$$
%% Below are very ill-defined categories
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%% Complex/Algebraic Geometry + Sheaf Theory
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%% Analysis
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%% Quantum Mechanics/Computing
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\DeclareMathOperator{\TIME}{TIME}
\DeclareMathOperator{\ccP}{P} % cc = complexity class
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%% Logic
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%% Machine Learning
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%% Diagrams
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%% Limit type things
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$$
My top five favorite posts:
All posts grouped by tags
-
Characteristic Classes
For a long time I’ve been telling myself that I should try and get a solid understanding of characteristic classes, so why not do that now?
-
A Tour of Some Number Theory: Part I: Elliptic Curves
I have a vague vision in my head of a series of posts designed to help me, and maybe also you, understand (parts of) the relationship between a few objects of interests in algebraic number theory: elliptic curves, selmer groups, binary quartic forms, and modular forms 1. I don’t know the whole story encompassing these objects, but allegedly, they are all linked in one way or another, and I have soft plans to understand these links over the next $n$ days. To aid in this, I hope to write (mediu...
-
Brown Representability
Let’s prove and discuss one of my favorite theorems from algebraic topology. It gives a characterization of when a contravariant “homotopy functor” from topological spaces 1 to sets is representable, i.e. is (isomorphic to one) of the form $F(X)=[X,S]$ where $S$ is $F$’s representing object, and the notation $[X,S]$ denotes the set of homotopy classes of maps $X\to S$. The nice part about this is that many nice/important spaces one encounters when studying algebraic topology are secretly just...
-
Spectral Sequences
Back when I was in high-school, I became really interested in this thing called “machine learning.” The main idea is that you bombard some algorithm with a ton of examples (of a task being performed or of objects being classified), and then you cross your fingers and hope it has managed to reliably learn how to do what the examples showcased. One big draw of this approach is that there are many tasks where it is not clear how to accomplish them, but where it feels like there has to be enough ...
-
A Nice Lemma about Dedekind Domains
Often times in these posts, the main focus is some big/nice theorem/result; however, this time there’s a nice lemma about Dedekind domains that I think merits its own post. This is partially because I’m still shocked that it’s true, and partially because I don’t know where else it is written down. After proving it, I will (maybe briefly 1) mention one of its uses 2. However, I think this use is less exciting than the lemma itself. The gist of the lemma is that lattices over Dedekind domains c...
