After a pretty long time (over two and half years or so), Mahmoud Zeinalian and I have finished our paper “Simplicial presheaves of Green complexes and twisting cochains” (arXiv:2308.09627). In this blog post I want to give a brief overview of one of the main technical tools that we use, which we call Čech totalisation. The full story involves model categories and homotopy limits and all this sort of machinery, but the main part of this post will try to keep this to a minimum, and just talk about a surprisingly useful roundabout way of describing principal bundles. There might be a second part to this, but for those interested or wanting for more details I recommend just delving into the paper — I spent a lot of time trying to make it as readable as possible!

One thing that the past few years have taught me is that I am not good at doing maths all by myself. In fact, I would go as far as to say I am completely useless and unmotivated. I do much better when I have co-authors to give me deadlines and friends to talk to, but, for obvious reasons, the past two years have not been good for this. Not really the ideal time for first postdocs, but alas, that’s life.

I recently found an old notebook with some vague questions and research ideas in it, and then realised that I have had no motivation to work on any of these alone, so why not put them out there for other people to see?

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.

Despite being an analytic/algebraic geometer by name (and title, and qualification, and academic upbringing, and …), there are so many gaps in my knowledge, even when it comes to simple foundational things. One thing which I have always tried to do during my academic “career”, however, is to be the person who asks the first stupid question, so that others can feel less nervous about asking their (certainly less stupid) questions. Thus: this blog post.

I am going to explain what I do know, talk about what I don’t, and then ask some semi-concrete questions that I’m hoping people will be able to help me out with!

I recently submitted a video to the Junior Mathematician Research Archive that gives a brief overview of the work in my PhD thesis. You can watch the video here but, be warned, it’s really probably not the most coherent (ha ha) narrative.

For the sake of it, or in case you’re the sort of person who prefers to read instead of watch, I’ve included the transcript of the whole video below.

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.

This week just gone I uploaded two preprints to the arXiv:

1. Simplicial Chern-Weil theory for coherent analytic sheaves, part I;
2. Simplicial Chern-Weil theory for coherent analytic sheaves, part II.

Both have been extracted from my PhD thesis (which I’ve just sent off to the referees) and contain about 90% of the main mathematical content of my thesis, but with about 90% fewer inane footnotes and digressions. There are also a few appendices in my thesis which explain the background of some of the subjects in a bit more detail, which I think are quite nice, but I’ll probably turn the good ones into blog posts at some point in the coming months.

So what are these two papers about? And why is it split into two parts?

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!