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Inculcation

Mar 27, 2023, 5 min read

Driven by infinite enthusiasm

This list was made with materials mostly found on the internet, to reduce repetition from my side. All opinions are strictly mine. Because of that, this list lacks topics like number theory, algebraic geometry, graph theory, algorithms, combinatorics, etc.

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. dive into new discoveries
  3. a need for clarity and details: Linear constructions
  4. the old discoveries in new light: new way of looking at things
  5. epilogue: starting again?

little things: groups, group actions

One should start their journey with groups!

More into doing more algebra

big things: vector spaces (linear algebra

starting out

Philosophy of linear algebra I - finding happiness in small things (mathematical minimalism)

  • We see with just the “little” definition of a vector space, we can have things like writing any vector as a unique linear combination of finitely many vectors from a smaller subset of the entire space.
  • This much of structure is enough to ask a lot of questions and a solve a whole lot of problems!
  • If you want more things, we can have more things! (oriented vector space, inner product spaces, normed vector spaces, etc.).

pre-first semester course essense

Start with 3b1b:

A very good place to start (basic) mathematics like linear algebra is to read chapters from Napkin:

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

Philosophy of linear algebra II - giving lore behind objects

  • origin story of matrices: they were actually “linear maps” all along!
    • As in every matrix gives a linear map between the vector spaces .
    • Now try learning how to write a matrix for a given linear map between aribtrary vector spaces
    • Rotations are linear maps! Trace is a linear map! Transposition of matrices is a linear map! Try writing all of them as matrices (if they are not defined as matrices anyways)
    • What about derivatives tho? They are linear maps too? Can we write them as a matrix?
  • origin story of tensors: they are actually multi-linear maps!
    • Maybe wait a while to start with tensors, but keep this idea in the back of your mind: tensors are multi-linear maps in a proper sense.
Link to original

And then for a first semester course.

things in the middle: analysis

  • from Napkin
    • Chapters 26-30 - Calculus 101
    • Chapters 42-45 - Total derivatives and differential forms
    • Chapters 2, 6-8 - Topology

Inside the reals and metric spaces

Real analysis task 1

We must set theoretically construct(!!), starting from , and then .

Real analysis task 2

  • Define sequences, and the definition of limits of sequences.
  • Try to prove that converges to as

  • Metric spaces with the intuition of

    • Sequences and series
    • 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

For “multi-variable calculus” AKA analysis on you can choose to start from:

mechanics

If you’re into physics, start here:

all of physics, all at once

Before doing anything, just watch this:

Mechanics using the Action principle

(ignore the Fourier expansion of electric field stuff, just the ideas behind GR, QFT matters!)

Continue with GR and action: General Relativity by Prof. Thanu Padmanabhan - YouTube

After that watch these lectures covering Newtonian, Lagrangian, Hamiltonian, Statistical mechanics, special relativity all at once!

  • These are nice as an intro “proper” physics, they will look fascinating, but my recommended levels of motivation and precision is absent. These are “Feynman lectures done right”. One may watch his non-linear dynamics and quantum mechanics lectures right after this.
  • But nothing is explained rigorously, although hinted at, lots of details are skipped and Balki name drops a LOT of stuff. You may choose to ignore them initially, because each term becomes a rabbit hole for math topics.

Balakrishnan's quantum physics

Link to original

This is how I started! Its the “non-conventional” route, study major building blocks of physics at once because

Quote

“Nature doesn’t work in semesters.” - T Paddy

dive into new discoveries

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

continue with algebra

So essentially what this should entail is more group theory, rings, and modules over them. A good way to bring all of these together is representation theory of finite groups

  • book Algebra Chapter 0 is a fat book that mixes in ideas of category theory while doing algebra
  • Discover the field of algebraic geometry while doing the last few chapters of Artin. Follow the book Harris - First course in Algebraic geometry for more.
  • You’ll quickly realise a lot of rings and commutative algebra is needed for this. Take help of an algebraist / algebraic geometer near you to continue further!

Another subject is Galois theory which is broadly missing from Artin. Try this fat book:

  • book Dummit and Foote - Algebra

continue with analysis

After the reals and metric spaces we move onto finite-dimensional vector spaces or just :

differential geometry of curves and surfaces

AKA what I would call ”spicy multi-variable calculus in dimension 3”!

  • book do Carmo - Differential geometry of curves and surfaces
  • lectures ICTP Diploma - Differential Geometry - Claudio Arezzo - YouTube
    • These lectures has pre-requisites of basic linear algebra, analysis in , knowing total derivatives and bilinear forms with introducing yourself a little topology (compactness, connectedness).
    • Does a bit of manifolds at the end!

start some cute topology

Visualizing higher dimensions

The meme is

A mathematician’s reply is

What I do is simply follow

  • step 0: Picture 1, 2, 3 dimensional vector spaces
  • step 1: Write instead of 1, 2 or 3 (imagine n=1,2 or 3 but don’t write it down, only use n) whenever I am generalizing. Note if we never use the fact that or , i.e. we write all the time etc, pretend we’re working with a natural number but visualize only 2 or 3 dimensions then our work is done: we have “visualized dim ”, by simply forgetting or , yay!
  • step 2: put whatever natural number you like!

That’s what a first semester course on linear algebra is supposed to do initially (after defining the “dimension” of a vector space). This is the algebraic way to visualize things: using symbols!

There’s a next step for doing functional analysis type stuff:

  • step 3: Don’t even think about n being a natural number anymore, it can be infinite.

This doesn’t help much working with infinite dimensional vector spaces, but alright!

We can generalize “dimension” and “spaces” to things beyond vector spaces: they are called ”manifolds”. There are many “different” manifolds of a fixed dimension even: a circle and (real line) are both manifolds of dimension 1, or “1-manifolds”.

2-manifolds are just surfaces: 2-sphere, torus, etc. is a 3-manifold, so is the “solid ball” 3-ball.

We can’t even all 3-manifolds like the 3-torus, 3-sphere (the sphere that lives in ) etc.

But I can visualize the 3-sphere upto removal of just a tiny a point! The joke/reason being 3-sphere minus a point is just or 3-ball (just like how the usual 2-sphere minus a point is just the disk or “2-ball”).

Link to original

dynamical systems and ODEs

interpret any general ODE as a vector field and study it visually

Did you ever think an ordinary differential equations book will have this picture:

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This is Perko’s book Differential Equations and Dynamical Systems, Third Edition (2006), a good read, if you’re familiar with Analysis in . Otherwise any physics text on “non-linear dynamics” works, for example Steven Strogatz’s Non-linear dynamics lecture videos and book.

Link to original

much elementary introduction

A short introduction with examples from models in population dynamics:

A full semester course:

Steven Strogatz's Non-linear dynamics

and his book Nonlinear dynamics and chaos : with applications to physics, biology, chemistry, and engineering

This playlist (and the channel) consists of shorter videos:

Link to original

formal theory of ODEs

Real analysis on and gives a foundation to general theory of ODEs.

ICTP's Dynamical systems

This is more a “dynamical systems” course, but has ODEs too. Lecture notes: SLI.pdf (bris.ac.uk)

And as I said above, this book has a pre-requisite of some analysis, but still one must cover the topics in the contents:

Tip Differential Equations and Dynamical Systems, Third Edition (2006)

|500

Link to original

More on this: inculcation-odes

probability and information

Well, I have no idea yet!

continue with handwavey physics

Be warned:

|400

but start with inculcation-all-of-mechanics

Linear constructions

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
    • studying groups
    • out on metric spaces
    • inside or vector spaces
      • out on normed -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
Link to original

new way of looking at things

With the language of manifolds/normed vector spaces we can work in geometry, topology and physics properly!

starting again?

How to study?

  • Make lecture notes.
  • Taking notes in Obsidian
    • you’ll need to learn bit of LaTeX and probably will need a drawing tablet
  • Taking notes in a physical notebook
    • Scan the lecture notes as PDF into one file and save it inside your vault. This 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

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