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 on my GitHub here.

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Jul 19

Oct 18

#### Torsors and principal bundles

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 therein^{1} 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.

Apr 18

#### Triangulations of products of triangulations

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 →Mar 18

#### Loop spaces, spectra, and operads (Part 3)

This post is a weird one: it’s not really aimed at any one audience, but is more of a dump of a bunch of information that I’m trying to process.

Continue reading →Dec 17

#### Loop spaces, spectra, and operads (Part 2)

In the previous post of this series, I talked a bit about basic loop space stuff and how this gave birth to the idea of ‘homotopically-associative algebras’. I’m going to detour slightly from what I was going to delve into next and speak about **delooping** for a bit first. Then I’ll introduce **spectra** as sort of a generalisation of infinite deloopings. I’ll probably leave the stuff about E_\infty-algebras for another post, but will definitely at least mention about how it ties in to all this stuff.

Dec 17

#### Loop spaces, spectra, and operads (Part 1)

I have been reading recently about spectra and their use in defining cohomology theories. Something that came up quite a lot was the idea of **E_\infty-algebras**, which I knew roughly corresponded to some commutative version of A_\infty-algebras, but beyond that I knew nothing. After some enlightening discussions with one of my supervisors, I feel like I’m starting to see how the ideas of spectra, E_\infty-algebras, and operads all fit together. In an attempt to solidify this understanding and pinpoint any difficulties, I’m going to try to write up what I ‘understand’ so far.