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What Are You Working On? May 23, 2022

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I just put a paper on the arxiv! It'll come out on Wednesday :)
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In between having a new sleeping problem where I wake up at six in the evening *regardless* of when I went to sleep the previous night and the current lectures being about numerical linear algebra which does my fucking head in, I did no work last week. My exam is next Monday, so I have to just do it this week.

My med student brother recommended nicotine gum as a stimulant, and it kind of works so far.

This year has been hell, and it won't even end until August when the rest of my exams are being taken.
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I’m reading a paper on directed algebraic topology with a professor, currently about defining topological spaces with some notion of partial order. One can think of it as using geometry to study processes that are directed in time. An example of such a process is a computing process, where resources are reserved, utilised and unreserved by concurrent processes.
I’m also going to start studying algebra.
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Exploring probability, combinatorics and set theory. It's fun and intuitive, I love it.
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Reviewing naive set theory before I begin an independent study of abstract algebra and real analysis
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Working on finite state transducers, specifically trying to apply transformations to one tape while leaving the relationship with the other tape intact.
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I am considering the product V\sub{n,m}(Q)=\prod\sub{i=Q+1}^{Q+m} (1-[1/(i,p\sub{n}\#)]) and the sum over this product S\sub n(m)=\sum\sub{i=0}^{p\sub{n}\#-1} V\sub{n,m}(i).

In particular, I have found that S\sub n(2m+2)-S\sub n(2m+1)=S\sub n(2m+1)-S\sub n(2m), and I am looking for similar identities for multiples of $m$ that are larger than 2.

Edit: not sure how it might translate, but it would be nice to be able to use MathJAX here.
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Trying to crack into category theory and homological algebra through Category theory for the working mathematican. Finding it quite challenging, especially coming from someone outside out algebra with more a topology background (no at). A lot of the exercises related to topology presupposes knowledge in algebraic topology (which is the thing I'm trying to work up to from category theory). Any suggestions?

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