$$
%% Below are very ill-defined categories
%% Linear Algebra
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%% Abstract Algebra
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\end{tikzcd}
}
\newcommand{\scmplx}[5]{ % single complex
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#1\arrow[r, "#2"]\& #3\arrow[r, "#4"]\& #5
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}
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\DeclareMathOperator{\ann}{Ann}
\DeclareMathOperator{\ztensor}{\otimes_{\mathbb Z}}
\DeclareMathOperator{\zHom}{Hom_{\mathbb Z}}
\DeclareMathOperator{\qz}{\mathbb Q/\mathbb Z}
\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\Spin}{Sp} % TODO (?): Change to \Sp
\DeclareMathOperator{\Ind}{Ind}
\DeclareMathOperator{\CoInd}{CoInd}
\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator{\Ext}{Ext}
\renewcommand{\hom}{\mathrm H}
\DeclareMathOperator{\ZG}{\Z G}
\DeclareMathOperator{\sat}{sat}
\DeclareMathOperator{\Torsion}{Torsion}
\DeclareMathOperator{\lead}{lead}
\DeclareMathOperator{\qtensor}{\otimes_{\Q}}
\DeclareMathOperator{\Stab}{Stab}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\SL}{SL}
\DeclareMathOperator{\PSL}{PSL}
\DeclareMathOperator{\PGL}{PGL}
\DeclareMathOperator{\SO}{SO}
\DeclareMathOperator{\ob}{ob}
\DeclareMathOperator{\Mor}{Mor}
\DeclareMathOperator{\Mod}{Mod}
\DeclareMathOperator{\Irr}{Irr} % Do I want a separate section for representation theory?
%% Algebraic Number Theory/Field Theory
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\newcommand{\qadjns}[1]{\qadjs{-#1}}
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\newcommand{\zadjs}[1]{\zadj{\sqrt {#1}}}
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\newcommand{\legendre}[2]{\left(\frac{#1}{#2}\right)}
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\newcommand{\zadjzeta}[1]{\Z\left[\zeta_{#1}\right]}
\newcommand{\sep}[1]{#1_{\mathrm{sep}}}
\newcommand{\nabs}[0]{|\,\cdot\,|} % norm + absolute value
\newcommand{\gnabs}[0]{|g^{-1}(\,\cdot\,)|}
\newcommand{\codiff}[1]{#1^* }
\newcommand{\compl}[1]{#1^{\wedge}} % Completion
\newcommand{\al}[1]{#1^{\mrm{al}}}
\DeclareMathOperator{\norm}{N}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\disc}{disc}
\DeclareMathOperator{\Gal}{Gal}
\DeclareMathOperator{\knorm}{\norm_{K/\mathbb Q}}
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\DeclareMathOperator{\ktrace}{\trace_{K/\mathbb Q}}
\DeclareMathOperator{\Char}{char}
\DeclareMathOperator{\denom}{denom}
\DeclareMathOperator{\Frac}{Frac}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\Fr}{Fr}
\DeclareMathOperator{\trdeg}{trdeg}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\Pic}{Pic}
\DeclareMathOperator{\atensor}{\otimes_A}
\DeclareMathOperator{\ord}{ord}
\DeclareMathOperator{\Frob}{Frob}
\DeclareMathOperator{\vol}{vol}
\renewcommand{\split}{\textrm{split}}
\DeclareMathOperator{\Qbar}{\overline\Q}
\DeclareMathOperator{\lcm}{lcm}
%% Modular Forms/Curves
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\newcommand{\glp}{\GL_2^+}
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%% Group cohomology/class field theory
\DeclareMathOperator{\cores}{cor}
\DeclareMathOperator{\tatehom}{\wh\hom}
%% Point-Set Topology
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\newcommand{\clos}[1]{\overline{#1}}
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\DeclareMathOperator{\Cl}{Cl}
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\DeclareMathOperator{\SP}{SP}
\DeclareMathOperator{\Homeo}{Homeo}
%% Differential Geometry/Topology
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\newcommand{\del}[0]{\partial}
\newcommand{\vft}[2]{#1\pderivd{x_1}+#2\pderivd{x_2}} % vector (field) in \R^2
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\newcommand{\dbz}[0]{\d\bar z}
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\renewcommand{\d}{\mathrm d} % This had a \, in the front before. Will I regret removing it?
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\DeclareMathOperator{\dvol}{dvol}
\DeclareMathOperator{\Todd}{Todd}
\DeclareMathOperator{\ch}{ch}
%% Algebraic Topology
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\DeclareMathOperator{\redhom}{\wt\hom}
\DeclareMathOperator{\RP}{\mathbb R\mathbb P}
\DeclareMathOperator{\CP}{\mathbb C\mathbb P}
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\DeclareMathOperator{\hofiber}{hofiber}
% Lie Theory
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\newcommand{\rs}[1]{#1^{\mrm{rs}}}
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\DeclareMathOperator{\Lie}{Lie}
\DeclareMathOperator{\ad}{ad}
\DeclareMathOperator{\Ad}{Ad}
%% Complex/Algebraic Geometry + Sheaf Theory
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\newcommand{\red}[1]{#1_{\mrm{red}}}
\newcommand{\cs}[1]{\parens{\ints{#1},\abs{#1}}} % Complex space
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\newcommand{\smp}[1]{#1^{\mrm{sm}}} % smooth points
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\DeclareMathOperator{\Spec}{\mbf{Spec}}
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\DeclareMathOperator{\PAb}{PAb}
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\DeclareMathOperator{\kod}{kod}
\DeclareMathOperator{\Sing}{Sing}
\DeclareMathOperator{\Alb}{Alb}
\DeclareMathOperator{\NS}{NS}
\DeclareMathOperator{\Ht}{ht}
\DeclareMathOperator{\codim}{codim}
\DeclareMathOperator{\Sch}{Sch}
\DeclareMathOperator{\Bl}{Bl}
%% Analysis
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\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\dabs}[1]{\left\|#1\right\|}
\DeclareMathOperator{\BV}{BV}
%% Quantum Mechanics/Computing
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%% Cryptography
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\newcommand{\uniform}{\xleftarrow R}
\DeclareMathOperator{\Adv}{Adv}
\DeclareMathOperator{\parity}{parity}
\DeclareMathOperator{\reverse}{reverse}
\DeclareMathOperator{\EXP}{EXP}
\DeclareMathOperator{\poly}{poly}
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\DeclareMathOperator{\Commit}{Commit}
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\DeclareMathOperator{\PRF}{PRFAdv}
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\DeclareMathOperator{\DDH}{DDHAdv}
\DeclareMathOperator{\pub}{pub}
\DeclareMathOperator{\priv}{priv}
\DeclareMathOperator{\key}{key}
%% Complexity Theory
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\DeclareMathOperator{\NL}{NL}
\DeclareMathOperator{\coNL}{coNL}
\DeclareMathOperator{\coNP}{coNP}
\DeclareMathOperator{\TIME}{TIME}
\DeclareMathOperator{\ccP}{P} % cc = complexity class
\DeclareMathOperator{\SAT}{SAT}
\DeclareMathOperator{\UNSAT}{UNSAT}
\DeclareMathOperator{\Perm}{Perm} % Permanent of a matrix
\DeclareMathOperator{\MAJPP}{MAJPP}
\DeclareMathOperator{\ccRP}{RP}
\DeclareMathOperator{\coRP}{coRP}
\DeclareMathOperator{\ZPP}{ZPP}
%% Logic
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%% Machine Learning
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\newcommand{\Layer}[2]{#1^{\left[#2\right]}}
\newcommand{\KL}[2]{\mathrm{KL}\left(#1\left\|\,#2\right.\right)}
%% Probability/Statistics
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\DeclareMathOperator{\Cov}{Cov}
\DeclareMathOperator{\Var}{Var}
\DeclareMathOperator{\Corr}{Corr}
%% Diagrams
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\begin{tikzcd}[ampersand replacement=\&]
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%% Limit type things
%% Letters/Fonts
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\newcommand{\mfm}{\mathfrak m}
\newcommand{\mfX}{\mathfrak X}
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\newcommand{\mfh}{\mathfrak h}
\newcommand{\mfH}{\mathfrak H}
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\newcommand{\ms}{\mathscr}
\newcommand{\mfq}{\mf q}
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\newcommand{\mc}{\mathcal}
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\newcommand{\mcS}{\mc S}
\newcommand{\mcM}{\mc M}
\newcommand{\mcL}{\mc L}
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\newcommand{\Lam}{\Lambda}
%\renewcommand{\S}{\mathbb S}
\newcommand{\F}{\mathbb F}
\newcommand{\Q}{\mathbb Q}
\newcommand{\Z}{\mathbb Z}
\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\E}{\mathbb E}
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\newcommand{\T}{\mathbb T}
\newcommand{\A}{\mathbb A}
\newcommand{\eps}{\varepsilon}
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\renewcommand{\P}{\mathbb P}
\DeclareMathOperator{\msE}{\ms E}
\renewcommand{\a}{\alpha}
\renewcommand{\b}{\beta}
%% Grouping Operators
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%% Misc
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\newcommand{\from}{\leftarrow}
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\newcommand{\xinto}[1]{\overset{#1}\into}
\newcommand{\too}{\longrightarrow}
\newcommand{\xtoo}{\xlongrightarrow}
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\newcommand{\into}{\hookrightarrow}
\newcommand{\onto}{\twoheadrightarrow}
\newcommand{\dashto}{\dashrightarrow}
\newcommand{\mapdesc}[5]{
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\renewcommand{\div}{\mathrm{div}} % Might regret this one day
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$$
My top five favorite posts:
All posts grouped by tags
-
Classification of (Complex) Elliptic Surfaces without Multiple Fibers
To some extent, every post I write is written for me moreso than for you, the proverbial reader. However, this post is especially written for me. I want to understand the construction of complex elliptic surfaces with prescribed global monodromy, but don’t want to type up the details of the parts of this story that are needed for this construction (e.g. the local classification of elliptic surfaces), so here we are. I’ll start by recalling the local picture 1, and then we’ll look at the globa...
-
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 ...
