zhant - Network

I recently finished my master's thesis, "2D Topological Quantum Field Theories, Frobenius Structures, and Higher...

I recently finished my master's thesis, "2D Topological Quantum Field Theories, Frobenius Structures, and Higher Algebra".
(Note that I am using "higher algebra" mostly to refer to n-categories, although there is a short section on Morita contexts).
This is a hard topic to write about — the sources are too well-written for me to improve on them much!
So I tried giving my own narrative focus on the matter; hopefully the end result fills some niche.

I am no expert, so there will be errors and inaccuracies.
But I hope whoever reads it enjoys their time.

https://hdl.handle.net/20.500.14352/105943

Edit: Updated the link to a more permanent location.

Hatcher's AT + personal issuesOkay they weren't that difficult to understand now that I've rested a bit, but they were pretty...

Hatcher's AT + personal issues

Okay they weren't that difficult to understand now that I've rested a bit, but they were pretty intimidating
Now they feel almost trivial and that makes me feel dumb for taking longer than I expected, but at least it's the first days and I can blame that on me being rusty and still building momentum
I hope that not having deadlines (for once) makes me more comfortable with my own pace of study, or at least reduces the impostor syndrome

Hatcher's ATI'm reading Hatcher's AT (still at chapter 0, working on building intuition) and it's funny how fast the...

Hatcher's AT

I'm reading Hatcher's AT (still at chapter 0, working on building intuition) and it's funny how fast the examples scalate in complexity

You start with "well, a 1-dim CW is just a graph" and later on the same page you have "I hope you can visualize spheres in dimensions ≥ 4, since that's the minimum dimension where this example is meaningful"

Or "a retraction is just squashing things!" and then "look at this fucked up house, it actually retracts to a point lol"

Some other projects which I brainstormed when it became apparent that I would have to take most of the year off from university and that I...

Some other projects which I brainstormed when it became apparent that I would have to take most of the year off from university and that I might or might not end up doing:
- Learning Haskell, which I've wanted for quite a while but didn't have any immediate use case for.
- Getting into game dev (enough to submit a tiny game to a jam, and see if I like doing it).
- Working on being better at self-care and developing healthy coping mechanisms (very hard, actually).
- Researching some other topics.

I just finished the last exams I had left from my bachelor's and will have the rest of the year off, so I'll probably be a bit more...

I just finished the last exams I had left from my bachelor's and will have the rest of the year off, so I'll probably be a bit more active here now.

I'm planning on self-studying, now that I have both the time and the knowledge to pick topics that interest me, and knowing myself I'll probably have an urge to toot about it from time to time.

One of my teachers recommended me Hatcher's Algebraic Topology, and it looks interesting and engaging enough for self-study, so I'll probably start there.

I just presented my bachelor's thesis and it went pretty well so I have (anxiety-less) free time again!

I just presented my bachelor's thesis and it went pretty well so I have (anxiety-less) free time again!

I had 5 hours of consultation time with my advisor today, I feel bad about taking up way too much time lolWe thought we were going to be...

I had 5 hours of consultation time with my advisor today, I feel bad about taking up way too much time lol
We thought we were going to be done by hour 3 but then we thought a bit more about a handwaved step in a proof and got carried away in the details

Okay, bachelor thesis deposited, finally! Every time I thought I was done I found out something else to expand upon or refine, I'm going...

Okay, bachelor thesis deposited, finally!
Every time I thought I was done I found out something else to expand upon or refine, I'm going to kindly refrain from reading it until a few months pass

Making slides is hard. How do I make sure not to cram them with too much information, while also making them dense enough so that I can...

Making slides is hard. How do I make sure not to cram them with too much information, while also making them dense enough so that I can remember what I wanted?
I should probably write some notes, but still–I personally prefer having enough text to be able to reconnect when I space out while attending a talk, so I err way too much into that while preparing one.

I just disabled microtype to test something and got 7 overfull hboxes (where there were previously none)This package is black magic

I just disabled microtype to test something and got 7 overfull hboxes (where there were previously none)
This package is black magic

By the way, the computations I talk about at the start aren't the cohomology ones.The example I had in mind was when I tried to compute...

By the way, the computations I talk about at the start aren't the cohomology ones.

The example I had in mind was when I tried to compute the (dual of the) Lie algebra of the Heisenberg group.

I was confused for an hour or two as to why my results didn't match the basis I knew; did I forget how to compute in coordinates, or did I make a mistake somewhere?

It ended up being that I switched notation for the points midways AND only in the lhs, so my left action was actually a right action.

(Sorry if this is way too rambly or venty, I needed to sort some thoughts and my usual avenues for doing so probably can't relate to...

(Sorry if this is way too rambly or venty, I needed to sort some thoughts and my usual avenues for doing so probably can't relate to this as much as people here.
I made the first and last (this one) posts public, and the rest unlisted, and *as far as I know* that should prevent flooding your timelines with a dozen of rambling toots. Please tell me if that worked!
Maybe I should have put some CWs for venting/academic stress/etc? I'm not sure of the etiquette, let me know if you miss them).

The joy of learning is starting to be overshadowed by the stress of the deadline.I've known since the start that I would have to rewrite...

The joy of learning is starting to be overshadowed by the stress of the deadline.
I've known since the start that I would have to rewrite a lot of sections, and that I would need to take compromises in order to fit everything in the page limit, but I understimated both the amount of prerequisites that I needed to talk about and my ability to write concesily (as this very thread shows).

But all of that feels undermined by a page limit that feels way too restrictive, and by the fact that this is going to be graded. Maybe we...

But all of that feels undermined by a page limit that feels way too restrictive, and by the fact that this is going to be graded.
Maybe we picked way too big of a scope, or maybe I need to learn how to be concise, or maybe I can cheat a bit and move the prerequisites to an appendix (which doesn't contribute to the page limit), or maybe I *can* fit everything without big compromises (there's a lot of redundancy in my writing, for one), but still...

Then we conclude with a subsection where we talk about some recent accesible research problems (see the paper...

Then we conclude with a subsection where we talk about some recent accesible research problems (see the paper https://arxiv.org/abs/2103.06090 ), and my advisor sent me some examples of his own that I'm trying to incorporate.
(In fact, we built the dissertation around that paper, since it's accesible for the level and can be tackled from interesting angles).
That's also meaningful, and it feels great to learn with a somewhat concrete goal in focus, while still being able to meander around other topics!

You do some computations, and you conclude that can't be the case; you always map some generator to zero.So it's a contradiction, and it...

You do some computations, and you conclude that can't be the case; you always map some generator to zero.
So it's a contradiction, and it stems from assuming that M accepted a complex structure.

Of course, the unproven theorems here are doing the bulk of the work, but it still feels great!
It lets me poke the depths of the field, while still being able to remain in scope.

I really like the proof of the last one:First, you assume that the nilmanifold M = Γ\G admits a complex structure, so it's a compact...

I really like the proof of the last one:

First, you assume that the nilmanifold M = Γ\G admits a complex structure, so it's a compact complex surface.
You want to compute the cohomology of M, but by Nomizu's that's the cohomology of the Lie algebra Lie(G) = 𝔤.
You see that it's first Betti number is 2, even, so by Buchdahl-Lamari's it must be Kähler.
But then, by Lefschetz's hard Theorem, there must be a symplectic form ω in H²𝔤 such that (–)∧ω is an isomorphism from H¹𝔤 to H³𝔤.

So I have to (superficially) talk about de Rham's cohomology and Kähler manifolds, which are deep and interesting topics on its own,...

So I have to (superficially) talk about de Rham's cohomology and Kähler manifolds, which are deep and interesting topics on its own, just to name some theorems which I'll use later.

Then they're used to give some examples: the torus T⁴ admits a canonic Kähler structure, the Kodaira-Thurston nilmanifold admits both a complex and a symplectic structure but not compatibly (i.e., it doesn't admit a Kähler one), and a filiform 4-nilmanifold doesn't admit a complex structure at all.

And the only prerequisite which I don't give a proper proof (or idea of proof) for is just Frobenius' Theorem!And then there's a...

And the only prerequisite which I don't give a proper proof (or idea of proof) for is just Frobenius' Theorem!
And then there's a second part, where I talk about more concrete examples, and that's when I start to introduce a lot of theorems that are far beyond the scope of the work, as well as some research problems (Lie's Correspondence, classification of nilpotent Lie algebras, Nomizu's, Buchdahl-Lamari's, Lefschetz's hard Theorem)

In the first part which is not just differential geometry prerequisites i get to talk about complex manifolds, the Hopf problem on the...

In the first part which is not just differential geometry prerequisites i get to talk about complex manifolds, the Hopf problem on the 6-sphere and the general problem of which real manifolds accept complex structure, why we want to consider *almost* complex structures in real manifolds (and then complexify them) to tackle that problem, and the newlander-nirenberg theorem.

And i have enough space to motivate each concept, and make a cohesive narration, and that feels great!