Not Even Wrong

A few math and physics items:

• Masaki Kashiwara is the winner of the 2025 Abel Prize, see an announcement here, a New York Times article here and a Nature article here.

  Kashiwara has worked an a wide range of topics in representation theory and has been one of the main developers of the field sometimes known as algebraic analysis. Many of his papers are available through the list of publications at his website. For more about his work, see this 2018 article by Pierre Schapira.

  My own encounters with his work include reading his 1978 papers with Michele Vergne (see here and here), getting a lot out of this wonderful survey of the geometric approach to representation theory of real Lie groups, and periodic partially successful attempts to better understand “algebraic analysis” through a couple of his co-authored books on my shelves (Foundations of Algebraic Analysis and Sheaves on Manifolds). Schapira tells me he has been writing some newer lecture notes with Kashiwara, see his web-page here.

• There was a conference last week at Harvard to mark the 100th anniversary of the birth of John Tate. The talks were intended to be not so technical and accessible to graduate students. Slides from many of the talks are now available on the conference webpage, and I hear that they were all recorded, with videos to be available at some point.
• Turning to physics, there’s an article about what’s happened to supersymmetry at Scientific American, and a survey from Copenhagen about attitudes of theorists towards some issues in fundamental physics. No numbers about how many physicists now take SUSY models seriously, other than that it’s clearly has been a monotone decreasing function of time for quite a while. On the “Which is the best candidate for a theory of quantum gravity?” question, string/M-theory is now at 21%, also likely a monotone decreasing function of time.
• Robinson Erhardt has a new podcast featuring Leonard Susskind. Susskind goes into more detail about his claims made here that, as currently understood, string/M-theory cannot be a theory of the real world. He remains hopeful that some new ideas inspired by string theory will come up that will provide such a theory, but acknowledges that right now, this is just a hope.
• Finally, if you haven’t gotten enough about me and my views so far, the First Principles website now has an interview (picture is from last week in Paris…).

The prospect of massive cuts in US federal government science funding has caused an increase in the usual heated internet debates over science funding. These typically involve people arguing either:

• “Funding for science research is an unalloyed good that leads to a more prosperous society. It’s a small fraction of the federal budget, should be much larger.”
• “Funding for science research is a scam that just lines the pockets of an entrenched and privileged elite. It comes out of the paychecks of hard-working people, should be much smaller or completely removed.”

I don’t want to enter into this kind of debate for lots of reasons, but thought it might be a good idea to write something about what I’ve seen of the effects of US federal government spending on research in the two subfields I know well (pure mathematics and theoretical physics). Both of these fields are far removed from the politically charged subfields of science (e.g. climate research), so opinions on whether research on them is inherently good/bad don’t follow the usual red state/blue state divisions. They’re also different in a very significant way from experimental sciences, where grant funding is completely crucial (you’re not going to do an experiment without money to fund the needed equipment).

The two subjects share other significant similarities: a researcher with a job doesn’t really need grant money to think about what they want to think about, the amount of grant money involved is relatively small, and what it can be spent on is a limited list of things (summer salary, travel, conferences, grad students, postdocs). In both cases, among these things what is most expensive is graduate students and postdocs. In the case of graduate students, university accounting charges grants for their tuition (which is something that would never otherwise be paid), so paying for a graduate student on a grant is a lot of money.

What I’ve always found remarkable is that despite all the close similarities, the situation is significantly different in these two fields (at least in the US). Oversimplifying a bit, the source of the difference is:

• In math departments (especially at top research institutions), graduate students are rarely paid as research assistants on grants, almost always as teaching assistants. The money to pay them comes from tuition. There are a few NSF-funded students, but the NSF only funds US citizens. When faculty have NSF math research grants, the size is not enough to pay the large sum a graduate student would cost.

  The typical academic position at a top research institution for someone fresh out of a math Ph.D. is a term-limited non-tenure track teaching appointment, with the amount of required teaching kept low enough to allow time for research. There are some NSF-funded postdocs, but significantly fewer of these than the teaching jobs. The next step on the career ladder would be a tenure-track teaching position.

• In physics departments it’s typically been the opposite: graduate students are mostly paid as research assistants out of grant money (perhaps in some years holding a teaching assistantship). The situation with postdocs is also the opposite from that in math: these are essentially always pure research positions funded with grant money, do not involve teaching and funding from tuition money. Federal grants for theoretical particle physics come from two different agencies, NSF and DOE, more from DOE.

While both mathematicians and theoretical physicists are hoping to end up at the same place (a tenure-track teaching position funded with tuition money), they are getting there in two very different ways, with the mathematicians mostly funded by tuition money, the physicists funded by NSF/DOE grant money. The way they look at grant money is significantly different: for mathematicians it’s a nice supplement and a bit of a help for their research, for physicists it’s existential: no grant money, no job. At the time of a tenure decision, physicists to a much greater extent will be judged on whether they have a grant and how big it is. Once they have tenure, the situation is again very different. An NSF research grant for a mathematician is rarely going to pay for grad students and postdocs. To have other more junior people around to work with, you just need to maintain good relations with your colleagues on the graduate admission committee and the junior faculty hiring committee. Things are very different for physicists: the only way you’re going to get junior people to work with is to get a grant to pay for them.

I spend most of my time in a math department, and the issue of grants doesn’t come up very much, it’s not a big concern for most people. Whenever I go to talk to people in a physics department I’m struck by how the grant issue quickly comes up, with “what would this mean for my grant” something people are clearly thinking about.

In mathematics, it’s pretty clear what the implications of huge cuts in NSF funding will be: individual researchers will lose summer salary money, travel money for themselves and their collaborators, money to organize conferences. The number of grad students and postdocs will go down a bit. Most mathematicians look at this and think it’s obviously a mistake for society: why save a small amount of money by targeting cuts at the richest source of new mathematical ideas, some of which might even ultimately be of significant societal benefit?

In physics, it’s also pretty clear what the implications of huge NSF/DOE funding cuts would be: huge cuts in the number of grad students and postdocs, as well as the number of people in the field that universities would be willing to hire to tenure-track positions. Again, the amount of money involved is not that big, so the attitude is “why should my field be decimated and my research career destroyed to save a little money?”

Note that I’m not at all here discussing experimentalists. For them, the situation is even more straightforward: no grant, no experiment. Big science funding cuts means many fewer experiments.

The other big difference I see between pure math and theoretical particle physics is the relative intellectual health of the subjects. There’s plenty of useless math research going on, but there’s also a lot of very significant progress going on and many subfields are quite healthy. You can argue about whether “crisis” is the right word, but I don’t think there’s an honest case to be made that theoretical particle physics is a healthy subfield making significant progress. While a lot of the reason for this is not the fault of the theorists (SM too good, no experimental hints of how to do better), arguably the way grants have worked in the subject is partly responsible for the problem. If what everyone is doing is not working, but to get a grant you need to be doing what others are doing, then having grants be necessary for your career makes a bad situation worse.

So, from what I can see it’s clear that losing NSF grant money would be a net negative for US math research, and math researchers look at this as being pretty annoying. For US particle theory research, losing NSF and DOE grant money would have much bigger implications and researchers see this as a very personal and existential threat. Those who have been concerned about the health of the field and the negative effects of grant money on it are not necessarily all that sympathetic.

If you just want to engage in the usual arguments about government-funded scientific research, please don’t do it here. On the other hand, I’d be quite interested to hear other perspectives, especially from those who know more about the details of how grant-funded research works (my own information is limited and mainly math department based, it’s quite hard to get one’s hands on good numbers for what is going on with this kind of funding).

Back in 2022 at the time of the rescheduled 2022 ICM I wrote here:

One decision already made there was that the 2026 ICM will be hosted by the US in Philadelphia. With the 2022 experience in mind, hopefully the IMU will for next time have prepared a plan for what to do in case they again end up having a host country with a collapsed democracy being run by a dangerous autocrat.

While I’m trying to protect my sanity by avoiding the news as much as possible, what’s happening now seems to me to raise important questions that the IMU and the ICM Organizing Committee need to seriously start thinking about:

• The ICM opening ceremony has often been chaired by the host country head of state. Will the Philadelphia opening ceremony be chaired by Trump or his representative?
• Given the attitude of the current administration towards foreigners, what assurances have the ICM organizers gotten that mathematicians from all countries will have no trouble getting visas and traveling to the US?
• In 2022 the St. Petersburg ICM was canceled due to the Russian military entering Ukraine and attempting to take over the country by force. The US has threatened to do the same thing to Greenland and to Panama. Will the Philadelphia ICM be canceled if this happens?
• The news from yesterday appears to be that the US has changed sides in the Ukraine conflict, now refusing to condemn the Russian invasion, and demanding economic reparations from the Ukrainians to compensate for past US military support. If the Russians are able to take over Ukraine and install a puppet government with US help, would that be a reason to cancel the ICM?

Given how fast things are evolving, it’s impossible to predict what the situation will be in July 2026. The ICM 2022 debacle was caused by the decision to hold the ICM in a country governed by a dangerous dictator, then hope for the best and not make contingency plans. The same mistake should not be made twice.

If, as now appears all too likely, the US government decides to join forces with the expansionist Russian dictatorship, everyone who can do anything about this has a moral issue to face. The ICM organizers need to start deciding on and making clear what their red-lines are, with a contingency plan if they get crossed.

One problem is that with fascism on the march world-wide, it’s unclear what alternate location would be safe. At least in this case, I’ll point out that for now France is looking pretty good, even if only through the spring of 2027.

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.

Jesper Grimstrup and Jarl Sidelmann have an interesting new paper up on the arXiv, entitled Competition and survival in modern academia: A bibliometric case study of theoretical high-energy physics. It uses bibliometric data to study career paths in hep-th, especially how many people who start out in the field are still in it at various later times.

If you think that things are going fine in hep-th, this kind of study is of limited interest. If you think the field is in trouble, it’s of interest as pointing to one source of the trouble. The problem with this kind of thing though is that on the whole the people making decisions about what to do are the “survivors”, for whom the current system has worked out just fine. They’re the least likely people to think there’s a crisis or to see any reason to do anything about it. As for the job situation (which has been terrible since 1970), I can report that when one doesn’t have a permanent job this seems to be an important and serious problem, but once one does have a permanent job all of a sudden it seems much less important.

What has struck me most in recent decades about hep-th is not the bad job environment, but the monotone-decreasing number of interesting new ideas, now so small that I don’t think “intellectual collapse” is an unfair characterization of what’s happened. I started carefully following the latest preprints in the field more than 40 years ago, pre-arXiv, when they were collected physically at a “preprint library” in one’s institution. Most preprints in hep-th have always been minor advances, not of much interest unless you’re working on much the same problem, but in the past there were always a significant number with something really new and significant to report. The arrival in the preprint library of something new from Witten or any number of other well-known figures in the field was an event, and there also was a steady stream of new ideas coming from people not so well-known. In recent years the situation has been very different, with something worth reading appearing in the arXiv hep-th section less and less often, to the point where it’s a rare occurrence.

This slow death of the field I believe is a very real phenomenon, although I’m not sure how one could quantify it. There are multiple reasons for why it has happened, some of which are just facts of life (the SM is too good, no unexpected experimental results). I do think though that one reason is the one the authors here are trying to get at: decades and decades of a difficult job situation where the only viable way to win the game of survivor is to publish lots of papers in a dwindling number of accepted research programs. This is one problem that the field actually could do something about, but chances of that happening seem remote.

I just noticed something new showing up in Google searches, summaries of the state of scientific research areas such as this one about String Theory And Quantum Gravity. They’re produced by Nature Research Intelligence, which has been around for a couple years, trading on the Nature journal brand: “We’ve been the most trusted name in research for over 150 years.” The business model is you pay them to give you information about the state of scientific research that you can then use to make funding decisions.

Each of their pages has a prominent button in the upper right-hand corner allowing you to “Talk to an expert”. The problem with all this though is that no experts are involved. The page summarizing String Theory and Quantum Gravity is just one of tens of thousands of such pages produced by some AI algorithm. If you click on the button, you’ll be put in touch with someone expert in getting people to pay for the output of AI algorithms, not someone who knows anything about string theory or quantum gravity.

It’s very hard to guess what the impact of AI on scientific research in areas like theoretical physics will be, but this sort of thing indicates one very real possibility. Part of Nature’s previous business model was to sell high-quality summaries of scientific research content produced by the best scientific experts and journalists who consulted with such experts. This kind of content is difficult and expensive to produce. AI generated versions of this may not be very good, but they’re very cheap to produce, so you can make money as long as you can find anyone willing to pay something for them.

The relatively good quality of recent AI generated content has been based on having high-quality content to train on, such as that produced by Nature over the last 150 years. If AI starts getting trained not on old-style Nature, but on new-style Nature Research Intelligence, the danger is “model collapse” (for a Nature article about this, see here). Trained on their own output, large language models start producing worse and worse results.

I’m no expert, you should probably consult an AI about this instead, but it seems to me that one possibility is that instead of superintelligence producing ever more impressive content, we may have already hit the peak and it’s all downhill from here. A thought that occurred to me recently is that back in the 80s when people were talking about string theory as science that anomalously happened to fall out of the 21st century into the 20th, they may have been very right, but not realizing what was going to happen to science in the 21st century…