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!

### Čech totalisation

23-09-08For the first time, I have released into the wild a preprint of which I am the sole author, and had no real supervision. This is a scary moment indeed — how do I know that I haven’t written complete made-up nonsense? It’s true that I talked with a couple of close colleagues about the results, and they nodded in vague agreement, but the responsibility of checking the actual formal details is all on me. Even worse, I wanted to include some results about something that I don’t really have any formal experience with. Anyway, I hope the resulting paper is at least mildly “good” (whatever that might mean). It’s called “Various notions of (co)simplicial (pre)sheaves”, and is now on the arXiv: 2205.15185.

Continue reading →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?

Continue reading →### Simplicial Chern-Weil theory

20-03-29This week just gone I uploaded two preprints to the arXiv:

- Simplicial Chern-Weil theory for coherent analytic sheaves, part I;
- 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?

Continue reading →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.

Continue reading →