Inculcation: Linear constructions

Now that you see how deep things are, go back and focus on the details.

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

Philosophy: construct spaces and do algebra, analysis, geometry, and whatever we can 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
  • logic: propositional
  • logic: first order
  • first order set theory
• build structures on sets (or beyond) and work inside
  • inside $\mathbb{Z},\text{Q},\mathbb{R}$
  • out on groups
  • out on metric spaces
  • inside ${\mathbb{R}}^n$
  • out on vector spaces
    • out on normed $\mathbb{R}$-vector spaces, inner product spaces
  • surfaces inside ${\mathbb{R}}^3$, in submanifolds of ${\mathbb{R}}^n$
  • out on rings and fields
  • out on modules
  • 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
• what do we do now? everything here have a tiny different vibes
  • analysis
    • ODEs
    • PDEs
  • geometry
  • dynamics
  • mechanics
  • algebra
    • finite groups
    • representations
    • Lie algebras
  • topology and algebraic topology
  • algebraic geometry
  • number theory

Inside reals and metric spaces §

• #book Tao Analysis vol I, II - this is the best reference for any beginner!
• #book Apostol - Mathematical Analysis
• #book Rudin Analysis (Baby Rudin)

• Analysis in $\mathbb{R}$ §

  • Construct(!!) from $\mathbb{Z}\to \mathbb{Q}\to \mathbb{R}$
  • Sequences, limits of sequences

• 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 spaces §

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

• In normed vector spaces §

  • Analysis in finite-dimensional normed vector spaces

• Differential 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…
  • better thing to do is directly jump to smooth manifolds altogether.

Living completely outside ${\mathbb{R}}^n$ §

Going outside of ${\mathbb{R}}^n$ for analysis: there are two routes

• analysis on (finite dim) manifolds (classical mechanics, ODEs, Hamiltonian systems, oscillations)
• analysis in function spaces(infinite dim complete normed/inner product spaces) (quantum mechanics, PDEs, waves)

Analysis in normed vector spaces was just the begining.

Measure spaces (measure theory) §

I’ve no idea yet!

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
• #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!

Of course, I’ve no idea about this, yet.

• #lectures Frederic Schuller - Quantum Theory