Finished Some Things

I’ve now finished with two things that I’ve been working on over the last year or so:

  • The paper explaining my proposal for “Twistor Unification” is now done and uploaded to the arXiv, see here.
  • I’ve finished lecturing for the course on quantum mechanics for mathematicians that I’ve been teaching this academic year. Because of the Covid-required online format for the lectures, they could easily be put on Youtube, where they’re available here. I’m hoping to never ever have to teach this way again, so don’t expect to ever again be producing Youtube lectures. The lectures pretty closely follow my book, and I had been hoping to work on improving and expanding the text. Unfortunately, partly due to laziness and partly due to the twistor stuff, while I found a lot in the book that needs improvement, I didn’t find the time to do the necessary rewriting and writing. I do however have a notebook full of notes on what needs to be done.

For the future, I’m hoping to go on some sort of vacation in a couple weeks, and soon get back to work on some of the major issues raised by the unification proposal (much of which is very sketchy, a lot to be done). I hope to do quite a bit of traveling the rest of this year, likely won’t be teaching in the fall, but probably will be teaching the quantum course again next year during the spring semester. At that point perhaps I’ll get finally get around to the project of rewriting and expanding the quantum book.

Posted in Uncategorized | 14 Comments

Hawking Hawking

There’s a very good new book about Stephen Hawking that just came out, Charles Seife’s Hawking Hawking. Some detailed reviews can be found at Prospect Magazine (Philip Ball) and the New York Review of Books (James Gleick). Seife has chosen to write the story of Hawking’s life starting at the end and ending at the beginning, which takes some getting used to, but provides a different perspective.

Hawking was a huge world-wide celebrity, widely considered by the public and the press to be the modern-day analog of Einstein, dominating the field of theoretical physics. His personal story, involving a long life battling a disease that left him quadraplegic and severely disabled, added greatly to the phenomenon he became. His life has been the subject of various books, films and TV shows, but only now, three years after his death, has something appeared that gives an account of this life corresponding not to myth but to reality.

The reality of this story is that Hawking was a very good theorist, with a high point of his career his work on Hawking radiation in 1974. I remember attending lectures by him at Princeton in the early 1980s, when he was actively working on Euclidean quantum gravity. His speech was hard to follow, so one of his graduate students or postdocs would translate for the audience. Unfortunately, the disease continued to take its toll, and after he nearly died from it in 1985, losing the ability to speak to a tracheostomy, all evidence I’ve seen is that he was no longer able to continue to do research at the highest level. From then on he lived a remarkable and full life for another 33 years, including some collaborative work with other theorists, but he was no longer the driving force behind any new research programs. Seife quotes extensively many physicists who worked with Hawking during this time, including Andy Strominger and Hawking’s student Marika Taylor, who give a fairly good idea of what it was like to work with him.

During the early 80s Hawking was quite fond of the idea that N=8 supergravity would be a successful unified theory, famously giving a talk about it entitled Is the end in sight for theoretical physics?. The advent of string theory coincided with the serious deterioration in his health and ability to communicate. From then on he was reliant on others to explain to him what was going on in string/M-theory:

[Marika] Taylor didn’t yet know how difficult the task ahead of her was. Her thesis was going to be on M-theory, but Hawking was not an expert on the subject. Taylor would largely have to guide herself straight to the frontier of an incredibly difficult branch of theoretical physics, digest all the important work of the past few years, and then teach Hawking what she had learned before even being able to come up with a thesis idea. On top of that, Hawking wasn’t particularly enthusiastic about the string-theoretic parts of the theory: he just cared about supergravity. “As I was starting to go into those areas, I wouldn’t say that he was skeptical,” Taylor says. “He was just not interested… Actually I think the real truth is that he didn’t want to engage with people on territory he was unfamiliar with.”

Soon after I started this blog in 2004, I wrote here and here about Hawking’s heavily publicized talk in Dublin announcing that he had figured out how to resolve the black hole information paradox. I was baffled by reports of his talk and his paper, and not the only one. Seife tells the story of this in some detail, and I think the consensus is that there was no there there.

A large part of Hawking’s celebrity and income derived from his work as a popular author. His 1988 popular book, A Brief History of Time, was a huge success. Seife tells the story of how that book came about, partly motivated by the need for a new source of income. An initial manuscript due to Hawking was edited and improved a great deal before the published version was done. Many other books followed, and if you go to any bookstore with a science section, you’re likely to find quite a few of them for sale. The problem is that, on the whole, they’re not any good, and they’re not written by Hawking. Seife documents this sorry tale in some detail.

I first noticed this when I ran across a copy of God Created the Integers, a thick anthology of writing on mathematics, supposedly edited by and with commentary by Hawking. At least he’s listed as the sole author. Given the topic and the volume of material, it seemed highly implausible to me that Hawking was actually the author. For a review of the book, see here. Seife explains in detail that much of it is essentially plagiarized from other sources, and that to this day, it seems to be unknown who wrote the material (just that it clearly wasn’t Hawking).

At least this sort of thing got little attention, which unfortunately was not true of his 2010 The Grand Design, co-written with Leonard Mlodinow. I wrote about this book in some detail here. Put bluntly, it was an atrocious rehash of the worst nonsense about M-theory and the string theory landscape, with an argument for atheism thrown in to get more public attention. This is the sort of thing that has done a huge amount of damage to both the public understanding of fundamental physics, and even to the field itself. James Gleick’s otherwise excellent review of the Seife book ends with

Hawking promoted the theory of everything with a vengeance. He made it part of his brand. It was the title of the 2014 biopic in which Eddie Redmayne played Hawking. The much-quoted ending of A Brief History of Time raised the prospect of a complete theory—a final theory: “It would be the ultimate triumph of human reason—for then we would know the mind of God.” At the 1998 White House event, Hawking told the assembled dignitaries:

We shall have to rely on mathematical beauty and consistency to find the ultimate Theory of Everything. Nevertheless I am confident we will discover it by the end of the 21st century and probably much sooner. I would take a bet at 50-50 odds that it will be within twenty years starting now.

He would have lost that one, too. It was hubris—but it sold, and it is part of his legacy. He showed younger colleagues how to chase grand theories and best-selling books. Hawking is not the only physicist guilty of hawking.

The theory of everything is a false idol. Why should the universe, which grows more gloriously complex the more we see, be reducible to one set of equations and formulae? The point of science is not the holy grail but the quest—the searching and the asking. Let us hope there will never be a final theory.

We now live in an environment where the idea that there may be a deeper, more unified theory has become completely discredited, through the efforts of many, with Hawking playing an unfortunate part.

If you have any interest at all in Hawking’s story, you owe it to yourself to read this book. It’s a rich and thoughtful examination of his life and work, pushing aside the myth and bringing out the much more interesting reality behind it.

Posted in Book Reviews | 16 Comments

Muon g-2 Result

The long awaited FNAL muon g-2 result was announced today, you can watch a video of the seminar here, look at the paper and a discussion of it at Physical Review Letters, or read stories from Natalie Wolchover at Quanta and Dennis Overbye at the New York Times. Tommaso Dorigo has an extensive discussion at his blog. In terms of the actual new result, it’s not very surprising: quite similar to the previous Brookhaven result (see here), with similar size uncertainties. It’s in some sense a confirmation of the Brookhaven result. If you combine the two you get a new, somewhat smaller uncertainty and ($a_\mu=\frac{1}{2}(g-2)$)
$$a_\mu(Exp)=116592061(41)×10^{−11}$$

The measurement uncertainties are largely statistical, and this is just using data from Run 1 of the experiment. They have accumulated a lot more data since Run 1, and once that is analyzed the FNAL experiment should be able to provide an experimental value with much lower uncertainty.

The big excitement over the g-2 experimental number has to do with it being in conflict (by 4.2 sigma now) with the Standard Model theoretical calculation, described here, which gives
$$a_\mu(Theory)=116591810(43)×10^{−11}$$
An actual discrepancy between the SM theory and experimental value would be quite exciting, indicating that something was missing from our understanding of fundamental particle physics.

The problem is that while the situation with the experimental value is pretty clear (and uncertainties should drop further in coming years as new data is analyzed), the theoretical calculation is a different story. It involves hard to calculate strong-interaction contributions, and the muon g-2 Theory Initiative number quoted above is not the full story. The issues involved are quite technical and I certainly lack the expertise to evaluate the competing claims. To find out more, I’d suggest watching the first talk from the FNAL seminar today, by Aida El-Khadra, who lays out the justification for the muon g-2 Theory Initiative number, but then looking at a new paper out today in Nature from the BMW collaboration. They have a competing calculation, which gives a number quite consistent with the experimental result:
$$a_\mu(BMW)=116591954(55)×10^{−11}$$

So, the situation today is that unfortunately we still don’t have a completely clear conflict between the SM and experiment. In future years the experimental result will get better, but the crucial question will be whether the theoretical situation can be clarified, resolving the current issue of two quite different competing theory values.

Update: Also recommended, as always: Jester’s take.

Posted in Experimental HEP News | 41 Comments

The God Equation

When I was out for a bike ride yesterday I stopped by a large book store and looked to see if they had a copy of Michio Kaku’s new book The God Equation. They didn’t, but did have plenty of copies for sale of his various previous efforts to promote string theory, such as 1987’s Beyond Einstein, 1994’s Hyperspace and 2005’s Parallel Worlds. If someone interested in fundamental physics walks into a bookstore, and looks in the Science section for something to read written by a well-known physics professor, these books are what they’re likely to end up taking home and reading.

When I got back from the bike ride, several people had forwarded me a link to this story from the Guardian which gives a good idea of what’s likely in the book, claims like:

Well, string theory has also created a tremendous amount of interest, as well as a backlash. People say, well, where is the proof? Quite frankly we don’t have the proof, in the same way that Newton did not have the proof of his inverse square law back in 1666. Sometimes, the mathematics and the ideas are ahead of the concrete experimental data. That’s where the Large Hadron Collider comes into play…

The Standard Model is the theory of almost everything. It works spectacularly well but it’s one of the ugliest theories proposed so far. There’s this avalanche of experimental numbers you have to put in by hand. But in string theory the Standard Model just pops right out. With just a few assumptions you get the entire Standard Model. So the point here is that we need experimental proof and the LHC may give us hints of a deviation in the Standard Model and that’s where this post-LHC physics comes into play.

This is just complete and unadulterated bullshit, of exactly the same sort Kaku and a host of others well-credentialed physicists have been heavily and successfully promoting for the last 35 years. I started writing about this 20 years ago, and there have been some changes since then (for one thing, we have Sabine Hossenfelder). I’m still waiting though for any of the leading figures in the physics community responsible for the string-theory hype campaign to do anything at all to try and stop Kaku and the rest of the Fake Physics onslaught that they unleashed.

Usually with books like this, once I get a copy of the book I try and write here a careful review quoting the writer accurately and explaining the problems with what they’ve written, but this time I think I’ll pass on the grounds that this would be a waste of time.

The funny thing though is that I probably agree with Kaku far more than most people about the possibility of unification, although I wouldn’t use the terminology “God equation” to describe a unified theory. Unfortunately Kaku has done far more than most physicists to discredit the search for a better unified theory, through the endless nonsense he has put out about the subject in books like this. I do think we’ll find a better, more unified theory, and I even think I know a couple of the crucial equations, which, leaving God out of it, are:
$${D\mkern-11mu/}_A\psi=0$$
and
$$F_A^+=0$$

Update: You can read the book’s introduction here. It seems that Kaku has conceptualized the book as a response to criticism of string theory. Near the end of the introduction, he assures us:

This book will hopefully give you a balanced, objective analysis of string theory’s breakthroughs and limitations.

This morning he’s on Morning Joe.

Posted in Book Reviews, This Week's Hype | 32 Comments

Twistor Unification

I’ve finally finished writing up a new version of some ideas that I first wrote about here last summer. The latest draft is here, I may set up a web page with more info here.

Several people had very helpful comments on what I wrote last summer, especially in pointing out that I wasn’t providing sufficient justification for the most radical claim I was making, that the problems with analytic continuation of spinor fields indicated that one could interpret one of the Euclidean space rotation group SU(2)s as an internal symmetry. I then spent a lot of time mastering aspects of Euclidean QFT I had never properly understood. Section two of the current paper is the result. It’s in some sense quite elementary, people may find it of independent interest, even if you’re not interested in the ideas involving twistors. Section three, an exposition of relevant aspects of twistors, is pretty much unchanged. Section 4 is an outline of the ideas about how to get a unified theory out of twistors, much there is still sketchy. I understand a lot better than last year how what I’m proposing fits into some standard ideas about “chiral” formulations of gravity, also have learned a bit more about previous attempts to formulate chiral gravity and gauge theory on twistor space. Some highly speculative remarks that this might all be somewhat related to N=4 super Yang-Mills have been added.

Here’s a little bit more here about the hardest to believe claim being made (about analytically continuing spinors). The standard assumption (this is what I always thought) has been based on the analytic continuation behavior of correlation functions: Schwinger and Wightman functions are analytic continuations of each other, and one might think there’s nothing more to analytic continuation between Euclidean and Minkowski space theories. After learning more about the Euclidean QFT literature, I was struck by how different this is from the physical Minkowski space formalism: states and fields don’t just analytically continue, they’re quite different sorts of objects in the Euclidean case. Anyway, this is all explained in detail in the paper…

Update: No, this is not an April Fool’s joke. I’ve now created a twistor unification page where I’ll try and maintain updated information about this unification proposal

Posted in Twistor Unification | 11 Comments

The Future of Fundamental Physics

IAS director Robbert Dijkgraaf will be giving the CERN colloquium tomorrow, with the title The Future of Fundamental Physics. Here’s the abstract:

The reports of the death of physics are greatly exaggerated. Instead, I would argue, we are living in a golden era and the best is yet to come. Not only did the past decades see some amazing breakthrough discoveries and show us the many unknowns in our current understanding, but more importantly, science in general is moving from studying `what is’ to `what could be.’ There will be many more fundamental laws of nature hidden within the endless number of physical systems we could fabricate out of the currently known building blocks. This demands an open mind about the concepts of unity and progress in physics.

I don’t know of any “reports of the death of physics”, but there are a lot of reports of the death of string theory (Dijkgraaf’s specialty) and of the larger subject of attempts to go beyond the Standard Model, experimentally or theoretically. CERN yesterday announced new results from LHCb testing lepton universality (a prediction of the Standard Model). LHCb sees a ratio of decays to muons vs. electrons in a certain process that is off from the Standard Model prediction by 3.1 sigma.

If this result is confirmed with better data and careful examination of the theory calculation, that will be a dramatic development, indicating a significant previously unknown flaw in the Standard Model. BSM theory and experiment would be very much undeniably alive (no known relevance of this though to the troubles of string theory). Unfortunately, the experience of the past few decades is that 3 sigma size violations of Standard Model always go away after more careful investigation (see for instance the 750 GeV diphoton excess). It’s exactly this pattern that has people worried about the health of the field of high energy physics.

Dijkgraaf’s claim that “we are living in a golden era” is an odd one to be making at CERN, which has seen some true golden eras and is now facing very real challenges. Even odder is arguing at CERN that the bright future of science is due to it “moving from studying `what is’ to `what could be.’” CERN is at its core a place devoted to investigating “what is” at the most fundamental level. I’m curious to hear what those at CERN make of his talk.

Dijkgraaf’s abstract to me summarizes the attitude that the best way to deal with the current problems of HEP theory is to change the definition of the goals of the field, thereby defining failure away. The failure of heavily promoted ideas about string theory and supersymmetric extensions of the Standard Model is rebranded a success, a discovery that there’s no longer any point to pursue the traditional goals of the subject. Instead, the way forward to a brighter future is to give up on unification and trying to do better than the Standard Model. One is then free to redefine “fundamental physics” as whatever theorists manage to come up with of some relevance to still healthy fields like condensed matter and hot new topics like machine learning and quantum computing. I can see why Dijkgraaf feels this is the way forward for the IAS, but whether and how it provides a way forward for CERN is another question.

Update: I just finished watching the Dijkgraaf talk, together with the question session afterwards. Dijkgraaf basically just completely ignored HEP physics and the issues it is facing. He advertised the future of science as leaving the river of “what is” and entering a new ocean of “what can be”, with the promising “what can be” fields biotech, designer materials and AI/machine learning. He hopes that theorists can contribute to these new fields by trying to find new laws governing emergence from complexity, perhaps via new ideas using quantum field theory tools.

With nothing at all to point to as a reason to be optimistic about HEP, a couple questioners asked whether his river of “what is” might be now hitting not an ocean but a desert, and he didn’t have much of an answer. All in all, I’m afraid that the vision of the future he was trying to sell is not one in which high energy physics has any real place. It fits well with the depressing increasingly popular view of the field, as one which had a great run during the twentieth-century, but now has reached an end.

Update: For more discussion of the reliability of the LHCb result, see comments here and here, as well as Tommaso Dorigo’s blog post.

Update: Tommaso puts his money where his mouth is.

Posted in Uncategorized | 30 Comments

New Spaces in Mathematics and Physics

Available online today (if your institution is paying…) from Cambridge University Press are two volumes well-worth spending some time with: New Spaces in Mathematics and New Spaces in Physics. These contain write-ups based on a workshop organized back in 2015 by Mathieu Anel and Gabriel Catren, the videos of which are available here.

It would be hard to write in any detail about the wealth of material in these volumes, so I’ll mainly just link to the essays that seemed especially interesting to me:

For several decades now one often hears from prominent theoretical physicists that “Space-time is doomed”, to be imminently replaced by something new coming out of the latest ideas about fundamental physics. For a long time the claims of this sort getting the most attention were from string theorists, and in these volumes Marcos Mariño explains these in his Stringy geometry and emergent space. More recently, Nima Arkani-Hamed has been making well-publicized claims along these lines, with space-time to be replaced with volumes of objects in Grassmanians such as the amplitudehedron.

A large fraction of the theory community is now working on things like “it from qubit”, which propose to somehow get space-time emergent out of things like qubits or quantum information theory. For most of this kind of thing, I’ve found it hard to figure out exactly what the proposal is for the more fundamental objects from which space-time is supposed to emerge. One recent extreme proposal, by Sean Carroll, has the virtue of specifying what the object is (a self-adjoint matrix acting on a complex vector space), but I don’t think there’s a plausible route from that to our observed physics.

As many of the articles linked to above should make clear, mathematicians have over the past centuries developed a range of deep and surprising ideas about new sorts of ways to think about space and geometry. This activity continues: Peter Scholze’s perfectoid spaces and condensed mathematics are examples of new directions of this kind, too new to make it into these volumes.

Of all of these ideas, the ones that at the moment I find most compelling are the twistor geometry ideas of Roger Penrose, and I’ll have much more to say about those in another blog post soon.

Posted in Book Reviews | 7 Comments

ABC is Still a Conjecture

Just a reminder that the abc conjecture is still a conjecture, there is no known valid proof (don’t believe what you might read in an EMS journal). For more about why one attempted proof doesn’t work, see here and here. For extensive background on this, you could start at this blog posting and work backwards, to the first announcement of a claimed proof back in 2012. By 2018 Scholze and Stix had shown that the claimed argument was flawed, and since then the math community has lost interest and moved on. Devotion to the idea that the proof is valid seems now restricted to a small circle of die-hards based in Kyoto and Nottingham who are doing what they can to try and pretend the hole pointed out in the proof does not exist. There will be an IUT Summit in Kyoto in September, but the organizers don’t seem to have found anyone from outside Kyoto or Nottingham willing to participate.

Update: Mochizuki today on his website has put out a 65 page manuscript dealing with criticisms of his proof, it’s entitled:
ON THE ESSENTIAL LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMULLER THEORY IN TERMS OF LOGICAL AND “∧”/LOGICAL OR “∨” RELATIONS: REPORT ON THE OCCASION OF THE PUBLICATION OF THE FOUR MAIN PAPERS ON INTER-UNIVERSAL TEICHMULLER THEORY

I’ve taken a quick look at this document, and I don’t think it will convince anyone Scholze is wrong about the flaw in Mochizuki’s proof. There’s a long third and final technical section, but the first two sections do a great deal of damage to Mochizuki’s credibility. Nowhere in the document do the names Scholze or Stix appear (they are referred to as “RCS: the redundant copies school”), but it starts off with statements such as

the response of all of the mathematicians with whom I have had technically meaningful discussions concerning the assertions of the RCS was completely uniform and unanimous, i.e., to the effect that these assertions of the RCS were obviously completely mathematically inaccurate/absurd, and that they had no idea why adherents of the RCS continued to make such manifestly absurd assertions.

and

the assertions of the RCS are nothing more than meaningless, superficial misunderstandings of inter-universal Teichmuller theory on the part of people who are clearly not operating on the basis of a solid, technically accurate understanding of the mathematical content and essential logical structure of inter-universal Teichmuller theory.

Before going on to the more technical third part, the second part is an extensive discussion of elementary mathematical errors, as some sort of “explanation” of what’s wrong with Scholze and Stix.

Essentially the claim Mochizuki is making in these first two sections is that the most accomplished and talented young mathematician in his field is an ignorant incompetent, and that everyone Mochizuki has consulted about this agrees with him. It’s hard to imagine a more effective way to destroy one’s own credibility and to convince people not to bother to try and make sense of the third section.

There’s no direct reference to the Scholze-Stix document, just a reference to Mochizuki’s own web-page about March 2018. Mochizuki has even gone to some trouble to stop anyone from accessing the Scholze-Stix document without first reading his own web-page.

As for the long discussion by Scholze and others of the problems with the proof that was hosted here and gathered here, the only apparent reference to this is

More recently, one mathematician with whom I have been in contact has made a quite intensive study of the mathematical content of recent blog posts by adherents of the RCS.

followed by

Despite all of these efforts, the only justification for th logical cornerstone RCS-identification of (RC-Θ) that we [i.e., I myself, together with the many mathematicians that I have discussed these issues with] could find either in oral explanations during the discussions of March 2018 or in subsequent written records produced by adherents of the RCS [i.e., such as the 10pp. manuscripts referred to above or various blog posts] were statements of the form

“I don’t see why not”.


Update
: To take a look at the preface, see here.

Posted in Uncategorized | 78 Comments

Yet More Geometric Langlands News

It has only been a couple weeks since my last posting on this topic, but there’s quite a bit of new news on the geometric Langlands front.

One of the great goals of the subject has always been to bring together the arithmetic Langlands conjectures of number theory with the geometric Langlands conjectures, which involved curves over function fields or over the complex numbers. Fargues and Scholze for quite a few years now have been working on a project that realizes this vision, relating the arithmetic local Langlands conjecture to geometric Langlands on the Fargues-Fontaine curve. Their joint paper on the subject has just appeared [arXiv version here]. It weighs in at 348 pages and absorbing its ideas should keep many mathematicians busy for quite a while. There’s an extensive introduction outlining the ideas used in the paper, including a long historical section (chapter I.11) explaining the story of how these ideas came about and how the authors overcame various difficulties in trying to realize them as rigorous mathematics.

In other geometric Langlands news, this weekend there’s an ongoing conference in Korea, videos here and here. The main topic of the conference is ongoing work by Ben-Zvi, Sakellaridis and Venkatesh, which brings together automorphic forms, Hamiltonian spaces (i.e classical phase spaces with a G-action), relative Langlands duality, QFT versions of geometric Langlands, and much more. One can find many talks by the three of them about this over the last year or so, but no paper yet (will it be more or less than 348 pages?). There is a fairly detailed write up by Sakellaridis here, from a talk he gave recently at MIT.

In Austin, Ben-Zvi is giving a course which provides background for this work, bringing number theory and quantum theory together, conceptualizing automorphic forms as quantum mechanics on arithmetic locally symmetric spaces. Luckily for all of the rest of us, he and the students seem to have survived nearly freezing to death and are now back at work, with notes from the course via Arun Debray.

For something much easier to follow, there’s a wonderful essay on non-fundamental physics at Nautilus, The Joy of Condensed Matter. No obvious relation to geometric Langlands, but who knows?

Update: Arun Debray reports that there is a second set of notes for the Ben-Zvi course being produced, by Jackson Van Dyke, see here.

Update: David Ben-Zvi in the comments points out that a better place for many to learn about his recent work with Sakellaridis and Venkatesh is his MSRI lectures from last year: see here and here, notes from Jackson Van Dyke here.

Update: Very nice talk by David Ben-Zvi today (3/22/21) about this, see slides here, video here.

Posted in Langlands | 14 Comments

Isadore Singer 1924-2021

I was sorry to hear this morning of the death yesterday at the age of 96 of Is Singer, a mathematician who led much of the interaction between mathematics and physics during the 1970s and 1980s. In the early stages of my career, among mathematicians investigating the amazing relations between mathematics and the quantum field theories describing fundamental physics there were three towering figures: Atiyah, Bott and Singer. That the last of them has now left us marks the end of an era.

Each of the three had a huge influence on me, both intellectually and personally. Reading their papers and listening to their lectures were great intellectual experiences, shaping early on my understanding of what is central to mathematics and how it fits together with physics. Especially inspirational was the way that they brought together very different fields of mathematics, with Atiyah having his roots in algebraic geometry, Bott in topology and Singer in analysis. Their work together makes a strong case for the unity of mathematics and the relation to physics makes an equally strong case for the unity of mathematics and physics.

On a personal level, at a time when I was tentatively moving from a career in physics to one in mathematics, getting to meet and talk to each of them had a big impact. Much as I respected the great theoretical physicists I had met, rarely had I found them to be particularly friendly or encouraging, and their attitudes influenced the general atmosphere of the field. Atiyah, Bott and Singer struck me each in their own way as wonderfully warm and enthusiastic personalities, and I believe this influenced the atmosphere among mathematicians working in their fields. They were among the most respected figures in the math community, so their enthusiasm for ideas coming out of physics generated a lot of interest in these ideas among a wide variety of mathematicians.

Singer had always had an interest in physics, majoring in physics as an undergraduate at the University of Michigan, then after the war going to graduate school in mathematics at the University of Chicago. I highly recommend reading or watching this long interview with him from 2010, where you can learn the story of his career.

A mathematical high point of this career was his work during the early 1960s with Atiyah that led to the Atiyah-Singer index theorem. A crucial part of this story was Atiyah in 1962 asking Singer why the A-roof genus was integral. Singer realized that this was because it counts the number of solutions of an equation, and that the equation was the Dirac equation. This example in some sense generates a huge amount of mathematics which is described by the index theorem, and which links together very different mathematical fields. On this and other topics, well-worth reading is the 2004 interview with Atiyah and Singer after they were awarded one of the first Abel Prizes.

One can trace much of the history of the modern interaction of mathematics and quantum field theory to an origin back in the summer of 1976, when Singer visited Stony Brook and talked to physicists there about gauge theories, geometry and the BPST instanton (Simons and Yang a year earlier had started to realize how gauge theory, geometry and topology were linked). The next year he was in Oxford working with Atiyah and Hitchin on instantons, which really set off an explosive development of new ideas, inspiring and fascinating both mathematicians and physicists.

Singer spent the years from 1977 to 1983 at Berkeley, which he turned into a major center for this new mathematical physics. During this time he was also one of the founders of MSRI, which to this day plays a major role in worldwide mathematical research. After 1983 he returned to MIT, from which he retired in 2010. I believe the last time I saw him was at his 85th birthday conference, which I wrote about here.

Update: The New York Times has an obituary here.

Update: Dan Freed (who was a graduate student of Singer’s) has a piece about Singer at Quanta magazine here.

Posted in Obituaries | 9 Comments