ℝ²

inculcation

Properties1
tags

12 min read

This list was made with materials found on the internet (mostly freely available), to reduce my pain of searching each time they are needed. All opinions, arrangements, organizations etc are strictly mine. Because of that, this list lacks topics like number theory, algebraic geometry, graph theory, algorithms, combinatorics, etc.

This list was recently (August ‘26) revamped!

I

We start with groups, vector spaces and metric spaces.

finite groups

The miniseries Essence of Group Theory - YouTube carries it forward.

vector spaces

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.).

Need help proving stuff? Try following the arrows to prove equivalent conditions for injectivity of linear maps on finite dim spaces:

metric spaces

real analysis I

Napkin

Evan Chen’s A Infinitely Large Napkin is an introduction to a lots of fields of math! However, it is NOT a textbook but a really nice introductory reference. It starts with groups and metric spaces!

The Prinston Companion to Mathematics

The Princeton Companion to Mathematics

history of mathematics

lecture videos MathHistory: A course in the History of Mathematics (although be aware, the instructor doesn’t believe that exists)

handwavy physics

We learn physics though 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 mechanics, 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.

Going ahead, we can look at quantum physics.

For more mechanics one may look at

and

  • book Steven Strogatz - Nonlinear dynamics and chaos : with applications to physics, biology, chemistry, and engineering

II

We continue with algebra, analysis, topology and geometry.

rings and modules

Looking back on linear algebra, we can reiterate the following.

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

I am still looking on how to understand two linear endomorphisms giving a structure on . 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 -module defined by two linear endomorphisms . Then we always have at least one simple non-trivial -submodule of .

representation theory of finite groups

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

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

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

curves and surfaces

  • book do Carmo - Differential geometry of curves and surfaces
  • lecture videos 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).

probability and information theory

illustrations in mathematics

elementary algebraic geometry and number theory

281

III

Life and mathematical adventure is very nonlinear, but arguments should not be circular. We linearly construct from the ground up to have a stronger foundation for mathematics.

logic

set theory

real analysis II

Total derivative of function

between finite-dim real normed vector spaces at a point is supposed to be a linear map

that “approximates” near . So if is a rotation (as a linear map), then

should “look” like a rotation “near ”.

  • Analysis by Herbert Amann and Joachim Escher Volume I

complex analysis

1

Complex analysis is the study of holomorphic functions on open subsets of .

After a first course on complex analysis, we can move onto functional analytic techniques. Using such tools we can prove the Riemann mapping theorem.

There some some number theoretic results that we can prove using the theory of holomorphic functions.

We can do even more functional analysis with holomorphic functions. We can ask: what does holomorphic images of disks look like? How large or small can they be? Do they contain a disk of some fixed radius?

One may study a generalization of conformal maps, called quasiconformal homeomorphisms. We can prove results about them similar to holomorphic maps. Using this theory, we can produce an explicit sequence of maps that converge to the Riemann mapping as in the Riemann mapping theorem.

We can even look at connections to random processes, fractals and so on.

Link to original

measure theory

topology

Galois theory

algebraic topology I

topological and geometric group theory

smooth manifolds

dynamics

This is more a “dynamical systems” course, but has ODEs too

And follows the notes

|246

And move onto more dynamical systems

everything in a dynamics pov

What is “dynamics”? A dynamical system, in general, is a monoid action on a set.

Ignoring the generality, let’s consider the definition of a discrete dynamical system: a function

on a set is considered to be a discrete dynamical system (autonomous) where the iterations of the map

are considered the time map for . Thus orbit of a point is defined to be

Simply put, just “iterations of a map” produces the dynamics.

Using this simple, extremely general definition, we reinterpret a lot of math in terms of “dynamics”:

  • Linear algebra is study of iterations of one linear map on a vector space. The canonical forms essentially decomposes the dynamics into indecomposable pieces.
  • Finite group theory studies group action on a (finte) set are by definition “dynamics”
  • A module of a ring is just a special group action by the additive group of the ring which also plays well with the multiplicative structure. Representation theory in most cases studies such modules.
  • There are various applications of theorems that are “dynamical”
    • proof of existence of ODEs using contraction mapping theorem
  • Solutions of an ODE (with unique solutions) on produces a flow map

on the phase space that map initial conditions to solution curves (this is the definition). Thus this is a “continuous-time dynamics”.

  • A common proof of existence of solutions to ODEs uses a fixed point theorem, which is in turn a “dynamical” theorem!
  • Machines have a natural monoid action.
Link to original

semi-Riemannian manifolds

geometric mechanics

Schuller’s lectures are a good place to start.

People have written up notes from these lectures: THE WE-HERAEUS INTERNATIONAL WINTER SCHOOL ON GRAVITY AND LIGHT (Spanish Translation Available!) (richie291.wixsite.com) and Maths with Physics: The WE-Heraeus International Winter School on Gravity and Light, Lectures, Tutorials and Solutions .

PDEs

symplectic manifolds

computation in mathematics

IV

analysis

We start with some Functional analysis.

Then look at some Fourier analysis.

Some even more hard-core analysis.

ergodic theory

ergodic theory and Lie groups

Link to original

Lie groups, Lie algebras and their finite dimensional representation theory

introductory

Link to original

  • book Fulton, Harris - Representation theory

Lie groups and Lie algebras

Link to original

mechanics

  • book Gregory L. Naber - Topology, Geometry and Gauge fields - two volumes
  • book Mikio Nakahara - Geometry, topology, and physics

algebraic topology II

symplectic dynamics

differential geometry

algebraic varieties

  • book Harris - Algebraic geometry

commutative, homological and categorical algebra

Well…

more complex analysis, Riemann surfaces, algebraic curves, hyperbolic surfaces

2

Riemann surfaces are connected complex 1-manifolds. Compact Riemann surfaces are same as -algebraic curves. We can study meromorphic functions and meromorphic forms on these spaces.

There are alternative ways to define Riemann surfaces: the nicest of which is as the ringed space of functions locally isomorphic to open subsets of .

And there are many more references… a hundred more books and notes available online https://mathoverflow.net/questions/313254/references-for-riemann-surfaces.

Standard texts prove the monodromy theorem for analytic continuations. However, that just says when two such continuations are bound to be the same. We don’t have any result about existence of analytic continuations. Why?

It is because they may behave very weirdly.

We however know what is the maximal extent of analytic continuations. Riemann asked this question originally. Ahlfors has the Etale space definition of domain of global holomorphic functions. Forster denotes this as . Its connected components actually exhaust the entire class of open Riemann surfaces (I think?).

  • ??

We can look at examples of such global domains of functions/gems of functions such as , or such fractional power of polynomials , and so on. These functions are roots of polynomials for example satisfies . Some people call such functions algebraic. Standard texts such as Forster talks about Riemann surfaces of algebraic functions, which are more generally roots of polynomials over meromorphic functions over a compact Riemann surface . Such functions have global domain a finite branched cover over .

Alright, we dealt with algebraic functions that satisfy a polynomial with coefficients in the field of meromorphic functions. What other holomorphic germs can we deal with? How about the elliptic integral

where . How may we describe these functions? Well there are two ways to talk about them.

First, we can construct their Riemann surface. Forster does this, even for higher degree polynomials hyperelliptic integrals.

Second, we can try to ask what’s the “inverse” of the function defined above. There is a nice answer to this! Its the Weirstrass elliptic function . Such elliptic functions are meromorphic on complex tori isomorphic to the elliptic curve defined by the polynomial . These are the periods of this elliptic curve.

Period of an elliptic curve uniquely reconstructs it as a Riemann surface. However, doing this with compact Riemann surfaces of genus higher than one is a bit tricky. Equivalent question is how may we “invert” integrals of the form

where is a polynomial of degree 5 or higher. The degree 3,4 case corresponds to the case of elliptic curves.

The works of Abel and Jacobi shows that we can think of these integrals as holomorphic 1-forms on compact Riemann surfaces and their inverses are functions on their Jacobian, a complex tori of dimension = genus of .

The theorem of Torelli shows that the period of a higher genus compact Riemann surface also reconstructs it uniquely.

This chain of thought somehow leads us to Hodge theory.

The uniformization theorem for Riemann surfaces state that any simply connected Riemann surface is biholomorphic to either , the unit disk or .

  • lecture videos Riemann Surfaces by Jacob Bernstein for MSRI summer school 2014: Complex geometry and geometric analysis on complex manifolds ^yg7hmg
    • Prerequisites:
      • Knowledge of basic complex analysis—at the level of Ahlfors, Complex Analysis, Chapters 1-5—will be assumed. Some basic familiarity with (abstract) surface theory and differential forms will be helpful. However, I will review this material as needed.
    • Reading:
    • Other useful references:
      • Farkas and Kra, Riemann Surfaces; a classical text on the subject.
      • Miranda, Algebraic Curves and Riemann Surfaces; a more algebraic perspective.
    • Week 1: Introduction to Riemann Surfaces
      • Surfaces and Topology
      • Riemann Surfaces and Holomorphic Maps
      • Maps between Riemann Surfaces
      • Calculus on Riemann Surfaces
      • De Rham Cohomology
    • Week 2: Geometric Analysis on Riemann Surfaces
      • Elliptic Functions and Integrals
      • Meromorphic Functions
      • Inverting the Laplacian
      • The Uniformization Theorem
      • Riemann Surfaces and Minimal Surfaces

By the uniformization theorem, every Riemann surface is a quotient of , the unit disk or by a discrete subgroup of their automorphism group acting freely. No such quotient exists for . We can easily classify the quotients of . The quotients of the unit disk by torsion free Fuchsian groups are a intersecting class of objects of study.

Riemann surfaces which are quotients of the unit disk by torsion free Fuchsian groups are biholomorphic if and only if the Fuchsian groups are conjugate inside the automorphism group of the unit disk. Oriented hyperbolic surfaces (connected, complete Riemannian 2-manifolds of constant sectional curvature -1) are also quotients of the unit disk by torsion free Fuchsian groups, with oriented isometries related to conjugacies. Therefore we may study, even classify Riemann surfaces through the theory of oriented hyperbolic surfaces.

The space of all marked Riemann surfaces of genus is called Teichmuller space of genus Riemann surfaces. These spaces are complex manifolds of dimension depending on the genus .

  • book Farb Margalit - Mapping class groups
Link to original

several complex variables and complex manifolds

Riemannian geometry

A second course in Riemannian geometry is of fashion.

We can specifically look at Ricci flows!

The fields of spectral geometry and geometric analysis are very interesting!

  • lecture videos Masoud Khalkhali - Spectral geometry
  • book Olivier Lablée - Spectral Theory in Riemannian Geometry
  • lecture videos Aaron Naber - Geometric Analysis
    • Prerequisites: Basics of manifolds, tensors, and differential forms. Basics of PDE theory, for instance Evans’s book Partial Differential Equations, in particular those chapters on second order elliptic and parabolic equations. Familiarity with exponential maps, injectivity radius, and geodesics would be helpful, for instance chapter one of Jost’s book Riemannian Geometry and Geometric Analysis is more than sufficient.
    • Reading: The main source will be Petersen’s book on Riemannian Geometry. We will also rely on Jost’s Riemannian Geometry and Geometric Analysis, and on the book by Cheeger Degeneration of Riemannian Metrics Under Ricci Curvature Bounds. More advanced topics will use relevant papers in the field.
    • Week 1: Introduction to Geometric Analysis
      • Review of Manifolds and Smooth Structure
      • Introduction to Curvature and Geodesic Coordinates
      • Laplacians and Harmonic Coordinates
      • Heat Kernels and Geometry
      • Sectional Curvature and Finite Diffeomorphism Theorems
    • Week 2: Topics in Regularity Theory
      • Ricci Curvature, Volume Monotonicity and Rigidity Theorems
      • Ricci Curvature and Almost Rigidity Theorems
      • Lower Ricci Curvature and Stratification Theorems
      • Bounded Ricci Curvature and ε-regularity Theorems
      • Outline of Regularity Theory for Einstein Manifolds
  • lecture videos 【MIT数学课程】 流形上的几何：18.966 Geometry of Manifolds Ⅱ 2021 Spring_哔哩哔哩_bilibili
  • MAT 6229: Geometric spectral theory | Institut des sciences mathématiques

We may also look at some Kahler geometry.

abstract harmonic analysis, operator theory

  • book Folland - Abstract harmonic analysis

Transclude of inculcation-lie-groups#banach-algebras-c--algebras

analytic group theory

Algebraic geometry

Well…

V

symmetric spaces, semisimple Lie groups, their unitary representations and discrete subgroups

We look at locally symmetric spaces and correspondingly discrete subgroups of semisimple Lie groups: their deformations and phenomenon of rigidity. We have Mostow’s strong rigidity.

Lie groups, symmetric spaces and beyond

Link to original

large events

Link to original

global, microlocal, infinite dimensional analysis

nilpotent Lie groups and their unitary representations

number theory

low dimensional topology and geometry

dynamics and rigidity

locally symmetric spaces of higher rank and hyperbolic manifolds in higher dimensions

symplectic topology

algebraic curves and Riemann surfaces, moduli and Teichmuller spaces

3

We can consider the algebraic perspective to Riemann surfaces.

We may define a weaker notion of isomorphism of Riemann surfaces: quasiconformal homeomorphism. The class of marked Riemann surfaces quasiconformal to a fixed Riemann surface generalise the Teichmuller space of compact Riemann surfaces.

  • book Hubbard - Teichmuller theory volume 1

A hyperbolic surface comes with its Laplacian. We may study its spectrum and eigenfunctions and ask interesting questions!

We may study the spectral theory of modular surfaces and extract number theoretic facts.

We can ask questions about moduli of Riemann surfaces, such as its spectra as genus goes to infinity and so on.

We can compactify the moduli space.

Link to original

VI

8 items under this folder.

Graph View