A few items of various kinds:
- A little while ago I did another podcast, this time with Hrvoje Kukina. The result is now available here.
- There’s a new French documentary out, available here, about the story of the campaign by a committee of mathematicians in the 1970s to get the Ukrainian mathematician Leonid Pliouchtch out of hands of the KGB. It’s directed by Mathieu Schwartz, whose great-uncle was Laurent Schwartz, one of the main figures in this story (see here). Another member of the committee, Michel Broué, also appears in the film.
One of the issues discussed in the film is how mathematicians could have pulled this off, and whether the devotion of mathematicians to rigorous truth makes them more likely to take a stand on principle on an issue like this (Cédric Villani is interviewed, and takes the position that mathematicians aren’t much different than others). Another aspect of this story is that it may have been influential by making more people on the French left aware of the true nature of the Soviet system, making cooperation between different leftist parties more possible. For more about this aspect of the film, there’s a debate here.
- Source Code is a new book just out, an autobiography of Bill Gates, dealing with his early years, up to the time Microsoft moved to Seattle in early 1979. An important theme of the book is the importance to Gates of mathematics during those early years:
Realizing early on that I had a head for math was a critical step in my story. In his terrific book How Not to Be Wrong, mathematician Jordan Ellenberg observes that “knowing mathematics is like wearing a pair of X-ray specs that reveal hidden structures underneath the messy and chaotic surface of the world.” Those X-ray specs helped me identify the order underlying the chaos, and reinforced my sense that the correct answer was always out there–I just needed to find it. That insight came at one of the most formative times of a kid’s life, when the brain is transforming into a more specialized and efficient tool. Facility with numbers gave me confidence, and even a sense of security.
There’s quite a lot about his years as a student at Harvard, especially about the freshman-year Math 55 class he took, which was taught by John Mather. This brought back a lot of memories for me of my experiences there a couple years later. Gates arrived as a freshman in the fall of 1973, which was two years before me. Something we had in common was not being the best students in Math 55, but somewhere in the middle. Our reactions to that however were very different, since Gates was extremely competitive:
In our Math 55 study sessions, even as we were helping each other, we were also subtly keeping score. That was true in our broader circle of math nerds as well. Everyone knew how everyone else was doing, for instance, that Lloyd in Wigg B aced a Math 21a test or that Peter–or was it someone else?–found an error in Mather’s notes. We all grasped who among us was quicker that day, sharper, the person who “got it” first and then could lead the rest of us to the answer. Every day you strived to be on top. By the end of the first semester, I realized that my ranking in the hierarchy wasn’t what I had hoped…
By most measures I was doing well. I earned a B+ in the first semester which was an achievement in that class. In my stark view however it was less of a measure of what I knew than how much I didn’t. The gap between B+ and A was the difference between being the top person in the class and being a fake…
I was recognizing that while I had an excellent math brain, I didn’t have the gift of insight that sets apart the best mathematicians. I had talent but not the ability to make fundamental discoveries.
In the book, Gates then explains how he ended up concentrating most of his effort on computer-related projects and describes those in detail. Other sources say he took the graduate course Math 250a from John Tate the spring semester of his sophomore year, but he doesn’t mention that. By that time he mostly wasn’t attending classes, getting by on cramming for finals, while spending all his time writing a BASIC compiler with Paul Allen, then heading out to Albuquerque to start Microsoft. The semester I arrived at Harvard (fall 1975) he was technically a student, but spending most of his time working for Microsoft, finally leaving Harvard halfway through his junior year.
- The KITP in Santa Barbara is now running a “What is Particle Theory?” program, talks here. Among the talks, one I can recommend is Simon Catterall’s Sneaking up on lattice chiral fermions, especially for its focus on what are called Kahler-Dirac fermions.
Update: The Math 55 textbook used during those years is available here.
Update: Looking through some old files, I see that I got a Math 55 grade of B in the fall, B+ in the spring. So, competitive with the likes of Bill Gates, but not with the best students in the class. My memory of the class and the significance of my grade is very different than his. I don’t remember being very aware of how other students in the class were doing, other than that there were a few of them sitting in the front row who had won Math Olympiads and the like, were clearly understanding things faster and better than I was. I also wasn’t that interested in how I was doing, being an average student in the class was fine with me. The main thing was to be learning as much math as fast as possible, and for that Math 55 was the perfect course.
Exactly what did Math 55 cover in that book? I was at Harvard as a grad student
(in Chemistry) fall 1966 to Jan 1971.
Doug McDonald,
The first semester was taught by Konrad Osterwalder, who followed the book closely from the beginning. We got through something like the first four chapters.
The second semester was taught by John Hubbard, who didn’t follow the book. A lot of what he did was dynamical systems, sometimes following Hirsch and Smale. I don’t remember much else, except that at the end of the course the way he dealt with manifolds and differential forms was to tell us to read Spivak’s Calculus on Manifolds during the reading period between classes and finals.
This was not the point in my life at which I understood differential forms…
I think Math55 is single handedly responsible for discouraging some of the best students away from mathematics. It teaches exactly the wrong lessons about what research mathematics, or creative persistence is about. Arguably one of the gifts of being a beginner is being able to build your own personal mental model of how things hang together; and this subjective perspective likely plays a role in achieving breakthrough results. It is also the power of having things “warm in your memory”. Now if this beginner has a speed-reading like tour of basically most of undergraduate (& then some) math, there isn’t much of a chance to deepen intuition, or bathe in it long enough to allow any kind of creative insight to strike. Grothendieck’s remarks in Recoltes et Semailles about not being able to perform tricks and learn fast like the other students resonate in this regard.
One mustn’t just learn mathematics. One must build it from the ground up for herself.
It is also true that many students of Math55 go on to do very well. They were doing well before it; and I suspect would’ve done well without it as well.
Deniz,
My impression is that the Harvard department long ago realized that there were problems with this as a course for all beginning mathematicians, and has made quite a few changes since the 70s. There’s an obvious problem with a course that can shake the self-confidence of someone like Bill Gates and convince them not to be a math major.
Personally, it worked for me for various reasons, but for many people it’s the wrong course. Furthermore, the course depended a lot on who was teaching it. My year the two semesters were very different experiences due to two very different instructors. Here’s something from Dick Gross (a truly first-rate research mathematician) about his experience (https://celebratio.org/Gross_BH/article/1096/)
Dick enrolled at Harvard in the fall of 1967, expecting to be a math major. He tried to take the honors freshman course Math 55, which was taught that year by Loomis and Sternberg out of their book. His roommate loved the text, where he said that the only numbers in it were subscripts. But Dick found the course way above his level.
“I didn’t know the theory of finite dimensional vector spaces and the book started with topological vector spaces! So I dropped the course, took multivariable calculus, and decided to become a physics major. In my sophomore year I was looking for an extra course and wandered into a room where Andrew Gleason was teaching Math 55. His style of lecturing was just entrancing and I enrolled in his course. Many times in my research career when I understood something fundamental, like exterior powers as functors on the category of finite dimensional vector spaces, I realized that it was something Andy was trying to teach us in Math 55.”
Looking at the book, saying it started with topological vector spaces is a bit of a slur (no topology), but it did start with a very abstract discussion.
https://en.wikipedia.org/wiki/Leonid_Plyushch
https://fr.wikipedia.org/wiki/Léonid_Pliouchtch
Good news: I was forewarned against Math 55. Bad news: I took 212 (Real Analysis taught by Loomis) and 213 (Complex Analysis taught by Ahlfors) instead. Only much later when teaching those subjects did I realize how little I had understood them as a student. Fortunately, most institutions didn’t recognize that a B was failing grade in a graduate course.
‘…there were a few of them sitting in the front row who had won Math Olympiads and the like,’
Do you have any idea what those students have become now? Superstars of the field?
Curious,
I don’t remember much about people in that class, but don’t think any became superstars of the field. Doing a little bit of research (list of 1974 MO winners), I’m pretty sure Stephen Modzelewski was in that class, possibly Paul Zeitz. Modzelewski went into finance, applying math to the bond market. Zeitz did go on to become a math professor, see
https://www.thegreatcoursesplus.com/paul-zeitz
This is interesting. In physics education research there is a lot of work done on how to teach introductory physics (especially to mathematically challenged students) but there is very little work published on how best to teach upper level courses to students who want to be physicists. Having taken and now taught physics for upper class undergrads, I expect nothing has changed since the mid 20th century. It sounds like the same is true for upper level math courses. If you were to change how these courses were taught, what would you do?
Amit,
That’s a large topic I’m not really competent to address. I’ve only taught a few different upper level math courses, and no physics. In general, it’s remarkable how the nominal content of these courses stays fixed, not evolving much over the years. The more leeway instructors have to teach their own perspective the better. From my own perspective, the world could use a lot more physics courses incorporating more math, math courses incorporating more physics.
On the topic of upper-level classes, the situation when I was a student at Harvard and Princeton was pretty weird. At Harvard, after getting through Math 55, students wouldn’t take upper-level undergrad courses, would go directly into graduate classes. When I got to Princeton as a physics grad student, the situation in the math department was that they assumed all their incoming grad students had already taken grad courses at their undergrad institution, so basically didn’t teach any grad courses (just research seminars). This wasn’t ideal for me, since while I had taken a couple math grad courses as a physics undergrad, I had a lot more to learn and would have enthusiastically taken a couple more math grad courses during my physics Ph.D. if that had been possible.
Peter, OT, but curious to know your comments about Sabine’s latest video.
Shantanu,
My views are in many ways quite different than Sabine’s, but I very much identify with her choosing to rant obscenely about the intellectual and moral collapse of a large part of the field of theoretical fundamental physics.
As for those who are upset that she’s getting a lot of attention and threatening their funding, I can’t help but wonder where they were the past few decades as the public was fed endless outrageous hype about fundamental physics. I wasn’t the only one pointing out that this was going to lead to both discrediting of the field and possibly a lot of collateral damage to science in general. The problem with “Michio Kaku and the rest of them are getting the public excited about my field and helping me get funding, so I’m not about to complain” is that the same public might someday realize they’d been had, with ugly results. Who knows what craziness is about to ensue with NSF/DOE funding, but I’m pretty sure what’s going to take the biggest hit is legitimate science, not the hypesters.
Checking Twitter, the good news from that sewer is that stringking42069 is back, attacking David Gross, Strings 2025 and the latest nonsense about quantum gravity far more vigorously than I ever could. Besides him, the site seems to be all Elon all the time. Every so often he takes a break from promoting fascism and pushing Russian propaganda and does things like ask his AI to come up with a fundamental new quantum gravity (results no worse than a lot of what’s popular among experts).
It’s tempting to follow and comment on this rapidly moving horror show, but seems best to try and ignore it for now and get some work done. So, that’s it from me now about Sabine etc. for the time-being and I encourage everyone to do the same.
Ah, Math55. I know what you mean about feeling I was not cut out to be a research mathematician but feeling a great sense of accomplishment nevertheless (I believe I got an A-, but I’m sure there had been some grade inflation by then). As students we were incredibly cooperative, not competitive, and we just wanted to learn more and more. I clearly remember the joy in learning what you could do with such a simple concept as a metric space! Clearly my lack of cutthroat competition meant I was not cut out to be a businessman of Bill Gates’ ilk either — I suspect his view was in the minority. My version was 1978-9. and was taught David Rohrlich who was a kind and decent young faculty member at the time, and who was quite clear in his explanations (interestingly, I googled his “rate my professor” ratings and those qualities are still noted by his students at BU). Yes, we started with point-set topology and a fairly standard analysis curriculum, moved on to manifolds, differential forms, manifolds, and also got some linear algebra and other topics thrown in. I believe this is where I learned what a pullback is. This came back in a useful way just last year when I was trying to understand all the hullabaloo about “Koopman operators” (aka koopmania) in dynamical systems — I was able to pretty much instantly grasp the concepts because of what I was taught in that course. No one else where I work had even HEARD about a pullback. How that survived 50 years buried in my brain I do not know. Now ask me about the two semester upper division seminar on differential equations taught by Phil Griffiths. That was a mind-bender.