#spectracled posts

Despite hell descending upon the UK again for the third time in two months, this week has been very productive! Sadly no seminars anymore cause term's over so I only have my project to talk about.

This calculation I've been doing has gone even quicker than I expected! Past me did a lot of good work setting up the tools to actually do this effectively. I actually spent Tuesday and Wednesday writing up everything I did since the last time I updated my document. I will say I've done a lot of combinatorics. Basically all of my arguments can broadly be classes as "let's count the number of dimensions". Though it did mean using the Rank-Nullity Theorem (ol' reliable).

My next goal is to generalise what I've done in the semifree case to the full algebraic model. This shouldn't actually take too long; I think it's mostly a case of bookkeeping and maybe introducing a bit more notation to do things effectively. Once this is done, that's my original goal achieved!! But this certainly isn't the end, there's another question I can tackle after this using what I've done. I've already been thinking about it a bit and it's already quite interesting!

This is all sounding rather close to analysis but I'm also thinking about a more broader context. Like can we treat the cellular approximation theorem in the same way? That is, are cellular maps between CW complexes dense in continuous maps such that approximates can be made to be homotopic to the original function? Or similarly the simplicial approximation theorem? Are there any other contexts where this is a meaningful question?

Despite literal hell descending upon us I've managed to have another pretty productive week!

Seminars:

The algebraic topology seminar this week was about Bredon sheaf cohomology which was pretty cool! It's always nice when there are talks on stuff that's actually vaguely related to things I'm interested in.

The postgrad seminar was interesting. It was about studying how planes intersect in weighted projective spaces. My main interest was about the topology of these weighted projective spaces but I didn't ask cause I don't think the speaker would've known the answer (I know him and I don't think he's thought much about CW complexes and most certainly not in this context).

No junior algebra seminar this week cause the speaker cancelled due to the heat (not that I'dve gone anyway cause I worked from home yesterday precisely because of the heat).

Summer Project:

I've made very good progress again this week! I managed to generalise my method for calculating things in the semifree case to the full case. I'm definitely glad I did the semifree stuff first because it made it way more obvious what I should try to do for the full case.

I'm not sure I'll make as good progress next week as I'll be moving onto a slightly different (but related) computation. But it's almost certainly going to be more fiddly but if I can get it in fully generality it might be something publishable. My next task will be to try to do this computation in very easy cases and hope it will illuminate a way to do it more generally.

My supervisor and I have decided to put aside this other part of my project that we were hoping to do cause it's not really related and would just distract from the first goal. Kinda sad but equally I'd rather make sure I do this stuff well.

I've also come up with another question actually related to what I'm doing that's quite interesting that'll probably also be publishable if I can answer!

Other Stuff:

This week's been pretty good! I've made decent progress and I'm understanding things a lot more! I did have to take yesterday off and work from home today because I've been to tired to go into uni. However, as I said the progress I've made has been good! My supervisor meeting this week was especially good because I got all my questions answered in a way I understood them. I also felt like I was more having a conversation with my supervisor. Like I actually had decent things to say rather than just listening and absorbing.

Seminars:

The algebraic topology seminar was very interesting and a well given talk. It was about trying to understand BDiff(M) for compact connected 3-manifolds in terms of the BDiff(P₁⊔⋅⋅⋅⊔Pₙ), where Pᵢ are the irreducible manifolds in the prime decomposition of M (except not S¹×S² though, even if the manifold has it as a connected summand I can't remember why). The jist was the speaker (and collaborators) construct a map using some ∞-categorical techniques and are abled to fully describe it's fibre in a nice combinatorial way.

The postgrad seminar was alright. It was a geometric analysis talk about the Plateau problem and results about regularity of solutions. It was a decently given talk, it just isn't my sort of thing.

Summer Project:

The extra stuff my supervisor had me working actually turns out to be a better way of doing the calculations I'd already done. It's a lot clearer what's going on and feels way less hand wavy. Originally he got me to work on a simple case because the goal was to get things done in more detail to allow us to calculate something slightly different. However it turns out the general method was very useful for the general case. Whilst I don't wish to pursue the full case in as much detail it's possible in theory (it just requires multiplying lots of polynomials together and I'm not sure it's worth it and my supervisor agrees).

A fun this is I had to produce a matrix whose entries are elements of ℚ[x,x⁻¹] which is fun! (Actually this part was my idea, when my supervisor was playing around with things he just wanted a matrix over ℚ but thought my idea was probably more fruitful!)