How to self study pure math - a step-by-step guide

this video is a collection of resources that you can use to self-study pure math there's a huge collection of books and videos many of which are freely available from which you can basically learn any branch of math you want at any level of difficulty we'll cover all the topics that one might encounter while in a pure math degree in first year one would normally see linear algebra and real analysis in second year one might see point set topology and complex analysis in a third year one might see group theory galwa theory algebraic topology and differential geometry that's a lot so let's get started with the first linear algebra this is not y equals mx plus b but something a lot more abstract it's the study of objects called vector spaces and the maps between them the best book to learn the subject is the book linear algebra done right by sheldon axler what i love about this book is the emphasis on examples so here for example the book introduces the concept of a linear map and it follows it up with several concrete examples of them this makes it really good for self-study it also has plenty of exercises here's a few of them to look at let's look at the table of contents so it starts by introducing vector spaces the basic object of study in linear algebra it then talks about linear maps which are maps between vector spaces it then talks about eigenvectors and eigenvalues so up to this point is the contents of a first course the rest of it is very interesting but it's a lot more specialized the philosophy of this book is that you shouldn't learn linear algebra using determinants the author of this book sheldon axler has actually written an article called down with determinants where he develops all of linear algebra from start to end without ever mentioning determinants this article is quite illuminating just to read on its own and it gives a lot of good context about why the book is written the way it is if you like videos the author sheldon axler has made curated playlists just for this book where for every concept in the book there is an associated video that explains that concept the videos are really short and they're in bite-sized chunks which makes them really nice for self-study next up is real analysis roughly speaking this is calculus but rigorously you'll learn about epsilon delta proofs limits derivatives and integrals all from a proof's perspective the best book for learning the subject by far is understanding analysis by stephen abbott so looking at the table of contents it starts out by introducing the real numbers it then talks about sequences and series which introduces the language of analysis in a familiar setting then it goes into the basic topology of the real number line and then calculus proper begins limits and continuity and then derivatives then sequences and series of functions and the integral if you'd like videos there's a youtube playlist full of lectures by francis sue which is one of the most amazing lecture series i've ever seen on youtube they are so clear and the professor is really really empathetic towards its students so you really feel like the professor is talking to you when you're learning the subject next up on the list is point set topology the subject is about taking concepts from calculus and linear algebra and putting them in a more abstract and general context an amazing place to learn this is these online notes from the university of toronto for every single concept there's a separate set of notes which explain the concept with a lot of examples and motivation there's also a gigantic list of problems which is appropriately called the big list when i took this course at the university of toronto i went through this list and did every single problem no joke it took a lot of time but it was immensely worth it these problems are incredibly illuminating and anyone who does them all will come away with an incredibly strong understanding of the subject next up is complex analysis so this is very roughly calculus but with complex numbers by far the best place to get an introduction for the topic is the book visual complex functions an introduction with phase portraits this is by far one of the most beautiful books i have ever read it uses color and face plots to illustrate the core ideas of complex analysis the pictures are just so beautiful and it's a great way to get a feel for complex functions before diving into the technical details if you really want to get into the details the book complex analysis by serge lang is quite good and as a plus it has 132 illustrations because somebody actually counted what i love about this book is that it has tons of nice pictures so it gives a wonderful geometric intuition for the subject it also just makes it really fun to learn so looking at the table of contents it starts with the basic definitions so what complex numbers are and the basics of how to do calculus with them the second chapter is about power series with complex functions and then the fun begins integration in the complex plane so here there are two big theorems the first is koshi's theorem which has two parts and the second is the residue theorem up to this point is the contents of a first course the rest of it is more specialized it's very interesting but it's not strictly necessary to understand the core of the subject if you're more of a video person you might like watching this playlist from wesleyan university above complex analysis all the links are in the description next up group theory roughly speaking group theory is the study of symmetry by far the best place to learn the subject is these lectures by professor benedict gross these lectures are so clear and understandable and professor gross is just so good at explaining ideas in an engaging and gripping way so if you're interested in learning group theory that's the number one place to go if you're into books you might want to check out the book topics in algebra by herstein so looking at the table of contents the part about group theory isn't the whole book but it's just chapter 2. it first introduces the basic definitions what is a group then it talks about subgroups and quotients two ways to build groups out of other groups and it culminates with the two big theorems of group theory the so-called cello theorems and the fundamental theorem of finite abelian groups what i love about this book is that there are lots of examples which makes it really nice for self-study and at the end of each chapter there is a truly gigantic list of exercises up next is galwa theory this is where group theory is applied to show that polynomials of degree 5 and higher don't have a general formula in radicals now there's a video on this channel that gives an introduction to the subject if you'd like to dip your toes in for a more in-depth explanation these notes by professor tom lanester are just amazing they are freely available on the internet and they are so understandable they have lots of exercises and examples throughout which makes them really good for self-study up next is differential geometry this is the study of curved surfaces in higher dimensional space called manifolds a wonderful book to learn the subject is the book introduction to differentiable manifolds and rimani and geometry this book is completely rigorous but it also has lots of pictures and intuitive discussions which makes it really fun to read so looking at the table of contents it starts out with an informal introduction to what manifolds are it then talks about calculus in n-dimensional space which is a prerequisite to knowing how to do calculus on curved surfaces then it introduces what manifolds are properly next it introduces the main technical tools vector fields and tensor fields the next chapter is integration on manifolds which forms the heart of the book there are two big theorems stokes's theorem which is a vast generalization of the fundamental theorem of calculus and deram's theorem which links topology and calculus in a stunning way i've made a video about these topics if you'd like to see them in more detail this part marks the end of the differential geometry portion of this book the second half of this book is about riemannian manifolds which roughly speaking are curved surfaces with a notion of distance next up is algebraic topology it's the study of how to use methods from group theory to study questions from topology a wonderful book to learn the subject is the book algebraic topology by alan hatcher the book is really really wonderful it has tons of pictures and tons of exercises it has three parts homotopy homology and co-homology which are the three main pillars of the subject another plus is that professor pierre albin has made a lecture series that follows this book pretty much chapter to chapter and he's made this lecture series freely available on youtube these lectures are also really great for understanding the material and i've put the link in the description that's all for now if you happen to know of any other resources for learning math that you found helpful type them below in the comments and i'll be sure to add them in the description thanks a lot for watching and i'll see you next video you

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Length: 9min 53sec (593 seconds)

Published: Fri Dec 24 2021