Noncommutative Analysis

Souvenirs from Amsterdam

(I am writing a post on hot trends in mathematics in the midst of war, completely ignoring it. This seems like the wrong thing to do, but my urge to write has overcome me. To any reader of this blog: I wish you a peaceful night, wherever you are).

Last week I returned from the yearly “International Workshop on Operator Theory and Applications”, IWOTA 2014 for short (see the previous post for the topic of my own talk, or this link for the slides).

This conference was very broad (and IWOTA always is). One nice thing about broad conferences is that you are able sometimes to identify a growing trend. In this talk I got particularly excited by a series of talks on “noncommutative function theory” or “free analysis”. There was a special session dedicated to this topic, but I was mostly inspired by a semi-plenary talk by Jim Agler, and also by two interesting talks by Joe Ball and Spela Spenko. I also attended nice talks related to this subject by Victor Vinnikov, Dmitry Kalyuhzni-Verbovetskyi, Baruch Solel, Igor Klep and Bill Helton. This topic has attracted the attention of many operator theorists, for its applications as well as for its inherent beauty, and seems to be accelerating in the last several years; I will only try to give a taste of some neat things that are going on, by telling you about Agler’s talk. What I will not be able to do is to convey Agler’s intense and unique mathematical charisma.

Here is the program of the conference, so you can check out other things that were going on there.

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The isomorphism problem: update

Ken Davidson, Chris Ramsey and I recently uploaded a new version of our paper “Operator algebras for analytic varieties” to the arxiv. This is the second paper that was affected by a discovery of a mistake in the literature, which I told about in the previous post. Luckily, we were able to save all the results in that paper, but had to work a a little harder than what we thought was needed in our earlier version. The isomorphism problem for complete Pick algebras (which I like to call simply “the isomorphism problem”) has been one of my favorite problems during the last five years. I wrote four papers on this problem, with five co-authors. I want to give a short road-map to my work on this problem. Before I do so, here is  link to the talk that I will give in IWOTA 2014 about this stuff. I think (hope) this talk is a good introduction to the subject. The problem is about the classification of a large class of non-selfadjoint operator algebras – multiplier algebras of complete Pick spaces – which can also be realized as certain algebras of functions on analytic varieties. These algebras all have the form

M_V = Mult(H^2_d)\big|_V

where V is a subvariety of the unit ball and Mult(H^2_d)  denotes the multiplier algebra of Drury-Arveson space (see this survey), and therefore M_V is the space of all restrictions of multipliers to V. The hope is to show that the geometry of the variety V is a complete invariant for the algebras M_V, in various senses that will be made precise below.

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An old mistake and a new version (or: Hilbert, Poincare, and us)

[Update June 28, 2014: This post originally included stories about Poincare and Hilbert making some mistakes. At some point after posting this I realised how unfair it is to talk about somebody else's mistake (even if it is Hilbert and Poincare) without giving precise references. Instead of deleting the stories, I'll insert some comments where I think I am unfair. Sorry!]

I was recently forced to reflect on mistakes in mathematics. The reason was that my collaborators and I discovered a mistake in an old paper (16 years old), which forced us to make a significant revision to two of our papers.

A young student of mathematics may consider a paper which contains a mistake to be a complete disaster. (By “mistake” I don’t mean a gap – some step that is not sufficiently well justified (where “sufficiently well” can be a source of great controversy). By “mistake” I mean a false claim). But it turns out that mistakes are inevitable. A paper that contains a mistake is a terrible headache, indeed, but not a disaster.

Arveson once told me: “Everybody makes mistakes. And I mean EVERYBODY”. And he was right. There are two well known stories about Hilbert and Poincare which I’d like to repeat for the reader’s entertainment, and also to make myself feel better before telling you about the mistake my collaborators and I overlooked.

First story: [I think I first read the story about Hilbert in Rota's "Ten Lesson's I wish I had been Taught" (lesson 6)]: When a new set of Hilbert’s collected papers was prepared (for his birthday, the story tells), it was discovered that the papers were full of mistakes and could not be published as they were. A young and promising mathematician (Olga Taussky-Todd) worked for three years to correct (almost) all the mistakes. Finally, when the new volume of collected (and corrected) papers was presented to Hilbert, he did not notice any change. What is the moral here? One moral, I suppose, is that even Hilbert made mistakes (hence we are all allowed to). The second is that many mistakes — say, the type of mistakes Hilbert would make — are not fatal: if the mistakes are planted in healthy garden, they can be weeded out and replaced by true alternatives, often-times leaving the important corollaries intact.

[Update June 28: A reference to Rota's "10 Lessons" is not good enough, and neither is reference to the Wikipedia article on Olga Taussky-Todd, which in turn references Rota's "Indiscrete Thoughts", where "10 Lessons" appear.]

Second story: actually two stories, about Poincare.  Poincare made two very important mistakes! First mistake: in 1888 Poincare submitted a manuscript to Acta Mathematica – as part of a competition in honour of the King of Norway and Sweden – in which, among other things (for example inventing the field of dynamical systems), he claimed that the solutions of the 3-body problem (restricted to the plane) are stable (meaning roughly that the inhabitants of a solar system with a sun and two planets can rest assured that the planets in their solar system will continue orbiting more or less as they do forever, without collapsing to the sun or diverging to infinity). After winning the competition, and after the paper was published (and probably in part due to the assistant editor of Acta, Edvard Phragmen, asking Poincare for numerous clarifications during the editorial process), Poincare discovered that his manuscript had a serious error in it. Poincare corrected his mistake, inventing Chaos while he was at it.

[Update June 28: This story is well documented. I learned it from Donal Oshea's book "The Poincare Conjecture: In Search of the Shape of the Universe" , but it is easy to find online references, too]. 

Second mistake: In 1900 Poincare claimed that if the homology of a compact 3 manifold is trivial, then it is homeomorphic to a sphere. He himself found out his mistake, and provided a counterexample. In order to show that his example is indeed a counter example he had to invent a new topological invariant: the fundamental group. He computed the fundamental group of his example and saw that it is different from the one of the sphere. But this led him to ask: if a closed manifold has a trivial fundamental group, must it be homeomorphic to the 3-sphere? This is known as the Poincare conjecture, of course, and the rest is history.

[Update June 28: Here I should have given a reference of where exactly Poincare claimed that trivial homology implies a space is a sphere. I don't know it (it probably also appears in Oshea's book)]

The moral here? I don’t know. But it is nice to add that after making his first mistake, Poincare and Mittag-Leffler (the editor) set a good example by recalling all published editions and replacing them with a new and correct version.

So that’s what I’ll try to imitate now.

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Daniel Spielman talks at HUJI – thoughts

I got an announcement in the email about the “Erdos Lectures”, that will be given by Daniel Spielman in the Hebrew University of Jerusalem next week (here is the poster on Gil Kalai’s blog). The title of the first lecture is “The solution of the Kadison-Singer problem”. Recall that not long ago Markus, Spielman and Srivastava proved Weaver’s KS2 conjecture, which implies a positive solution to Kadison-Singer (the full story been worked out to expository perfection on Tao’s blog).

My immediate response to this invitation was to start planning a trip to Jerusalem on Monday – after all it is not that far, it’s about a solution of a decades old problem, and Daniel Spielman is sort of a Fields medalist. I highly recommend to everyone to go hear great scientists live whenever they have the opportunity. At worst, their lectures are “just” inspiring. It is not for the mathematics that one goes for in these talks, but for all the stuff that goes around mathematics (George Mostow’s unusual colloquium given at BGU on May 2013 comes to mind).

But then I remembered that I have some obligations on Monday, so I searched and found a lecture by Daniel Spielman with the same title online: here. Watching the slides with Spielman’s voice is not as inspiring as hearing and seeing a great mathematician live, but quite good. He makes it look so easy!

In fact, Spielman does not discuss KS at all. He says (about a minute into the talk) “Actually, I don’t understand, really, the Kadison-Singer problem”. A minute later he has a slide where the problem is written down, but he says “let me not explain what it is”, and sends the audience to read Nick Harvey’s survey paper (which is indeed very nice). These were off-hand remarks, and I should not catch someone at his spoken word, (and I am sure that even things that Spielman would humbly claim to “not understand, really”, he probably understands as well as I do, at least), but the naturality in which the KS problem was pushed aside in a talk about KS made we wonder.

In the post I put up soon after appearance of the paper I wrote (referring to the new proof of KS2) that “… this looks like a very nice celebration of the Unity of Mathematics”. I think that in a sense the opposite is also true. I will try to reformulate what I wrote.

“The solution of KS is a beautiful and intriguing manifestation of the chaotic, sticky, psychedelic, thickly interwoven, tangled, scattered, shattered and diffuse structure of today’s mathematics.”

I don’t mean that in a bad way. I mean that a bunch of deep conjectures, from different fields, most of which, I am guessing, MSS were not worried about, were shown over several decades to be equivalent to each other, and were ultimately reduced (by Weaver) to a problem on the arrangement of vectors in finite dimensional spaces (Discrepancy Theory), and eventually solved, following years of hard work, by three brilliant mathematicians using ingenious yet mostly elementary tools. The problem solved is indeed interesting in itself, and the proof is also very interesting, but it seems that the connection with “Kadison-Singer” is more a trophy than a true reward.

It would be very interesting now to think of all the equivalent formulations with hindsight, and seek the unifying structure, and to try to glean a reward.

 

 

Interesting figure

I found an interesting figure in the March 2014 issue of the EMS newsletter, from the article by H. Mihaljevic´ -Brandt and O. Teschke, Journal Profiles and Beyond: What Makes a Mathematics Journal “General”?

See the right column on page 56 in this link. (God help me, I have no idea how to embed that figure in the post. Anyway, maybe it is illegal, so I don’t bother learning.) One can see the “subject bias” of Acta, Annals and Inventiones.

On the left column, there is a graph showing the percentage of papers devoted to different MSC subjects in what the authors call “generalist” math journals (note carefully that these journals are only a small subclass of all journals, chosen by a method that is loosely described in the article). On the right column there is the interesting figure, showing the subject bias. If I understand correctly, the Y-axis is the MSC number and the X-axis represents the corresponding deviation from the average percentage given in the left figure. So, for example, Operator Theory (MSC 47) is the subject of about 5 percent of the papers in a generalist journal, but in the Annals there is a deviation of minus 4 from the average, so if I understand this figure correctly, that means that about 1 percent of papers in the Annals are classified under MSC 47. Another example: Algebraic Geometry (MSC 14), takes up a significant portion of Inventiones papers, much more than it does in an average “generalist” journal.

(I am not making any claims, this could mean a lot of things and it could mean nothing. But it is definitely interesting to note.)

Another interesting point is that the authors say that of the above three super-journals, Acta “is closest to the average distribution, though it is sometimes considered as a journal with a focus on analysis”. That’s interesting in several ways.

 

The dominated convergence theorem for the Riemann and the improper Riemann integral (Measure theory is a must – part II)

(Hello students of Infi 2 – this post is for you).

In this post I will describe the dominated convergence theorem (DCT) for the Riemann and improper Riemann integrals. The previous post can serve as an introduction (a slanted one, beware) to this one. My goal is to convince that the important and useful convergence theorems in integration theory can (and therefore, needless to say, should) be taught in a first course on Riemannian integration.

The bounded convergence theorem for the Riemann integral is also known as Arzela’s Theorem, and this post does not contain anything new. In preparing this post I used as reference the short note “A truly elementary approach to the bounded convergence theorem”, J. W. Lewin, The American Mathematical Monthly. This post can be considered as a destreamlinization of that note. I think my presentation is even more “truly elementary”, since I avoid introducing inner measure. Warning: this post will really truly be at a very elementary level. Read the rest of this entry »

Souvenirs from the Black Forest

Last week I attended a workshop titled “Hilbert modules and complex geometry” in MFO (Oberwolfach). In this post I wish to tell about some interesting things that I have learned. There were many great talks to choose from. Below is a sample, in short form, with links.

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Measure theory is a must

[This post started out as an introduction to a post I was planning to write on convergence theorems for the Riemann integral. The introduction kind of got out hand, so I decided to post it separately. Since I have to get back to my real work, I will postpone writing that post on convergence theorems for the Riemann integral for another time, probably during the Passover break (but in any case before we need them for the course I am teaching this term, Calculus 2)].

 

Mathematicians love to argue about subjective opinions. One of the most tiresome and depressing subjects of debate is “What should an undergraduate math major curriculum contain?”

 

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K-spectral sets and the holomorphic functional calculus

In two previous posts I discussed the holomorphic functional calculus as part of a standard course in functional analysis (lectures notes 18 and 19). In this post I wish to discuss a slightly different approach, which relies also on the notion of K-spectral sets, and relies a little less on contour integration of Banach-space valued functions.

In my very personal opinion this approach is a little more natural then the standard one, and it would be even more natural if one was able to altogether remove the dependence on Banach-space valued integrals (unfortunately, right now I don’t know how to do this completely).

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Advanced Analysis, Notes 19: The holomorphic functional calculus II (definition and basic properties)

In this post we continue our discussion of the holomorphic functional calculus for elements of a Banach algebra (or operators). The beginning of this discussion can be found in Notes 18. Read the rest of this entry »

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