Inculcation

Driven by infinite enthusiasm §

All opinion are strictly mine.

This list was made with materials mostly found on the internet, to reduce repetition from my side.

What to study? Where to study from? §

(order is not strict, life is very non-linear)

This has five major sections

1. journey starts: start with the little things and the big things
2. discoveries: dive into math physics ✨
3. a need for clarity and details: Linear constructions
4. the old discoveries in new light: apply your new-found knowledge new way of looking at things never ends ✨
5. epilogue: starting again?

start with the little things and the big things §

• little things: groups (group theory, group actions) §

  • One should start their journey with groups!
  • Group theory, abstraction, and the 196,883-dimensional monster - YouTube
  • Essence of Group Theory - YouTube
  • Chapter 1 and 3 from evan-chen-napkin

• big things: Vector spaces (linear algebra) §

  • start here: inculcation-linear-algebra

A very good place to start is this

Evan Chen Napkin

Tip 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

• things in the middle: Analysis §

  • definitely start with Tao’s Analysis volumes 1 and 2, but for more: [[inculcation#inside-reals-and-metric-spaces|Analysis in $\mathbb{R}$]]

  • Anaysis and linear algebra in ${\mathbb{R}}^n$ §

    • from evan-chen-napkin
      • Chapters 26-30 - Calculus 101
      • Chapters 42-45 - Total derivatives and differential forms
      • Chapters 2, 6-8 - Topology
    • shifrin-multivariable-mathematics - a 2 semester course/book that covers linear algebra and (proper) multivariable calc
  • more in Rigorous constructions

• Mechanics §

  • Start here: inculcation-all-of-mechanics

dive into math/physics ✨ §

The first section had groups, vector spaces and analysis. Now we shall continue and add some geometry, dynamics and physics too!

continue with algebra §

• lectures Abstract Algebra by Benedict Gross - YouTube
• book Artin - Algebra
• book Algebra Chapter 0

continue real analysis §

After

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

---

Link to original

we move onto finite-dimensional vector spaces or just ${\mathbb{R}}^n$.

cute topology and geometry §

• differential geometry of curves and surfaces in ${\mathbb{R}}^3$ §

  • AKA what I would call spicy multi-variable calculus in dimension 3!
  • lectures ICTP Diploma - Differential Geometry - Claudio Arezzo - YouTube
    • These lectures has pre-requisites of baisic linear algebra, analysis in ${\mathbb{R}}^n$” knowing total derivatives and bilinear forms with introducing yourself a little topology (compactness, connectedness)

• cute topology §

  • Topology & Geometry by Dr Tadashi Tokieda - YouTube
  • Knot theory
    • How The Most Useless Branch of Math Could Save Your Life - YouTube
    • Knot Theory - YouTube
• how topology affects and interacts with geometry, analysis, algebra (Lie groups, say) and physics
  • Gauss–Bonnet theorem - Wikipedia
  • Dirac’s belt trick, Topology, and Spin ½ particles - YouTube
  • The derivative isn’t what you think it is. - YouTube

dynamical systems and ODEs §

vector fields and ODE dictionary §

What we do is, write a differential equation like

$${\dot{x}}_i=f_i(x_1,x_2,\dots ,x_n)$$

for all $1\le i\le n$ so we have $n$ equations for $n$ variables $x_i$ and make it even more compact by

$$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$$

where $\mathbf{f}:\mathcal{U}\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$ is a vector field on the open domain $\mathcal{U}$.

This gives us a geometric pov on ODEs in ${\mathbb{R}}^n$, and we have a

solving differential equations	analysis and geometry of vector fields
an equation $\dot{x}=\mathbf{f}(x)$	a vector field $\mathbf{f}:\mathcal{U}\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$
solutions of the equation $\begin{array}{rl}\dot{x}(t)& =\mathbf{f}(x(t))\\ x(0)& =x_0\end{array}$	integral curves of the vector field $t\mapsto {\Phi }_t^{\mathbf{f}}(x_0)$
how solutions depend on initial conditions $x_0\mapsto x(t)$	flows of the vector field $x_0\mapsto {\Phi }_t^{\mathbf{f}}(x_0)$
conserved quantities	integrals of the vector field
(linearly) decoupling the differential equation $\dot{z_i}=f_i(z_i)$	(linear) coordinate transformation such that $\mathbf{f}={\sum }_i{f}_i(z_i){\hat{z}}_i$

We may convert ordinary differential equation of any order to first order by taking enough independent variables and defining them to be higher derivatives.

Link to original

More on this: inculcation-odes

probability and information §

Well, I have no idea yet!

handwavey physics §

• Start with inculcation-all-of-mechanics
• Definitely do more math: lie groups, representations, fluid dynamics, and classical mechanics etc before quantum.

Linear constructions §

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

Link to original

apply your new-found knowledge: new way of looking at things never ends ✨ §

With the language of manifolds/normed vector spaces we can work in geometry, topology and physics properly! (no handwaves!)

• Riemannian geometry, semi-Riemannian geometry, GR
• ODEs Inculcation: ODEs, smooth dynamical systems
• Ergodic theory, dynamical systems in measure spaces
• PDEs
• talk Mathematics of Turbulent Flows: A Million Dollar Problem! by Edriss S Titi - YouTube
• Hamiltonian systems and symplectic geometry
• Thermodynamic systems, ODEs and contact geometry
• QM
• lecturenotes Use measure theory to do Classical Equilibrium Statistical Mechanics
• Differential topology, or algebriac topology with a differential viewpoint
• Probability, information theory

starting again? §

• lectures MathHistory: A course in the History of Mathematics

How to study? §

• How to do lectures? §

  • Use https://obsidian.md to organize the amount of content/
    • Copy the lecture note
    • Complete each lecture and tick the check box.
  • Make lecture notes
    • in Obsidian.
      • you’ll need to learn LaTeX and probably will need a drawing tablet
    • in physical notebook.
      • Scan the lecture notes as PDF into one file and save it inside your vault.
      • Might seem/be a waste of time, but loosing hard worked lecture notes/not organizing them is a bigger loss.
    • Filling pages upon pages and never returning back onto them is NOT the process, it seems ro me. Return to the notes, think, and solve your own questions.

more references, roadmaps §

Stuff missing from here

• Number theory!
• Graph theory, algorithms, combinatorics, etc.

Newsletter?, channel, chat?

You may want to join academic curiosity whatsapp community here for more math content!

• Look here:
  • The fast track – Sheafification
  • How to become a GOOD Theoretical Physicist (goodtheorist.science)
  • http://theportal.wiki/wiki/Read
  • 20-prerequisites-for-quantum-mechanics
  • book Paddy - Theoretical Astrophysics volumes I, II & III
• Watch (any one or all even) these one lecture and tell me how can you not love this!! (don’t pay attention to the name of the video)
  • Partial Differential Equations - Giovanni Bellettini - Lecture 01 - YouTube
  • History of Algebraic Topology - Pierre Albin - YouTube