What Are You Working On? June 13, 2022

I have been making a solution manual for Nakahara's ''Geometry, Topology, and Physics,'' 2nd edition textbook.... I don't know if that makes me a sadist or dedicated... maybe both since I'm typing them up in latex....
I've been learning more homological algebra and it's a lot of fun. There's certainly plenty of neat constructions involved and learning some algebraic topology along the side has helped a lot. Unfortunately, I haven't actually learned singular homology yet so constructions like mapping cones/cylinders for morphisms of complexes are things I can understand but haven't completely grasped in a topological context. The nice homological properties they have make everything feel motivated enough though.

I'm currently trying to understand how I might have developed homotopy of chain complexes on my own from an algebraic perspective, just for the fun of it. I think it's fair to operate under the assumption that we care about quasi-isomorphisms but they can be a little too difficult to study on their own, in part because they aren't always invertible. On the other hand, homotopy equivalences as a special case of quasi-isomorphisms have properties that *are* nice to work with, like forming an equivalence relation. As for actually recreating the definition, I'll have to think more about what properties I want that might force the definition of a homotopy.

Aluffi has begun hinting at things like homotopy categories and derived categories and it's very exciting to start learning about these because they always seemed so distant.
my mental health
I got nerd sniped into thinking about numbers that express their own proper factors. Ex: 25 fulfills this since it’s proper factors are 5, while 8 is not since it’s proper factors are 2 and 4.

What about numbers that express all of their factors? What about numbers that only express their prime factors?

Now I’m trying to see if I can prove if there are an infinite number of prime that contain a 1. I have zero training in number theory but hey, I’m having fun
This morning i flunked a Differential equation exam, so 99% chance i have to retake Calc 2 next semester. On the other hand i have a linear algebra test and i'm feeling confident, if i get >60% i pass the class
Started working on my first paper in higher category theory. Getting to typeset a lot of big diagrams!
Someone asked me the following problem earlier and I think I figured it out, but I'm still working out the details: let G be the disjoint union of an n-clique for each n and G' the disjoint union of G with a countable clique. Is G elementarily equivalent to G'?

(edit: if anyone is curious, my solution was to add a predicate Pn to the language for each n and let T be the theory of graphs where the adjacency relation is transitive and Pn(x) <-> the connected component of x has size at most n. Then argue that T admits quantifier elimination, so is model complete)
I'm working on understanding *Oscillatory Integrals of the First Kind* (in one variable) from Stein's book on Harmonic Analysis.
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Just something I thought was interesting:
I was playing around with summations and integrals,  especially summations. I was looking at the summation from n=1 to infinity of n^y x^(n-1). I was testing out if I could solve it for difference values of y, assuming |x|<1, and I got to a recursive way to solve it. The summation, I’ll call it A(x,y), equals A(x,y-1)+x*d/dx(A(x,y-1)). Just cool I thought.
Struggling hard with my complex analysis homework.

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