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.

Peter

If you don’t know it, take a look at Milnor’s book on Morse Theory. In about 40 pages, he covers essentially everything anyone needs to know about Riemannian geometry. Milnor is a wonderful expositor. When I decided I wanted to do a thesis in geometric analysis, having never taken a differential geometry course (actually having never taken a geometry course of any kind, I went to a weird high school), that’s where my advisor pointed me.

Have you looked at Thorne and Blandford’s Modern Classical Physics, which is all physics but with a strong geometric orientation?

Jeff M and CIP,

The books you refer to emphasize classical Riemannian geometry, and I’m leaving that to Brendle. From that point of view about geometry, you really can’t understand two of the most fundamental geometrical structures in modern physics:

1. gauge fields, which are connections, but on some general principal bundle, which has nothing to do with the tangent bundle.

2. spinor fields, which can’t be constructed just using a vector bundle (the tangent bundle or its tensor bundles). To get spinors, one way is to use principal bundles: consider the principal bundle of orthonormal frames of the tangent bundle, then find a spin double cover, use the spin representation to get spinor fields (as an associated vector bundle).

What’s remarkable is that the general story about connections and curvature in Kobayashi/Nomizu was developed with completely different motivations (that of understanding the relation of geometry and symmetry groups), ended up miraculously being exactly the right formalism for fundamental physics. This is a story both physicists and mathematicians should know about.

This is of course, not directed at you or your course, since I’ve never met you. However, in general, one problem many physicists have with talking to the general (pure) mathematical audience today is that they assume too much knowledge of differential equations.

I am an extreme example, but all my knowledge of differential equations comes from teaching the standard first undergraduate course on linear ODEs, and I learned that by TAing the course, not by ever having taken it. If pressed, I might be able to recall the solution to the heat equation.

Worse yet, as an algebraist, I usually think of a (partial) derivative as an abstract operator on elements of an algebra (over a field) that is linear, satisfies the Leibniz rule, and sends elements of the ground field to 0. Never mind limits or all that. (Actually, I’m a combinatorial algebraic geometer, which means I have colleagues tell me about statistical mechanics blah blah blah and other colleagues tell me about symplectic blah blah blah, none of which I really understand.)

If you start talking about the Lagrangian or Hamiltonian formulation of classical mechanics, you’ve already lost me.

So – this is a request for you to be nice to those mathematicians – but it’s also a request for recommendations for me to pick up some of the physics point of view on differential geometry.

Off-topic: Have you seen the autobiography of Polchinski: https://arxiv.org/abs/1708.09093?

murmur,

See previous posting and comments there.

quasihumanist,

The main problem with understanding gauge theory and GR, for both mathematicians and physicists, is that the differential geometry needed is rather sophisticated, and often not taught as part of the standard math curriculum, even at the graduate level. If you are comfortable with Riemannian geometry, GR is not hard. Purely as differential equations, the Einstein equations in coordinates are very complicated PDEs, but they have a fairly straightforward description in terms of the Riemann curvature tensor.

Are you thinking of writing up your notes as a sort of K&N for physicists?

Oops, sorry, I forgot that you already answered this.

Ah. I have always liked the tensor calculus centipede being intoxicated by a plethora of indices. Although if you want the full expressiveness of tensor calculus in index-free notation, you would be intoxicated by a plethora of definitions instead.

As a physicist I too learned most of my differential geometry from Eguchi, Gilkey and Hanson’s review. Kobayashi and Nomizu is a beautiful book which I now appreciate but I found it frustrating when I was learning the subject and it took me many years to understand why — it is deceptive because they prove some of the most beautiful theorems in 2 lines. The real work goes into many pages of definitions which are given almost without motivation. I guess this is a standard pure mathematics style, but I don’t find it useful pedagogically. If Kobayashi and Nomizu is a work of art, Eguchi, Gilkey and Hanson is a box of paints!

“I just took a look again at the 1980 review article by Eguchi, Gilkey and Hanson…It really is very good, and anything I’d write would spend a lot of time just reproducing that material.”

Think about how much easier this would be if the norm was for physicists to release all their work under a license that allowed re-use with attribution (e.g., Creative Commons ShareAlike). You could just immediately start building.

To me, the main disconnect is that there is an extensive physics literature on instantons, monopoles, and other topological phenomena, in which many interesting phenomena are computed (instanton contribution to effective lagrangians and the OPE, axial charge diffusion in an EW plasma, defect formation in phase transitions, baryon number violation, etc), and then there is a mathematical (or mathematical physics) literature in which a beautiful formalism is laid out (bundles, forms, etc), but nothing is really computed (or if something is calculated it is done by choosing coordinates, and writing things out in components). The only case that I am really aware of where, historically, sophisticated tools played a role is the ADHM construction, although even in that case these days it is usually presented as a clever ansatz for the gauge potentials. What would be nice is a review where one can really see the power of sophisticated methods in doing calculations.

Thomas,

I don’t know of a review of the kind you want, maybe someone else can suggest something. In general though, I think the power of the abstract geometrical formalism is that it tells you what the general coordinate independent features of solutions will be. Sometimes, especially with enough symmetry, you can calculate these things without choosing particular coordinates. But, in any particular case, to actually calculate you may need to choose coordinates, better, coordinates adapted to the problem.

Dear all, I remember the remark by Weinberg in his beautiful book about GR etc., i.e. that he thinks that the value of geometry in e.g. gauge theory is overtestimated. While I think he is not right, there is a grain of truth in his remark. Classical gauge theory as fibre bundle mathematics is certainly beautiful, however when quantizing the occurring fields transforms this into completely different entities. What one perhaps needs is some sort of quantum fibre bundles.

Hey Peter,

After preparing for this course, have you had any thoughts about studying synthetic differential geometry?

Peter,

What are the pre-requisites for your course in real analysis, algebra, geometry, linear algebra? I’ve always wanted to study some differential geometry, but my background is limited to linear algebra at the level of Serge Lang, modern algebra at the level of Fraleigh, Calculus at Stewart’s level, and some analysis that I vaguely remember.

TME,

I don’t really see the point of that, it seems to invoke a lot of abstract formalism, and get little for it. Definitely not appropriate for students.

I have been intrigued by the idea of formulating differentiable manifolds in a formalism more parallel to the definitions in terms of a sheaf of functions common in algebraic geometry and topology. Looking into this though, if you try and define the “sheaf of differentiable functions”, you end up going through the same apparatus as the usual definition, to which you’ve just added a lot of extra formalism.

Justin,

You should start with an advanced undergraduate course in geometry, specifically one dealing with differentiable manifolds. A good typical textbook is Loring Tu’s An Introduction to Manifolds. The course I’m teaching is supposed to be more advanced, assuming an undergraduate course as background.

I’m really glad to see a shout out to EGH which functioned as a ‘bible’ for understanding the consequences of the Wu-Yang dictionary to which Simons and Singer also contributed. The Eguchi, Gilkey and Hanson ‘article’ is really an example of a phenomena we don’t discuss much but which I think is fairly significant and highly conserved.

This famous article was really more of a book of a very special kind that appeared as a ‘paper.’ Sometimes, it becomes necessary for an entire technical field to take up a new toolkit; at those moments, it falls to someone or some group to write up the new tools as an enticement to colleagues to change their thinking. For some reason, in these situations, what gets written as a pitch or a sales job is often far clearer than what will later be written to introduce the toolkit to future students. Then, mysteriously, the old text is forgotten as new pedagogical texts attempt to reach students rather than professors. Yet, often, those more ‘professional’ texts aimed at students are often not as well or as deeply motivated.

As a related example, I believe I found a bound set of dusty notes from Dieudonné on Grothendieck’s scheme theory written at some point early in the development of the theory that attempted to convert people used to thinking about Algebraic Varieties to think in terms of Scheme Theory. Strangely, this old book (or set of notes) seemed much clearer and better motivated than the treatment in the leading contemporary pedagogical text of the time by Robin Harthshorne. This aroused my curiosity around a simple question: do people write most clearly when pitching a new toolkit to their colleagues and in a less well motivated way once the professors have been converted?

I have since concluded that there is something magical about the early days of a new toolkit where people are thinking more clearly than they are likely to think later about the motivation for the ‘kit’. As a consequence, it is often worth going back and looking for the text(s) which transitioned professors into a more modern viewpoint as they often have far more motivation and clarity than later introductory texts.

The best explanation that I can offer is this: only briefly is there a window to write for senior people (i.e. the most discerning of possible students) without fear of insulting their intelligence and while authors still remember what is counter-intuitive about the new kit. Even a short time later, people forget their beginners mind-set and thus what made the subject counter-intuitive enough to need a motivated pitch so that the new tools would be adopted.

I wish more beginning students would go back to look at those special moments where everything suddenly changed. There are very few of them in any career and each epiphany comes but once.

@Eric Weinstein:

The difficulty I’ve found surround good older texts is converting their ideas to modern notation. As ideas get more solidified, notations (sometimes) improve, and make things clearer. For that reason, I think results are somewhat mixed, as with any pedagogical text. (What I’ve seen of the early discussion of tensor products falls into this trap somewhat. I’d imagine it’d be less large for more recently-developed fields, though.)

I also wonder if the original paper might benefit from being longer [neglecting problems and the like] for the same material (or, more precisely, the same length for less material). More time to talk about things will improve your writing for free, in some ways, although this is perhaps more true of those who would argue some variant of “read the original literature” than those who support some sort of summary paper.

That said, I do think an “updated [in notation] original” of works of that variety can sometimes be an outstanding pedagogical tool, depending, because I do agree you get better insight into the original motivation by looking at those papers.

Eric,

That’s an interesting comment on the EGH story. I’ve been kind of surprised to look around not readily find something better than that. I would have expected that over the years many people would write up versions of this material, but there aren’t that many (suggestions welcome, does anyone know something that covers the material of EGH, but better?)

In this case I think part of what happened is that the 1984 advent of string theory turned attention to different areas of geometry/physics, so we ended up with people writing expository material not about gauge theory and its geometry, but about geometry of Riemann surfaces (for string world-sheets) and objects like Calabi-Yaus.

Magnema,

In this case it isn’t really not-up-to-date notation that is the problem, one of the great virtues of EGH is that they use quite good modern notation.

“In this case I think part of what happened is that the 1984 advent of string theory turned attention to different areas of geometry/physics, so we ended up with people writing expository material not about gauge theory and its geometry, but about geometry of Riemann surfaces (for string world-sheets) and objects like Calabi-Yaus.”

I have the opposite impression that the mathematical structure of gauge theories is from the standard model perspective “nice to know”, but became heavily utilized in the context of string theory.

To give some random examples, consider localization in non-Abelian gauged linear sigma models, the Kapustin Witten story or bundle constructions for heterotic models.

On a slightly different note i would love to understand what insights are to be gained from Urs Schreibers “higher pre-quantum geometry”.

I was a bit surprised that “Modern Geometry” means classical differential geometry.

Cobi,

Yes, the examples you give are random, wildly different sorts of mathematics, connections/non-connections to physics, and no connection to observed physics.

What do you think “Modern Geometry” is? Can you point to a graduate-level mathematics textbook covering whatever you think it is?

“Have you looked at Thorne and Blandford’s Modern Classical Physics, which is all physics but with a strong geometric orientation?”

My initial foray into this book suggests that it is very much written in physicist-speak rather than mathematician-speak.

Look at Sternberg’s recent textbook (title starts with the word “Curvature”). He makes some effort to relate differential geometry to physics.

Since you mention Loring Tu in one of your comments, I’d like to point out that Tu has a new textbook out on differential geometry. It seems to cover the kinds of things you want to touch upon (connections on principal bundles).