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

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

Over fields §

AKA fields and galois theory

Out on measure spaces §

AKA abstract measure theory, probability theory

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.

Inside infinite dim spaces §

AKA functional analysis!

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

Out on algebraic varieties §