\chapter{Basic Concepts and Results}
Here's a detailed outline of basic grounding in mathematics.
This is skeletal, in the sense that a real education
includes more topics and examples, but these are the bones.
conformal geometry of the sphere / projective geometry of complex line
/ hyperbolic geometry (heavenly sphere etc.)
contour integrals
Cauchy integral formula
\section{Analysis}
- partitions of unity (essential in manifold theory)
\subsection{Measure theory}
Lebesgue decomposition
Radon-Nikodym
\subsection{Functional analysis}
- Dirac delta (and its derivatives)
- lots of function spaces
\section{Algebra}
groups, rings, fields
\subsection{Lie theory}
I put this under algebra rather than geometry because it is fundamentally very rigid and algebraic.
\subsection{Commutative algebra and algebraic geometry}
- classification fg mod over PID
- Spec etc.
(Atiyah-MacDonald)
\subsection{Homological algebra}
- chain complex
- homology of a chain complex
- projective and injective resolutions
- homological dimension (of ring, of module, of chain complex)
\subsection{Category theory}
Basic concepts
- category (objects, maps)
- functor
- natural transformation of functors
(remark about issues in generalizing to $n$-categories)
- universal properties
- basic constructions:
- products and coproducts
- initial and final objects
- equalizers and coequalizers
- direct limit and inverse limit (injective/projective limit) of sequence
- generally: limits and inverse limits of diagrams
- adjoints and monads
- (Kan extensions)
Examples
- groupoids
- concrete categories and skeletal subcategories
(formalizing notions of concrete examples/writing down)
- reflective and coreflective subcategories (and reflections/coreflections)
...and many, many explicit examples of categories, adjunctions, etc.
\section{Geometry and Topology}
\subsection{Point-set topology}
Nutshell: objects and maps (topological spaces, continuous maps) and basic properties
(connected and compact)
- metric topology: metric, convergence, open/closed, neighborhood, continuity (\epsilon-\delta)
- general topology
topology
open/closed
neighborhood
basis for a topology (esp. in metric form)
(nets)
- maps
continuity (several definitions, including in terms of basis, yielding $\epsilon$-$\delta$)
homeomorphism (isomorphism in category of topological spaces and continuous maps)
- topological properties: connected, compact, (notion of a local property: locally )
\subsection{Homotopy theory}
- homotopy of maps
- homotopy group
- covering spaces: both to compute fundamental group, and interesting example themselves
\subsection{Manifold topology}
- vector fields
...as derivations and Lie bracket
- differential forms
\subsection{Manifold geometry}
- (Pseudo)-Riemannian metric
- connection
- curvature
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\section{Set Theory and Logic}
- cardinal
- Cantor's diagonalization
- ordinal
- axiom of choice
- ZF axioms
(rarely need to remember them precisely)
\subsection{Deep theorems}
Some big results in logic that non-logicians are aware of (and respect),
but don't know the proofs unless they've studied logic seriously,
as these are deep and technical.
- Goedel's incompleteness/inconsistency theorem
- Cohen's proof of the independence of Axiom of Choice and (Generalized) Continuum Hypothesis
(via set forcing)
\subsection{Computation}
increasingly important
See Nabutovsky-Weinberger,
especially as written up in
``Computers, Rigidity, and Moduli - The Large-Scale Fractal Geometry of Riemannian Moduli Space''
- halting problem
- unclassifiability of
\chapter{Other branches of math}
By ``other'' I do not mean to slight these fields
(from what little I know of them they are quite beautiful and deep),
peripheral, less central
Thus rather than outline them (since
*consumers*
applied math: consumers
discrete math: a different kind of math
stats to algebraic number theory
set theory:
\section{Probability and Statistics}
Probability is deeply related to analysis
Information and entropy is deep
\section{Combinatorics and Discrete Math}
some parts very algebraic
\section{Numerical Analysis}
Numerical solutions to PDEs
Matrix math
Optimization: linear programming, non-linear programming