Once again I feel the urge to type something nice, but have nothing new of my own to share. I did recently find, however, some notes I once wrote after hearing the phrase “Hochschild homology” for what felt like the hundredth time, so I thought I’d share them here. They’re not particularly enlightening, and I can’t claim to add any insight, but I often use my own blog as a reference for definitions that I once knew but later forgot, so this will at least serve that purpose!

Continue reading →Jul

2020

Jul

2019

#### Twisting cochains and twisted complexes

26th July, 2019This 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.

Continue reading →Oct

2018

#### Torsors and principal bundles

31st October, 2018In 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

2018

#### Triangulations of products of triangulations

11th April, 2018At 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

2018

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

12th March, 2018This 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

2017

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

11th December, 2017In 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

2017

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

8th December, 2017I 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.