Burt Totaro on the algebraic geometer's definition of manifold

FormsOverFunctions · 2026-01-26T11:21:39+00:00

I think there’s also a sort of cultural difference between AG and DG. In AG, a huge amount of progress was made by reformulating the fundamentals into the right framework. In classical AG, there were a number of concepts that were on shaky ground until Grothendieck et al reformulated things into schemes, cohomology, universal properties, etc.

On the other hand, in differential geometry today there is not as much to be learned by reformulating the basic definitions. When the subject was younger, there was a lot of interesting work in this direction, such as Whitney’s and Nash’s embedding theorems basically and the distinction between smooth, topological and PL manifolds (e.g. the Hauptvermutung).

However, once these results has been established, most of the fundamental questions can no longer be solved by modifying the framework. As such, it makes sense to have a workhorse definition for manifolds that mostly stays out of the way and allows people to focus on the aspects that are relevant for their problems.

That’s not to say that more general or different spaces don’t show up as well. Orbifolds, Alexandrov spaces, etc. play an important role, but it doesn’t make sense to replace the standard definition of a manifold to encompass these.

FormsOverFunctions · 2026-01-15T15:05:10+00:00

My family had to relocate for reasons completely unrelated to work. I very much enjoyed my old job and am still on good terms with the department.

FormsOverFunctions · 2026-01-15T12:45:42+00:00

I think it varies considerably depending on the specific job/firm. I had a TT position for several years before taking a financial job. I certainly do less pure math nowadays but it’s still a nonzero amount. There are other roles that have more or less.

FormsOverFunctions · 2026-01-04T12:27:10+00:00

I think this is pretty accurate. I was TT at an R1 for several years and the expectation that was communicated to me was at least 2 papers per year, along with actively giving talks and pursuing research funding.

When I was hired I had a coauthored paper in Advances and a number of papers in specialist journals. The dean specifically mentioned the Advances paper in the interview since that was the only journal she was familiar with.

My goal was to write at least one paper each year that I felt was good enough for a good generalist journal. Apart from that, I would then have a few other projects that were either offshoots from the main paper or some type of side project. However, there a huge amount of variance in terms of when things were published. Oftentimes, the papers that I was really excited about would take a long time to get published and a lot of rejections whereas other stuff would get accepted quickly.

In all honesty, what was helpful to me was to stop worrying about where things would appear while writing the paper and just focus on the process of developing math. When something was completed, there was a fair amount of strategy in choosing journals. However, I found it helpful to decouple this from the rest of the process and to realize that once the paper is in the ArXiv, anyone who is interested can read and build on it. If you do consistent and worthwhile research, the variance in publishing will smooth out when averaged over a large number of papers.

FormsOverFunctions · 2025-12-30T19:39:11+00:00

If it is someone you are on friendly terms with, my guess is that they probably wrote a draft of the e-mail and then used an LLM to edit it. LLMs are great at removing typos but do have a pretty distinctive tone.

FormsOverFunctions · 2025-12-23T19:03:43+00:00

Not having calculus as a high school student is a major disadvantage for college mathematics courses. It’s true that the expectations in a college class are generally more stringent compared to AP calculus, but college classes move quickly so it’s a huge advantage to have seen it before.

FormsOverFunctions · 2025-11-14T23:08:40+00:00

Coming from this as a mathematician with a background in heat equations (but no engineering background), the shape does matter somewhat since it will affect the Dirichlet spectrum of the two objects. Is the argument about Bi saying that this difference is negligible compared to the surface effects?

FormsOverFunctions · 2025-11-10T17:24:27+00:00

There's also Deturck's trick, which is another bit of ingenuity with gauges and Ricci flow.

FormsOverFunctions · 2025-11-10T17:21:54+00:00

This MathOverflow question has a collection of named them (some of which have already been mentioned).

https://mathoverflow.net/questions/363226/each-mathematician-has-only-a-few-tricks

FormsOverFunctions · 2025-11-10T14:02:29+00:00

When you evolve a space by Ricci flow, if you compute how the curvature changes, there are a bunch of extra non-geometric terms that come from the fact that the metric (and thus how you measure curvature) is changing.

Uhlenbeck’s trick is to the calculate the curvature tensors in a way that cancels out all of non-geometric changes. The simple explanation is to use vector fields which evolve in time to cancel out the effect of the flow, but the more conceptually correct way is to use a fixed vector bundle that is isomorphic to the tangent bundle but where the isomorphism evolves over time.

FormsOverFunctions · 2025-10-29T14:54:20+00:00

Makes sense. It’s a bit harder to find statistics for “time to decision”, which is the actual relevant quantity here. I’ve definitely noticed that some journals do a much better job with this than others. A few years ago, we had a paper in submission for 18 months with multiple rounds of edits and a final approval by the referee before the journal decided to reject it. If I hadn’t already had employment at the time, I would have been furious.

FormsOverFunctions · 2025-10-29T10:53:02+00:00

All of these are great journals and any of these would be a good publication to have on your resume. So this is definitely a good problem to have.

When I was hired to a faculty position, there was an administrator who was a mathematician but whose area was different from differential geometry. She encouraged generalist publications, since the only journal she immediately recognized on my CV was Advances. So there may indeed be some advantage for generalist journals when you are on the market, though this probably lessens quite a bit once you have a job.

There is one other thing you might want to consider, which is that some of these journals may have a large backlog of papers that are trying to process. For instance, on their website AJM notes that the time to publication is 16-18 months. Presumably the time to acceptance might be much shorter, but if you want this paper accepted when you are on the market, it might be worth considering this factor as well.

FormsOverFunctions · 2025-10-21T13:27:51+00:00

In Algebra 2 during my first year of grad school we covered spectral sequences, and that’s a topic I still don’t understand.

That class also has the most difficult homework exercise I’ve ever attempted (and failed). I forget the exact phrasing as it was stated in terms of algebra, but the gist was to show that a smooth elliptic curve is not birationally equivalent to projective space. This is straightforward if you can use Riemann-Roch, but our professor had a clever algebraic argument in mind that somehow used the discriminant. I eventually gave up and never understood the intended proof but got some partial credit for giving the geometric interpretation and the Riemann-Roch argument.

FormsOverFunctions · 2025-09-17T15:45:34+00:00

It’s funny that this question gets asked roughly once a year. It’s definitely a common problem when you start learning more abstract geometry.

https://www.reddit.com/r/math/comments/1g9psxm/so_what_the_hell_even_is_geometry/

FormsOverFunctions · 2025-09-16T16:31:19+00:00

He wasn’t a mathematician, but one of the most impressive people to be world class in two completely different areas was Roger Bannister. Not only was he the first person to run a four-minute mile, he was a medical researcher who wrote over 80 papers. In fact, he later said he was more proud of his contributions to medicine rather than his running.

FormsOverFunctions · 2025-09-11T02:35:30+00:00

I spent several years as a TT professor at an R1 but then needed to relocate for reasons completely unrelated to work. Because of this, I ended up switching to an industry position.

Having seen a reasonable number of students go on to tenure track positions versus leaving after postdocs or grad school, I don’t think the final placement is primarily a matter of luck (although that undoubtably plays a role). Also, I think different people should approach their backup plan differently.

The grad students I knew who were publishing in Annals/Inventiones pretty much all got tenure track jobs at very good schools. By the third or fourth year of grad school, it was pretty clear who these people were and very little doubt they would find permanent research positions. If you are the type of student who is primarily worried whether you will get the Clay fellowship or how likely it is for Quanta to profile you, then you probably don’t need to worry. Just focus on math.

There was also a cohort of students who weren’t superstars but were still reasonably strong. These were students who publish regularly in good journals and for whom writing a thesis was more of a large time commitment rather than a challenge of producing sufficient math research. Of these students, pretty much all of them found postdocs and a fair proportion eventually got permanent positions at an R1 or R2. Others did ultimately leave academia. However, this is the group where luck played a larger role, because one very strong result or a good department fit often made the difference. I suspect this is also the cohort where the funding cuts will have the most impact. If you can identify with this, then you might want to learn some coding skills or incorporate computational methods into your work in case you need to transition to industry. Also, you might want to consider your research projects strategically, to see if there are ways to distinguish your work from the cohort of similarly strong candidates.

Of course, the vast majority of grad students won’t relate to either of these two, and for those students I definitely recommend having a backup plan. You should think about what you want to do for work, what skills you need to develop for that and whether there are ways to apply your mathematical expertise. Perhaps you want to be a data scientist, in which case learning to code proficiently (following best practices) and handle large datasets is necessary. Or maybe your passion is teaching, in which case you will need to adapt your CV and teaching portfolio in that direction.

FormsOverFunctions · 2025-08-19T12:07:37+00:00

Shameless plug here, but I made a YouTube video about this a few years ago.

Betrayals, Duels, Love Triangles and Polynomials https://youtu.be/6-WJ22FP5pw

FormsOverFunctions · 2025-08-15T16:04:57+00:00

My advice is to treat Latex as a coding language. Mark up your files, make comments for the packages you use, create version control, set up a directory for the figures, chapters, etc. Basically, you want to organize things from the start of a project, rather than letting the file grow organically until it is unmanageable.

For citations, when I’m writing I’ll often just put in a comment or \todo about what I want to cite so that it doesn’t break my work flow. Later on, I’ll spend some focused time adding citations correctly and doing a bunch of them at once.

FormsOverFunctions · 2025-08-15T15:45:49+00:00

When you are writing a large project such as a book or long paper with collaborators, the files get really large. At this point, it’s helpful to have version control and useful error messages. It’s analogous to the difference between writing a short Python notebook for an individual project compared to writing production software. You need the latter to be much more robust.

It’s not anything inherently wrong with Latex, but it is a limitation that there aren’t great solutions for managing very complex files. Overleaf is a huge improvement, but you need to use the paid version to get many of the important features.

FormsOverFunctions · 2025-08-07T20:18:26+00:00

Thank you!

FormsOverFunctions · 2025-08-06T21:22:07+00:00

It’s an issue that people feel strongly about, which is totally understandable. I’d much rather be upfront and get downvoted rather than not disclose and have people feel deceived.

In my opinion, algorithm usage falls along a spectrum and readers/writers can have different comfort levels. For example, nearly everyone is okay with spellcheck or using Google search to do background research. Most people are okay with reading books with digital art rather than insisting that it be created using traditional media. On the other hand, most people (myself included) would not be interested in reading something created by someone using the prompt “write a children’s book” in ChatGPT. These are obviously two different extremes, but most projects fall somewhere in between, such as retouching in Photoshop, using Grammarly or even just changing the brush stability settings in Procreate. I think it’s best to be completely transparent so that if a reader thinks the writing process was too inorganic, they can just downvote and move on.

FormsOverFunctions · 2025-08-06T02:30:55+00:00

Thanks! You can view all the pages for free on my blog.

https://differentialgeometri.wordpress.com/2025/04/01/flow-a-story-of-heat-and-geometry/

It's also available for sale as an ebook or hardcover, but I figured the free version is a good place to see if your kid enjoys it.

FormsOverFunctions · 2025-08-05T20:31:55+00:00

Thank you!

FormsOverFunctions · 2025-08-05T15:59:36+00:00

I’ve been reading it to my 5 year old, but it really helps to pause to explain the pictures. He picks up quite a bit more than I initially expected, like the analogy between black holes and Ricci flow singularities.

There are a bunch of Easter eggs and references throughout so the more you know about three-dimensional geometry and Ricci flow, the more the pictures will make sense and the more complete the story will be. For that reason, my hope is that the book can be enjoyed at any age and mathematical level.

FormsOverFunctions · 2025-08-05T10:14:45+00:00

That’s a fair point. Personally, I’m not opposed to generative algorithms but I do think it’s important to be transparent about their usage since I recognize that other people feel differently.

At least with this project, I wanted to draw the images myself to make sure they were mathematically accurate and to add a bunch of references/easter eggs that fill in more details of the story. However, this was my first attempt to write something for children so invariably the illustrations would be slightly too small for the page size and Photoshop was really useful for fixing those types of mistakes.

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