Briefly: two very short translations concerning geometry, and the final two seminars of FGA (thus finishing a complete first draft of the entire FGA seminar series).

Hopefully one of my co-authors and I will be uploading a long-awaited preprint on dg-categories, twisting cochains, and homotopy limits to the arXiv “soon”. Until then, here are the small handful of translations that I’ve finished in the year since I last wrote

Just a very short post (you don’t even need to click “Continue reading” if you’re looking at this on my blog archive) — I finally managed to do some more maths (but only because I had some very hard working and very good coauthors), and I’ve written about it on the Topos blog as a two-part series: Part 1 and Part 2.

I would love to write more, but after having written these I’m afraid I’m all blogged out for the minute, so ciao for now!

I haven’t blogged about it in a while, but I’ve been working on just making my translations a bit better, both in terms of content and accessibility. Let’s have a look at what I’ve done, shall we?

This is really just a cross-post announcement: I wrote a post over on the Topos blog, but it’s something I’ve been thinking about a lot, so I wanted to share a link to it here as well. It’s basically the result of me, knowing a bit about derived categories and model categories, trying to digest this lovely bit of Australian category theory by Bryce Clarke concerning internal lenses. In fact, it’s really a fermented and distilled version of an old blog post from here, namely Cauchy completion and profunctors.

I haven’t posted anything here in well over a couple of months, so I thought it’d be a nice idea to just say “hello, I am still here, and here’s why I’ve been too tired to post anything”, whence this post.

I’ve had a bit of free time during various quarantines and lockdowns as of late, so I’ve been able to do some more translations. Here’s a quick summary of what I’ve uploaded to thosgood.com/translations recently, as well as what I’m now working on.

I’ve been having quite a bit of fun translating some maths papers into English recently. Of course, I’m still chugging away at the EGA translation project, having just finished another section of EGA II, leaving “only” about 60 pages left before that chapter will be done as well! However, I’ve started putting some of my other translations all together, and they can now be found at https://thosgood.com/translations. There are a lot still in progress, but here’s a summary of what I’ve done so far.

Having finished my defence (entirely virtually, and hence, of course, plagued with many technical difficulties), I have now been able to share my thesis with the world. Most of the technical content can already be found in my two preprints (part I and part II), but I think the thesis is a much more self-contained and leisurely read, with a lot more examples and (hopefully) helpful appendices. It also has geese! You can find a copy of it on here on TEL.

Edit. I’m in the process of moving over from YouTube to Vimeo (no ads, less tracking, and no Google), so you can find my videos there now instead. In particular, the connections and curvature videos can be found here.

This is a very short post just to say that I’ve started uploading some maths-related videos to my YouTube page. At the moment there’s a talk I recently gave at a graduate seminar about connections and curvature from the point of view of somebody trying to avoid differential geometry, and a series I’ve been working on called Nice Analytic Sheaves For All which aims to discuss motivations for coherence conditions of complex-analytic sheaves from various points of view. Hopefully there will be more updates to the latter soon!

I am a big fan of Stephen Bruce Sontz’s book Principal bundles: The Classical Case, and cannot recommend it enough, as somebody who usually finds differential geometry (a) dull, and (b) incomprehensible. Anyway, there’s a lovely quote from this book about how confusing the terminology surrounding connections can be, so let’s try to clear some of that up today.

As always, when I haven’t written anything on here in a while, I’m going to write about something entirely irrelevant to anything I’ve written about before (instead of continuing any of the many series of posts I’ve started or planned to start). Today we’re going to look at a ‘fun’ little calculation, which might be the most algebraic-geometry-like calculation I’ve done in a long time: we’re going to compute the determinant of the cotangent sheaf of the complex projective sphere. Note that even the statement of this problem makes me uneasy: I am nowhere near as comfortable with classical algebraic geometry as I should be!

On a particularly bad day, when I’d gone into the maths department on a public holiday by accident¹, I saw nobody around. Coming down the staircase to leave, however, I saw a blue tit trapped inside, loitering by the photo of Grothendieck.

I find myself in a mathematical rut more often than I would like. It is very easy, especially as a PhD student, I think, to become disillusioned with maths, the work that one can do, and academia as a whole. I realised only the other day, after talking to my family, some of the things that can contribute towards this. Hopefully this post can serve as a reminder to myself that I am human being who is trying to be a mathematician, and not the other way around.

This week was the Young Topologists Meeting at EPFL in Lausanne, and had courses by Julie Bergner (on 1- and 2-Segal spaces) and Vidit Nanda (on the topology of data), as well as a bunch of great talks by various postdocs and PhD students. I was lucky enough to get the chance to talk about twisting cochains and twisted complexes for half an hour, and you can find the slides here.

I have just come back from CT2019 in Edinburgh, and it was a fantastic week. There were a bunch of really interesting talks, and I had a chance to meet some lovely people. I also got to tell people about #math.CT:matrix.org, and so hopefully that will start to pick up in the not-too-distant future.

I haven’t posted anything in a while, and rather than trying to write about maths, I wanted to just share some lovely photos of Skomer island (which I recently visited). I am even less knowledgeable about birds than I am about maths, but I do love them, and this was the first time in my life that I’d actually seen a puffin in the flesh!

Having recently been thinking about twisting cochains (a major part of my thesis) a bit more, I think I better understand one reason why they are very useful (and why they were first introduced by Bondal and Kapranov), and that’s in ‘fixing’ a small quirk about dg-categories that I didn’t quite understand way back when I wrote this post about derived, dg-, and A_\infty-categories and their role in ‘homotopy things’.

This isn’t a long post and could probably instead be a tweet but then this blog would be a veritable ghost town.

In my thesis, switching between vector bundles and principal \mathrm{GL}_r-bundles has often made certain problems easier (or harder) to understand. Due to my innate fear of all things differentially geometric, I often prefer working with principal bundles, and since reading Stephen Sontz’s (absolutely fantastic) book Principal Bundles — The Classical Case, I’ve really grown quite fond of bundles, especially when you start talking about all the lovely \mathbb{B}G and \mathbb{E}G things therein¹ Point is, I haven’t posted anything in forever, and one of my supervisor’s strong pedagogical beliefs is that ‘affine vector spaces should be understood as G-torsors, where G is the underlying vector space acting via translation’,² which makes a nice short topic of discussion, whence this post.

So I’m currently at the Max Planck Institute for Mathematics in Bonn, Germany, for a conference on ‘Higher algebra and mathematical physics’. Lots of the talks have gone entirely above my head (reminding me how far behind my physics education has fallen), but have still been very interesting.

Just a small post to point out that I’ve uploaded some new notes, including some I took at the Derived Algebraic Geometry in Toulouse (DAGIT) conference last year. I’ve been hard at work on thesis things, so haven’t been able to write up all the blog stuff that I’ve wanted to, but hopefully will get a chance sometime in the near future.

This post assumes that you have seen the construction of derived categories and maybe the definitions of dg- and A_\infty-categories, and wondered how they all linked together. In particular, as an undergraduate I was always confused as to what the difference was between the two steps of constructing the derived category of chain complexes was: taking equivalence classes of chain homotopic complexes; and then formally inverting all quasi-isomorphisms. Both of them seemed to be some sort of quotienting/equivalence-class-like action, so why not do them at the same time? What different roles were played by each step?

At a conference this week, I ended up having a conversation with Nicolas Vichery and Eduard Balzin about why simplices are the prevalent choice of geometric shape for higher structure, as opposed to e.g. cubes or globes.