Inculcation: Linear constructions

Coming from the main article: inculcation, you have seen how deep things are. So now we go back and focus on the details.

starting out §

Life is very non-linear, but arguments should not be circular.

Philosophy: construct spaces and do algebra, analysis, geometry, and whatever we want to do!

• the idea is to create meaning, objects out of nothing…, even in familiar spaces and then going to unfamiliar ones!
• reminder: nothing (even rigor) is more scary than doing wrong stuff!

We start with logic and set theory, do analysis, algebra, geometry and topology: there is no actual distinction between their build-up, only in their vibes.

• build the grounds
  • construct propositional logic
  • construct first order based on propositional logic
  • construct ZFC set theory based on first order logic
• build structures on sets (or beyond) and work inside
  • inside $\mathbb{Z},\text{Q},\mathbb{R}$
  • studying groups
  • out on metric spaces
  • inside ${\mathbb{R}}^n$ or vector spaces
    • out on normed $\mathbb{R}$-vector spaces, inner product spaces
  • out on rings and fields
  • in graphs
  • out on categories
  • Out on topological spaces
  • Out on measure spaces
  • out on smooth manifolds
    • smooth manifolds with more structures
  • out on infinite dim spaces

finite groups §

little things: groups, group actions §

One should start their journey with groups!

• Essence of Group Theory - YouTube
• Chapter 1 and 3 from

  Napkin §

  Evan Chen's A Infinitely Large Napkin

  • an introduction to a lots of fields of math! (NOT a textbook but a really nice introductory reference)
  • starts with groups and metric spaces!

  Link to original

More into doing more algebra

Link to original

Inside the reals and metric spaces §

Real analysis, task 1

We must set theoretically construct(!!), starting from $\mathbb{Z}$, $\mathbb{Q}$ and then $\mathbb{R}$.

Real analysis, task 2

• Define sequences, and the $ϵ-N$ definition of limits of sequences.
• Try to prove that $\frac{1}{n}$ converges to $0$ as $n\to \infty$

• book Tao Analysis volume I, II - this is the best reference for any beginner!
• book Apostol - Mathematical Analysis
• lectures Real Analysis on YouTube – Francis Su
• lectures Real Analysis by Prof. S.H. Kulkarni, Department of Mathematics, IIT Madras - NPTEL
• book Rudin - Principles of Mathematical Analysis (Baby Rudin)
• book Stein, Shakarchi - Fourier analysis

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• Metric spaces with the intuition of $\mathbb{R}$ §

  • Sequences and series
  • $ϵ-\delta$ limits of functions, continuous functions
  • get motivated for topological spaces: prove the theorem that a function is continuous if and only if preimage of open sets is open

Inside finite dim vector spaces §

We do more inside ${\mathbb{R}}^n$ and venture slightly out to do analysis in finite-dim real normed vector spaces, which are of course linearly bijective to ${\mathbb{R}}^n$.

• Inculcation: Linear algebra
• Inculcation: Analysis in finite-dimensional vector spaces: total derivatives, measure theory, differential forms, flows, holomorphic functions

Diffferential geometry of submanifolds of ${\mathbb{R}}^n$

lecturenotes diffgeo.pdf (ethz.ch) One can distinguish extrinsic differential geometry and intrinsic differential geometry. The former restricts attention to submanifolds of Euclidean space while the latter studies manifolds equipped with a Riemannian metric. The extrinsic theory is more accessible because we can visualize curves and surfaces in ${\mathbb{R}}^3$ , but some topics can best be handled with the intrinsic theory…

But a better thing to do is directly jump to smooth manifolds altogether…

Over rings §

AKA rings and modules

There are usually three

• first course on rings and modules

  • linear algebra after doing rings and modules §

    A $k$-vector space $V$ with a endomorphism $T$ (fancy name for linear map $V\to V$) gives a $k[X]$-module structure on $V$, so we can directly use $k[X]$-modules classification theorems to construct the theorems on canonical forms.

    I am still looking on how to understand two linear endomorphisms giving a $k[X,Y]$ structure on $V$. At the least, I can re-interpret the theorem that says “we always have a common eigenvector of two commuting linear maps” as the following

    Let’s say we have a $k[X,Y]$-module $V$ defined by two linear endomorphisms $V\to V$. Then we always have at least one simple non-trivial $k[X,Y]$-submodule of $V$.

    Link to original

• commutative algebra
• (non-commutative) algebraic structures

representation theory of finite groups §

There are different levels to study representation theory of finite groups:

• before doing finite groups when one has basic ideas of matrices/linear algebra one may try to ponder on
  • consider a finite set of $n\times n$ matrices which are closed under multiplication and has inverses:
  • easy example is for an invertible matrix $A$ consider the set

$$\{I,A,A^2,\dots \}$$

this set might be finite or infinite, if this is finite then $A^n=I$ for some $n$, then what can we say about the matrix $A$ - Try to prove that if the field is $\mathbb{C}$ (or any algebraically closed field) $A$ must be diagonalizable. - Find what the eigenvalues of $A$ may be if the field is $\text{C}$. -

representation theory is just spicy linear algebra §

A group homomorphism from a (say, finite for now) group $G$ to the general linear group on a vector space $V$

$$G\to GL(V)$$

is called a representation of the group $G$. One can classify and study such homomorphisms (upto an equivalence ofcourse) and it’s called representation theory (of finite groups). This “helps” in doing linear algebra when we have a invertible linear map $V\to V$, in my opinion.

Link to original

• after doing finite groups and linear algebra it is easy to study representation theory of finite groups any text like
  • book Artin
  • book Fulton Harris
  • book Dummit and Foote covers all the introductory theory
• after doing rings and modules the same theory hits different
  • rephrase everything about representations of a group $G$ with its group algebra $\mathbb{C}[G]$
• after doing commutative algebra
  • …

Over fields §

AKA fields and galois theory

Out on measure spaces and infinite dim vector spaces §

AKA functional analysis, abstract measure theory, probability theory

• lectures Frederic Schuller - Quantum Theory
• https://www.kryakin.site/am2/Stein-Shakarchi%5D-4-Functional-Analys.pdf
• Papa and grandpa Rudin

Out on topological spaces §

AKA the fields of topology, algebraic topology

First semester course on Topology (AKA general topology/point set topology): as opposed to cute topology, we prove more content here

• book Munkres
• http://www.math.toronto.edu/ivan/mat327/?resources
• http://math.iisc.ac.in/~gadgil/topology-2021/all-lectures/
• Topology (MTH-TOP) - YouTube
• For a quick one lecture introduction with motivation: Lecture 1: Topology (International Winter School on Gravity and Light 2015) - YouTube

First semester course on Algebraic topology: study of holes in topological spaces

We learn homotopy groups, homology groups and at last cohomology groups!

• book Hatcher’s textbook
• lectures Algebraic Topology - Pierre Albin - YouTube

Out on smooth manifolds §

AKA “intrinsic differential geometry” or analysis on manifolds.

May try lectures Frederic Schuller -International Winter School on Gravity and Light 2015 without any other context or to get into it fully: inculcation-smooth-manifolds.

Out on Riemannian manifolds §

Working with topological groups and Lie groups §

Lie groups are famous. There are different levels to study Lie groups:

• after doing finite groups one may try to study matrix Lie groups, using knowledge of multivariable calculus, it is not ideal
• after doing topology, smooth manifolds you must already know definitions of topological and Lie groups along with their Lie algebras and exponential map
• after doing algebraic topology you will start understanding what simply connected Lie groups allow that its quotients by discrete subgroups don’t

There are references here: inculcation-lie-groups

Out on algebraic varieties §