# This and That

Various things that may be of interest:

• MSRI in Berkeley has announced a \$70 million dollar gift from Jim and Marilyn Simons, and Henry and Marsha Laufer. This gift will make up the bulk of a planned endowment increase of \$100 million and is the largest endowment gift ever made to a US-based math institute. The success of the Renaissance Technologies hedge fund is what has made gifts on this scale possible. This summer MSRI will be renamed the “Simons Laufer Mathematical Sciences Institute”, and the directorship will pass from David Eisenbud to Tatiana Toro.
• The journal Inference has just published an article by Daniel Jassby, which gives a highly discouraging view of the prospects for magnetic confinement fusion devices. Jassby, who worked for many years at the Princeton Plasma Physics Lab, argues that performance of magnetic confinement fusion systems has not much advanced in a quarter century, making for very bleak prospects that such designs will lead to a workable power plant in the forseeable future. He sees inertial confinement fusion systems like the National Ignition Facility at Livermore as making some progress, but ends with:

The technological hurdles for implementing an ICF-based power system are so numerous and formidable that many decades will be required to resolve them—if they can indeed be overcome.

• I’ve been spending some time reading Grothendieck’s Récoltes et Semailles, which is a simultaneously fascinating and frustrating experience. I’ve made it almost to the end of the first part, except that there will be another forty pages or so of notes to go. To get to the first part involved starting by reading through about two hundred pages of four layers of introduction. It seems that basically Grothendieck did no editing. Once he was done writing the first part, as he thought of more to say he’d add notes. He distributed copies to various other mathematicians, and then kept adding new introductions, with various references to how this fit in with more technical mathematical documents he was working on (La “Longue Marche” à Travers la Théorie de Galois, À la poursuite des champs).

After the first part, looking ahead there’s the daunting prospect of 1500 pages with the theme of examining his deepest mathematical ideas and what he felt was the “burial” that he and his ideas had been subjected to after his leaving active involvement with the math research community in 1970. Quite a few years ago I did spend some time looking through this part to try and learn more about Grothendieck’s mathematical ideas. I’ll see if I can try again, with the advantage of now knowing somewhat more about the mathematical background.

Besides the frustrating aspects, what has struck me most about this is that there are many beautifully written sections, capturing Grothendieck’s feeling for the beauty of the deepest ideas in mathematics. One gets to see what it looked like from the inside to a genius as he worked, often together with others, on a project that revolutionized how we think about mathematics. This material is really remarkable, although embedded in far too much that is extraneous and repetitive. The text desperately needs an editor.

There are various places online one can find parts of the book and other related material, sometimes translated. Two places to look are the Grothendieck Circle, and Mateo Carmona’s site.

• For an up-to-date project on reworking foundations of mathematics (with an eye to eliminating analysis…), Dustin Clausen and Peter Scholze are now teaching a course on Condensed Mathematics and Complex Geometry, lecture notes here.
• I noticed that the Harvard math department website now has an article on Demystifying Math 55. The past couple years this course has been taught by Denis Auroux, and one can find detailed course materials including lecture notes at his website.

The current version of the course tries to cover pretty much a standard undergraduate pure math curriculum in two semesters, with the first semester linear algebra, group theory and finite group representations, the second real and complex analysis. The course has gone through various incarnations over a long history, and has its own Wikipedia page. For various articles written about the course over the years, see here, here (about a Pavel Etingof version) and here (about a Dennis Gaitsgory version).

I took the course in 1975-76, when the fall semester was taught by mathematical physicist Konrad Osterwalder, who covered some linear algebra and analysis rigorously, following the course textbook Advanced Calculus by Loomis and Sternberg. The spring semester was rather different, with John Hubbard sometimes following Hirsch and Smale, sometimes giving us research-level papers about dynamical systems to read, and then telling us to read and work through Spivak’s Calculus on Manifolds over reading period.

My experience with the course was somewhat different than that described in the articles above, partly due to the particular instructors and their choices, partly due to the fact that I was more focused on learning as much advanced physics as possible. I don’t remember spending excessive amounts of time on the course, nor do I remember anyone I knew or ran into being especially interested in or impressed by my taking this particular course. What was a new experience was that it was clear the first semester that I was a rather average student in the class, not like in my high school classes. The second semester about half the students had dropped and I guess I was probably distinctly less than average. The current iteration of the course looks quite good for the kind of ambitious math student it is aimed at, and it would be interesting if a new textbook ever gets written.

Update: One more related item. This week Chapman University is hosting a conference about Grothendieck. Kevin Buzzard has posted his slides here.

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### 21 Responses to This and That

1. Analyst says:

What do Clausen and Scholze’s notes have to do with “reworking the foundations of mathematics,” much less about eliminating analysis therefrom? I’m not familiar with their work but their stated goal is just to develop some of the basic theory of the constructions from their research papers and to illustrate it by proving some old theorems.

2. Peter Woit says:

Analyst,
My reading of what Clausen/Scholze are trying to do is revamp the foundations of complex analysis/complex geometry so the subject is not based on analysis (e.g. differentiable functions), but on algebra. They explicitly advertise that their way of proving basic theorems of the subject is “analysis-free”. A motivation for this is arithmetic geometry, where Scholze would like to prove things like real local Langlands by the same p-adic methods used at finite primes:
“Part of our goal is to develop foundations for analytic geometry that treat archimedean and non-archimedean geometry on equal grounds; and we will proceed by making archimedean geometry more similar to non-archimedean geometry.

3. Joseph Healy says:

I think one thing missing from Daniel Jassby’s commentary is a current trend in managing tokamak plasma in real-time with AI.
.
https://www.nature.com/articles/s41586-021-04301-9

4. anonymous says:

That’s not an accurate characterization of the notes. They aren’t revamping “the foundations of complex analysis/complex geometry,” they’re just proving some core theorems about compact complex manifolds. (Note that complex analysis and complex geometry encompass far more than compact manifolds…) Further, it’s no surprise this can be done algebraically, because GAGA tells you that compact complex manifolds are equivalent to complex algebraic varieties. One could already prove all of these things from an algebraic perspective. The point of the notes seems to be to give new proofs along these lines that illustrate their new techniques.

This is an interesting project (especially as a way to make their techniques more accessible), but I feel it is inappropriate to claim it is “reworking foundations of mathematics (with an eye to eliminating analysis…).” There’s nothing about foundations in the notes, and the fact that these theorems can be “algebra-ized” is known to any PhD student in complex geometry.

5. mls says:

@ anonymous

Dr. Woit has a history of misrepresenting anything in mathematics and related subjects which does not conform to his personal beliefs. This has been especially true with every mention of “foundations.”

At least the last paragraph of his present post includes an honest history explaining his profound ignorance of mathematics.

6. Will Sawin says:

I’m going to continue the trend of every commenter having a slightly different interpretation. In particular, I think one should look at this in the context of Scholze’s previous work in condensed mathematics.

I think it’s important to begin with the observation that in the theory of complex manifolds one often wants to use analytic methods and one also often wants to use algebraic methods potentially involving derived categories and such. A basic concern is that, in some future arguments, one could run into trouble when these two strands don’t weave together properly.

For example, if one wants to keep track of size of something using Banach spaces, and also use derived categories, one could be stymied by the fact that Banach spaces do not form an abelian category, and therefore can’t be used to construct a derived category. Scholze’s earlier work on condensed, liquid, and solid vector spaces provides a fix for this by defining an abelian category that includes Banach spaces.

I believe Scholze’s work in this course is intended to provide further tools on the same lines that are potentially useful in future research. Specifically this is for research that already intersects analysis and algebra – I don’t think there is any intention to replace analysis in the proof of existence of solutions of some PDE or the prime number theorem or something like that. But it’s not specific to the point that there is some Weil-conjectures-like goal in mind.

@anonymous

> because GAGA tells you that compact complex manifolds are equivalent to complex algebraic varieties.

This is not true at all. It only tells you that, for a manifold that is already algebraic, studying it using analytic and algebraic tools will give you the same answers (the same compact submanifolds, the same cohomology, …)

> One could already prove all of these things from an algebraic perspectiv

This seems like a silly point of view when one of the things being proven in the notes is GAGA, i.e. exactly the bridge that links the algebraic and analytic perspectives. One can’t use GAGA to prove GAGA.

According to Laurent Fargues, one of the goals is the Hodge decomposition. Do you consider this something that can be proven entirely algebraically?

7. Johan says:

If I remember correctly, Columbia Library has one of the original copies of Récoltes et Semailles (sent by Grothendieck to Sammy Eilenberg), no? It may even be autographed, in any case it really belongs in the rare books collection.

8. Jonathan Chiche says:

I believe most, if not all, original copies of “Récoltes et semailles” are inscribed by Grothendieck. In 2008/2009 I read one of the two copies owned by Paris N university (with N=7 probably). One of them was inscribed by Grothendieck to Faltings, the other to Leray. If my memory serves me right, the latter had a note written by Leray pointing to an error in the text (regarding himself), and one of the two copies (I cannot recall which) was incomplete pf roughly one half. It would be interesting to have a list of institutionally and privately owned original copies, together with the text of Grothendieck’s dedication. I disagree that the text needs an editor and remember it as a great reading for an aspiring mathematician.

9. Dustin Clausen says:

For what it’s worth, I agree with what Will Sawin says (minus the attribution of everything solely to Scholze, of course!). I never thought of the goal of condensed math, or this approach to analytic geometry, as being to eliminate analysis. Actually, a lot of analysis shows up in the foundations. Rather, the goal is to put parts of analysis and topology in a new framework, one which allows to mix more easily with algebra. Then we can make formal arguments of a certain algebraic style, which however lead to analytic and topological conclusions.

In the example of the theorems we aim to reprove in the course, this means that the analysis is black-boxed into some foundational material about liquid vector spaces, and then the rest of the argument is in a sense purely algebraic. But I don’t really think of that as eliminating analysis. In fact, in my view it is neither desirable, nor even possible, to elminate analysis from the study of complex geometry!

10. Peter Woit says:

mls,
There was a small element in my post of trolling of analysts/foundations of math aficionados. I won’t really apologize since it was kind of successful, and in the spirit of Clausen/Scholze’s claim to be making the subject “analysis-free”. I see that Clausen has written in here to clarify, which is great.

There is some connection of this to my math education experiences at Harvard. Doing about average in a Math 55 class that had a bunch of the top Math Olympiad performers in it even though I wasn’t putting much time into it (much more of my time was going into the quantum mechanics class I was taking) wasn’t such a bad performance. More relevant, the next math class I took was a graduate course in analysis, taught by Andrew Gleason. This course extensively covered set theory, point set topology and measure theory, and had us spending lots of time puzzling out questions like whether a space was $T_{2\frac{1}{2}}$. Around the same time I also took a set theory course from Quine (I did pretty well in both from what I recall).

In retrospect taking these courses (Gleason and Quine) was a big mistake and a waste of time, caused by my idea that, both in physics and in math, what I should be doing was focusing on learning the “foundations”, from which understanding of everything else would flow. I should have been taking other courses which were not “foundational”, but would have taught me about some of the great unifying concepts that bring together a wide range of beautiful mathematical structures (as well as physics!). Someone should have told me to take representation theory…

11. Łukasz says:

@Analyst @Peter Woit

In my opinion, the phrase “reworking the foundations of mathematics,” connotes rather research in mathematical logic or set theory. For example, classical Arithmetic is based on classical first-order logic (some people prefer to tell in this context, about classical functional calculus).

12. Peter Woit says:

Łukasz,
I did clarify this later by specifying “foundations of complex analysis/complex geometry” and note that Clausen/Scholze talk about “a new foundation for combining algebra and topology”. I’m well aware that many people identify “foundations of mathematics” with mathematical logic/set theory. That identification led to the misguided educational experiences of my youth that I explained, so I rather intentionally used the term to refer to a broader (and, if you ask me, more interesting) set of issues.

13. Will Sawin says:

@Dustin Clausen

Sorry about that! Maybe next time I will make up for it by crediting it entirely to you and not at all to Scholze…

14. Richard says:

What does “GAGA” stand for?

15. Peter Woit says:

Richard,
For most people, a well-known performer. For some mathematicians, more relevant is
https://en.wikipedia.org/wiki/Algebraic_geometry_and_analytic_geometry

16. Richard says:

Peter Woit,

For * me * , it was what I was going trying to figure out what it means!
Thanks for the reference! [For the linklazy among us, “GAGA” refers to “Geometrie Algebrique et Geometrie Analytique”, a foundational paper in algebraic geometry published by the redoubtable Jean-Pierre Serre in 1956 (Full disclosure: my personal knowledge of such things is a trifle on the nonexistent side.).]

17. David Brown says:

Jassby’s 2018 article might be worth studying:
https://thebulletin.org/2018/02/iter-is-a-showcase-for-the-drawbacks-of-fusion-energy/
In any research, the yea-sayers who are overly optimistic might have a strong tendency to drive out the nay-sayers — because cash and career advancement are at stake.

18. martibal says:

And for those who do not speak french, let us not forget that “gaga” in popular langage means “senile” (more or less, maybe Peter would have a more accurate translation), and may also be for “to have a crush on someone”

19. Paul D. says:

The big problem with these DT fusion schemes is that the volumetric power density is just terrible. An existing PWR fission reactor’s primary pressure vessel might have a power density of 20 MW/m^3; the volumetric gross fusion power density of ITER is 0.05 MW/m^3. It’s very difficult to see how DT fusion can ever be cheaper than fission, given that the reactor itself will be at least an order of magnitude larger (and much more complex).

Note that this problem has nothing to do with plasma physics, but rather is due to limits on heat and radiation transfer at the wall of the reactor vessel and the square-cube law. Totally solve plasma confinement and the problem is still there, at least for reactors burning DT.

This issue has been known for approaching 40 years, if not longer. I find it incredible the press is still spouting glowing nonsense about DT fusion.