it doesn’t. Suppose that you can either walk or run, but you can only run for a certain small amount of the time. Where is it better to spend that amount of time running: on the moving walkways or in between the moving walkways? Does it matter?

Another question that continues to puzzle me (and to which I still don’t have a complete answer to) is: why do I continue to inflict upon myself the tortures of international travel, such as ten hour jet lag or trans-atlantic flights? More generally, I spent a lot of time wondering: why do I continue going to conferences? Is it worth it for me? Is it worth the university’s money? Is it worth it for mankind?

Last week I attended the Joint Mathematics Meeting in San-Diego. It was my first time in such a big conference. I will probably not return to such a conference for a while, since it is not so “cost effective”. I guess that I am a small workshop kind of person.

I spoke in and attended all the talks in the Free Convexity and Free Analysis special session, which was excellent. Here is the abstract and here are the slides of my talk (the slides).  I also attended some of the talks in the special sessions on Advances in Operator AlgebrasOperators on Function Spaces in One and Several Variables, and another one on Advances in Operator Theory, Operator Algebras, and Operator SemigroupsI also attended several plenary talks, which were all quite entertaining.

I am happy to report that the field of free analysis and free convexity is in really good shape! There was a sequence of talks in the first day (Hartz, Passer, Evert and Kriel) by three very young researchers on free convexity that really put me into high spirits! The field is blossoming and the competition is healthy and friendly. But the talk that got me most excited was the talk by Jim Agler, who gave a preliminary report on joint work with John McCarthy and Nicholas Young regarding noncommutative complex manifolds. Now, at first it might seem that nc manifolds will be hard to make sense of, because how can you take direct sums of points in a manifold, etc. Moreover, the only take on the free manifolds that I met before was Voiculescu’s construction of the free projective plane, which I found hard to swallow and kind of ruined my appetite for the subject.

However, it turns out that one can define a noncommutative complex manifold as topological space $X$ that carries an atlas of charts $(U,f)$ where $U$ is an open subset of $X$ and $f : \Omega \to U$ is a homeomorphism form an nc domain $\Omega$ onto $U$, such that given two intersecting charts $(U,f), (U',f')$, the map $f^{-1} \circ f'$ going from $f'(U \cap U')$ to $f(U \cap U')$ is an nc biholomorphism. This definition is so natural and clear that I want to shout! Agler went on and showed us how one can construct a noncommutative Riemann surface, for example the Riemann surface corresponding to the noncommutative square root function. How can one not want to hear more of this? I am looking forward very enthusiastically to see what Agler, McCarthy and Young are up to this time; it looks like a very promising direction to study.

Among the plenary talks that I attended (see here for description), the one given by Avi Wigderson struck me the most. I went to the talk simply for mathematical entertainment (a.k.a. to broaden my horizons), but I was very pleasantly surprised to find completely positive maps and free functions in a talk that was supposed to be about computational complexity. I went to the first two talks but missed the third one because I had an opportunity to have lunch with a friend and collaborator, which in any respect was more important to me than the lecture. The above link (here it is again) contains links to a tutorial and papers related to Wigderson’s talks, and I hope to find time to study that, and at least catch up on what I missed in the third talk.

One more thing: there was one quite eminent operator theorist who is long retired, and came to several of the sessions that I attended. At some point I noticed that after every talk a came up to the speaker and said several words of encouragement or advice. Seeing such a pure expression of kindness and love of humanity was touching and inspiring. Upon later reflection, I noticed that such expressions were happening around me all the time, for example when another “celebrity” in our field arrived and a hugging (!) session began. This memory brings a smile to my face. Well, maybe going to San-Diego was worth it, after all.