Not Even Wrong

This semester I’m teaching the first semester of Modern Geometry, our year-long course on differential geometry aimed at our first-year Ph.D. students. A syllabus and some other information about the course is available here.

In the spring semester Simon Brendle will be covering Riemannian geometry, so this gives me an excuse to spend a lot of time on aspects of differential geometry that don’t use a metric. In particular, I’ll cover in detail the general theory of connections and curvature, rather than starting with the Levi-Civita connection that shows up in Riemannian geometry. I’ll be starting with connections on principal bundles, only later getting to connections on vector bundles. Most books do this in the other order, although Kobayashi and Nomizu does principal bundles first. In some sense a lot of what I’ll be doing is just explicating Kobayashi and Nomizu, which is a great book, but not especially user-friendly.

A major goal of the course is to get to the point of writing down the main geometrically-motivated equations of fundamental physics and a few of their solutions as examples. This includes the Einstein eqs. of general relativity, although I’ll mostly be leaving that topic to the second semester course.

Ideally I think every theoretical physicist should know enough about geometry to appreciate the geometrical basis of gauge theories and general relativity. In addition, any geometer should know about how geometry gets used in these two areas of physics. I’ve off and on thought about writing an outline of the subject aimed at these two audiences, and thought about writing something this semester. Thinking more about it though, at this point I’m pretty sick of expository writing (proofs of my QM book are supposed to arrive any moment…). In addition, I just took a look again at the 1980 review article by Eguchi, Gilkey and Hanson (see here or here) from which I first learned a lot of this material. It really is very good, and anything I’d write would spend a lot of time just reproducing that material.

Update: Steve Bryson sent me another excellent suggestion for a book covering these topics, aimed at the physical applications: David Bleecker’s Gauge Theory and Variational Principles.

• The Stacks Project (see an earlier post here) had a very successful workshop in Ann Arbor earlier this month. This is a remarkable effort pioneered by Johan de Jong to produce a high quality open source reference for the field of algebraic geometry. It now is over 6000 pages, with an increasingly large number of papers citing it (according to data from Pieter Belmans, 85 citations in the arXiv so far in 2017 alone). During the workshop plans were discussed for the future of the project, with work on a new version of the project infrastructure underway (see slides and a blog post from Belmans).
• The latest AMS Notices has a wonderful article by my Barnard/Columbia colleague Dusa McDuff about her remarkable family history and reflecting on her equally remarkable mathematical career. A post earlier this year discussed a Quanta article about her recent work with Katrin Wehrheim on technical issues in the foundations of symplectic topology. Kenji Fukaya has recently written something for the Simons Center website (see here) explaining his take on this story.
• The Stanford Encyclopedia of Philosophy has a new entry about the fine-tuning problem, by Simon Friedrich.
• The LHC operators have run into some difficulty in recent weeks (reflected in the accumulated luminosity plots here and here), with problems centered around an unknown source of gas in the beam pipe at a specific location, leading to losses of the beam. Some information about this is available here. The past few days they seem to be having success running the machine with around 1500 bunches, much less than the 2500 or so of earlier in the summer. The target for the year is 40 inverse fb which may still be achieved, while more optimistic numbers that looked plausible earlier now seem less likely.

Update: Joe Polchinski has put on the arXiv a long autobiographical document, with a detailed discussion of his scientific career.

Update: As mentioned in the comments, Go Yamashita has posted a long document surveying Mochizuki’s claimed proof of the abc conjecture. Experts may find that this makes it more possible to understand and check the claimed proof, we’ll see.

Update: Also at the Simons Center website, there’s an interview with Michael Green. It’s interesting to see that in recent years his research interests have led him to getting closer to mathematics and to an appreciation of what mathematicians do. As for his claim about string theory that

I don’t think there is a substantial antagonism to it among those who have studied it, other than a few individuals who enjoy publicizing their views.

I think he’s quite wrong if you properly take “it” to refer to the aspect of string theory there is widespread antagonism to among physicists, the overhyped claims about a unified theory based on string theory.

Blogging will be light to non-existent for the next ten days or so, as I head out west on a road trip to see next Monday’s solar eclipse. Current plan is to fly to Denver tomorrow, pick up a vehicle, and head up to Wyoming the next day. If weather projections look good for the Wyoming/Idaho part of the track, that’s where we’ll plan to end up, likely camping out somewhere (accommodations along the track have long been booked up).

This will be the ninth eclipse I’ve traveled to see, and I urge anyone thinking of making a trip to the eclipse track to do so. A total solar eclipse is something quite different than a partial one, and this is a very rare opportunity to see this in the US. Besides the eclipse, a major motivation for these trips has always been that of getting to visit a more or less random place on Earth that one wouldn’t otherwise have any excuse to see. I’ve driven quickly through Idaho and Wyoming a few times over the years, look forward to spending more time in that part of the country this coming week (unless the weather there looks bad, in which case maybe we’ll end up in Oregon or Nebraska).

Some other random advice about eclipses:

• Be very careful about use of binoculars or telescopes, improper use of these at any time other than the period of totality is what can cause serious eye damage (by itself the eye is pretty good about automatically protecting itself).
• Don’t put a lot of effort into photography during totality, since that’s likely to lead to you spending the time you should be enjoying the experience fiddling with camera equipment (and not getting a good result anyway…). A simple thing to do is to set up a camera to take video of the overall eclipse scene as it happens, turn it on at some point then ignore it.

If you miss this one, next couple are far south in South America, there will be another chance in the US relatively soon, April 2024.

Update: Now back in New York. Had a very good view of the eclipse from a spectacular location: Stanley, Idaho, up in the Sawtooth mountains. Only not quite optimal part of the plan was camping out not not well-equipped for the the unexpected fact that it gets down to about freezing at night in that part of Idaho, even in August…

• For representation theory aficionados, George Lusztig has put on the arXiv a long document with comments on his papers (for a bit more about him, see this).
• For a new idea exemplifying the potential grand unification of mathematics and physics, Minhyong Kim and others have been developing arithmetic Chern-Simons theory (see here and here). There was a recent workshop on the topic, videos of talks available here.
• The editors of the Journal of Algebraic Combinatorics are leaving the Springer journal, setting up a new journal, Algebraic Combinatorics. For more about this, there’s a press release, a story at Inside Higher Education, and a blog entry by Timothy Gowers.
• Landon Clay, founder of the Clay Mathematics Insitute, passed away last week, more about him available here.
• There’s a profile and interview with Carlo Rovelli here.
• While Google continues to develop our new machine overlords, Google money will fund a new IAS program to address their theoretical underpinnings.

I’ve looked at the talks from a few of the HEP experiment and phenomenology summer conferences. If anyone can point me to anything interesting that I’ve missed, please do so. The lack of new physics beyond the Higgs at the LHC has left the field in a difficult state.

One conference going on this past week and next is the IAS PiTP summer program aimed at advanced grad students and postdocs. This year the topic is HEP phenomenology, and talks are available here. If you want to understand the conventional wisdom on the state of the subject, you can watch Nima Arkani-Hamed’s three and a half hour lecture (here and here) which he starts off by describing as on the topic “What the Hell is Going On?”.

A lot of the first part is historical, starting off with the Georgi-Glashow GUT and the arguments for SU(5) or SO(10) GUT unification first put forward 43 years ago. He then walks the audience through the sequence of steps theorists have taken to solve the problems of such models, after an hour ending up at the landscape, spending a half an hour promoting the anthropic solution to the CC and other problems. The second part of the talk is largely devoted to making the case for his favored split SUSY models, with anthropics and the landscape taking care of their naturalness problems. By the end of the three and a half hours, Arkani-Hamed admits that this scenario is not that convincing, while arguing that it’s the only thing he can see left that is consistent with the idea that theorists have been following a correct path since 1974:

It’s the only picture of the world that I know where everything that we learned experimentally and theoretically for the last 30 years has some role to play in it. But my confidence in it is not so super high, and I definitely think its worth thinking about completely radically different things.

The disadvantage to the trajectory of going with what works and then changing a little and changing a little is that you might just be in the basin of attraction of the wrong idea from the start and then you’ll just stay there for ever.

To me by now the evidence is overwhelming that HEP theory has been in the wrong basin of attraction for quite a while, and the overriding question is what can be done to get out of it. If you’re in the wrong basin of attraction, you need to get out of it by going back to the point where you entered it and looking for another direction. I think Arkani-Hamed is right to identify the 1974 GUT hypothesis as the starting point that led the field into this wrong basin. HEP theory has progressed historically by identifying new more powerful symmetry principles. The move in 1974 was to go beyond the SM symmetries by picking a larger gauge group, then breaking it at a very high energy scale with new scalar fields. The history of the last 43 years is that this idea isn’t a successful one: as this talk shows, it leads to an empty theory that explains nothing. Can one find different new ideas about symmetry that are more promising?

I’ve finally found some time to look around the web to see what has been happening at conferences this summer. In this blog post I’ll point to a few on the math/physics interface featuring interesting talks. This area now (I think it may be Greg Moore’s fault) has started to acquire the name of “Physical Mathematics”, to distinguish itself from old-school “Mathematical Physics”. At this point though I’d be hard-pressed to provide a useful definition of either term.

• Talks from last month’s 2017 Bonn Arbeitstagung are available here. This conference was in honor of Yuri Manin and supposedly devoted to Physical Mathematics (although I suspect some of the speakers might not realize that they are doing Physical Mathematics). Dan Freed and Jacob Lurie gave two characteristically lucid series of talks, well worth watching.

  A very active area of physics these days with significant overlap with mathematics (of the sort discussed by Freed) is the study of topological superconductors and other materials in which topology plays a large role. For an introduction to this topic, Davide Castelvecchi at Nature has a new article The strange topology that is reshaping physics.

• CERN has just finished running an institute on the topic of the Geometry of String and Gauge Theories. It included a colloquium talk by Greg Moore on d=4 N=2 Field Theory and Physical Mathematics. I’ve always been fascinated by the d=4 N=2 super Yang-Mills theory in its “twisted” topological version. The mathematics involved is deep and amazing, and it is frustratingly close to the Standard Model…
• Pre-string math 2017 was this past week, and String Math 2017 will be next week. All sorts of interesting talks at both of these, relatively few of which have much to do with string theory. That’s of course also true of Strings 2017, but I’ll write about that elsewhere.

Other suggestions of interesting mathematically related summer schools with talks available are welcome. On the physics side, please wait for a succeeding blog entry on that topic.