My main hobby is reading textbooks, so I decided to go beyond the scope of the question posed. I took a look at what I have on my shelves in order to recommend particularly good or standard books that I think could characterize large portions of an undergraduate degree and perhaps the beginnings of a graduate degree in the main fields that interest me, plus some personal favorites.

Neuroscience: Theoretical Neuroscience is a good book for the field of that name, though it does require background knowledge in neuroscience (for which, as others mentioned, Kandel's text is excellent, not to mention that it alone can cover the majority of an undergraduate degree in neuroscience if corequisite classes such as biology and chemistry are momentarily ignored) and in differential equations. Neurobiology of Learning and Memory and Cognitive Neuroscience and Neuropsychology were used in my classes on cognition and learning/memory and I enjoyed both; though they tend to choose breadth over depth, all references are research papers and thus one can easily choose to go more in depth in any relevant topics by consulting these books' bibliographies.

General chemistry, organic chemistry/synthesis: I liked Linus Pauling's General Chemistry more than whatever my school gave us for general chemistry. I liked this undergraduate organic chemistry book, though I should say that I have little exposure to other organic chemistry books, and I found Protective Groups in Organic Synthesis to be very informative and useful. Unfortunately, I didn't have time to take instrumental/analytical/inorganic/physical chemistry and so have no idea what to recommend there.

Biochemistry: Lehninger is the standard text, though it's rather expensive. I have limited exposure here.

Mathematics: When I was younger (i.e. before having learned calculus), I found the four-volume The World of Mathematics great for introducing me to a lot of new concepts and branches of mathematics and for inspiring interest; I would strongly recommend this collection to anyone interested in mathematics and especially to people considering choosing to major in math as an undergrad. I found the trio of Spivak's Calculus (which Amazon says is now unfortunately out of print), Stewart's Calculus (standard text), and Kline's Calculus: An Intuitive and Physical Approach to be a good combination of rigor, practical application, and physical intuition, respectively, for calculus. My school used Marsden and Hoffman's Elementary Classical Analysis for introductory analysis (which is the field that develops and proves the calculus taught in high school), but I liked Rudin's Principles of Mathematical Analysis (nicknamed "Baby Rudin") better. I haven't worked my way though Munkres' Topology yet, but it's great so far and is often recommended as a standard beginning toplogy text. I haven't found books on differential equations or on linear algebra that I've really liked. I randomly came across Quine's Set Theory and its Logic, which I thought was an excellent introduction to set theory. Russell and Whitehead's Principia Mathematica is a very famous text, but I haven't gotten hold of a copy yet. Lang's Algebra is an excellent abstract algebra textbook, though it's rather sophisticated and I've gotten through only a small portion of it as I don't plan on getting a PhD in that subject.

Computer Science: For artificial intelligence and related areas, Russell and Norvig's Artificial Intelligence: A Modern Approach's text is a standard and good text, and I also liked Introduction to Information Retrieval (which is available online by chapter and entirely). For processor design, I found Computer Organization and Design to be a good introduction. I don't have any recommendations for specific programming languages as I find self-teaching to be most important there, nor do I know of any data structures books that I found to be memorable (not that I've really looked, given the wealth of information online). Knuth's The Art of Computer Programming is considered to be a gold standard text for algorithms, but I haven't secured a copy yet.

Physics: For basic undergraduate physics (mechanics, e&m, and a smattering of other subjects), I liked Fundamentals of Physics. I liked Rindler's Essential Relativity and Messiah's Quantum Mechanics much better than whatever books my school used. I appreciated the exposition and style of Rindler's text. I understand that some of the later chapters of Messiah's text are now obsolete, but the rest of the book is good enough for you to not need to reference many other books. I have little exposure to books on other areas of physics and am sure that there are many others in this subreddit that can give excellent recommendations.

Other: I liked Early Theories of the Universe to be good light historical reading. I also think that everyone should read Kuhn's The Structure of Scientific Revolutions.

If you like the online course lectures, you should definately look at those. I know tons of great schools such as Yale, UCLA, MIT, Stanford etc. etc. offer full lecture series on youtube. Usually the syllabi are online for you to look at so you can get a feel for it.

I am more of a book learner myself so I will try to make some recommends, but when looking for books try googling, reading stackexchange posts and Amazon reviews.

I'm going to disagree with /u/Orion952 on Fraleigh's book, its an alright book but I have seen much better. For Abstract Algebra, I would recommend Nicholson's book. Its a very gentle introduction to the subject. There are lots of computation problems as well as proofs you can work through so you can get a nice feel for the subject. I would also hunt down the pdf for Dummit and Foote's book as well, I thought it was pretty gentle for the most part as well as comprehensive.

For analysis and topology, I have encountered some decent books.

Strichartz for analysis is very wordy and conversational, so I didn't care for it myself hence didn't read very much of it (I much prefer the style of Walter Rudin) but it might be good for starting out.

Bhatt has written a very nice book for analysis and covers a lot of material on metric space topology. I actually know the author pretty well so if you are interested in the book I may be able to hook you up.

Simmons has written a book that has a pretty conversational style, but I wasn't a big fan of his style. Bhatt's book will have a more "traditional" approach, but thats not to say it isn't readable. The first half of the book will cover the same stuff Bhatt's book does and the second half will be more advanced stuff including some concepts from Functional Analysis (which is a pretty interesting topic).

For Topology, if you have read some of the analysis books above, I would say Munkres' book is nice and it has tons of examples. But try googling beginner topology books if you want to get into the subject sooner, I know I have seen a few stackexchange threads on this.

These are really the topics one needs to know to really dive into mathematics beyond rote computation. I'm sure there are more books out there but these come off my head at this moment.

Machine learning is largely based on the following chain of mathematical topics

Calculus (through Vector, could perhaps leave out a subsequent integration techniques course)

Linear Algebra (You are going to be using this all, a lot)

Abstract Algebra (This isn't always directly applicable but it is good to know for computer science and the terms of groups, rings, algebras etc will show up quite a bit)

General Topology (Any time we are going to deal with construction of a probability space on some non trivial manifold, we will need this. While most situations are based on just Borel sets in R^n or C^n things like computer vision, genomics, etc are going to care about Random Elements rather than Random Variables and those are constructed in topological spaces rather than metric ones. This is also helpful for understanding definitions in well known algorithms like Manifold Training)

Real Analysis (This is where you learn proper constructive formulations and a bit of measure theory as well as bounding theorems etc)

Complex Analysis (This is where you will get a proper treatment of Hilbert Spaces, Holomorphic functions etc, honestly unless you care about QM / QFT, P-chem stuff in general like molecular dynamics, you are likely not going to need a full course in this for most ML work, but I typically just tell people to read the full Rudin: Real and Complex Analysis. You'll get the full treatment fairly briefly that way)

Probability Theory (Now that you have your Measure theory out of the way from Real Analysis, you can take up a proper course on Measure Theoretic Probability Theory. Random Variables should be defined here as measurable functions etc, if they aren't then your book isn't rigorous enough imho.)

Ah, Statistics. Statistics sits atop all of that foundational mathematics, it is divided into two main philosophical camps. The Frequentists, and the Bayesians. Any self respecting statistician learns both.

After that, there are lots, and lots, and lots, of subfields and disciplines when it comes to statistical learning.

A sample of what is on my reference shelf includes:

Real and Complex Analysis by Rudin

Functional Analysis by Rudin

A Book of Abstract Algebra by Pinter

General Topology by Willard

Machine Learning: A Probabilistic Perspective by Murphy

Bayesian Data Analysis Gelman

Probabilistic Graphical Models by Koller

Convex Optimization by Boyd

Combinatorial Optimization by Papadimitriou

An Introduction to Statistical Learning by James, Hastie, et al.

The Elements of Statistical Learning by Hastie, et al.

Statistical Decision Theory by Liese, et al.

Statistical Decision Theory and Bayesian Analysis by Berger

I will avoid listing off the entirety of my shelf, much of it is applications and algorithms for fast computation rather than theory anyway. Most of those books, though, are fairly well known and should provide a good background and reference for a good deal of the mathematics you should come across. Having a solid understanding of the measure theoretic underpinnings of probability and statistics will do you a great deal--as will a solid facility with linear algebra and matrix / tensor calculus. Oh, right, a book on that isn't a bad idea either... This one is short and extends from your vector classes

Tensor Calculus by Synge

Anyway, hope that helps.

Yet another lonely data scientist,

Tim.

I study topology and I can give you some tips based on what I've done. If you want extra info please PM me. I'd love to help someone discover the beautiful field of topology. TLDR at bottom.

If you want to study topology or knot theory in the long term (actually knot theory is a pretty complicated application of topology), it would be a great idea to start reading higher math ASAP. Higher math generally refers to anything proof-based, which is pretty much everything you study in college. It's not that much harder than high school math and it's indescribably beneficial to try and get into it as soon as you possibly can. Essentially, your math education really begins when you start getting into higher math.

If you don't know how to do proofs yet, read How to Prove It. This is the best intro to higher math, and is not hard. Absolutely essential going forward. Ask for it for the holidays.

Once you know how to prove things, read 1 or 2 "intro to topology" books (there are hundreds). I read this one and it was pretty good, but most are pretty much the same. They'll go over definitions and basic theorems that give you a rough idea of how topological spaces (what topologists study) work.

After reading an intro book, move on to this book by Sutherland. It is relatively simple and doesn't require a whole lot of knowledge, but it is definitely rigorous and is definitely necessary before moving on.

After that, there are kind of two camps you could subscribe to. Currently there are two "main" topology books, referred to by their author's names: Hatcher and Munkres. Both are available online for free, but the Munkres pdf isn't legally authorized to be. Reading either of these will make you a topology god. Hatcher is all what's called algebraic topology (relating topology and abstract algebra), which is super necessary for further studies. However, Hatcher is hella hard and you can't read it unless you've really paid attention up to this point. Munkres isn't necessarily "easier" but it moves a lot slower. The first half of it is essentially a recap of Sutherland but much more in-depth. The second half is like Hatcher but less in-depth. Both books are outstanding and it all depends on your skill in specific areas of topology.

Once you've read Hatcher or Munkres, you shouldn't have much trouble going forward into any more specified subfield of topology (be it knot theory or whatever).

If you actually do end up studying topology, please save my username as a resource for when you feel stuck. It really helps to have someone advanced in the subject to talk about tough topics. Good luck going forward. My biggest advice whatsoever, regardless of what you study, is read How to Prove It ASAP!!!

TLDR: How to Prove It (!!!) -> Mendelson -> Sutherland -> Hatcher or Munkres

hey nerdinthearena,

i too find this area to be fascinating and wish i knew more on the upper end myself. i'm just going to list off a few resources. in my opinion, graduate school will concentrate a lot on progressing your technical knowledge, but will likely not give you a lot of time to hone your intuition (at least in the first few years). so, the more time you spend in undergraduate school doing so, the better.

helpful for intuition and basic understanding

• geometrical vectors by gabriel weinreich - why there's more to vectors and how to draw them. this basically shows visual examples of differential forms and the like.
  
• advanced calculus: a differential forms approach by harold edwards - a fantastic approach to differential forms. it's a treasure, so just read it. :) it is actually well built for skimming and skipping around.
  
• vector calculus, linear algebra, and differential forms: a unified approach - this is a big book, but you could just jump ahead to the later sections covering differential forms for an interesting approach and technical skill building.
  
• div, grad, curl are dead by william burke - pure intuition and similar to geometrical vectors. don't look for anything else here, but definitely read it.
  
• applied differential geometry by william burke - a more complete work than the above and showcases a lot of the connections between geometry and physics.
  
  

  more advanced but still intuitive
  

• an introduction to manifolds - this is at the same level and covers essentially the same material as lee, but in my opinion, it is much better. it is much more concise and to the point with more focused exercises.
  
• differential forms in algebraic topology by raoul bott and loring tu - the sequel to the above, but i haven't read beyond the first chapter. it is well known as a must read classic, but having some topology background will help with this book.
  
• the laplacian on a riemannian manifold: an introduction to analysis on manifolds - an interesting introduction to geometric analysis.
  
• differentiable manifolds: forms, currents, harmonic forms by georges de rham - a classic by one of the greats and also one of the only sources i've found that covers de rham currents (besides research papers that just use them).
  
• riemannian manifolds: an introduction to curvature by john lee - the sequel to lee's smooth manifolds book.
  
  hopefully this helps. if i were to revisit geometric analysis, i would basically use the above books to help bone up my understanding, intuition, and technical skill before moving on. these are also mainly geometry books, so learning analysis (like functional analysis) would be good as well. i mainly have three suggestions there.
  
  

  three general analysis favorites
  

• introductory functional analysis with applications by erwin kreyszig - used everywhere.
  
• calculus of variations by gelfand and fomin
  
• linear differential operators by cornelius lanczos - a geometrical look at the subject. also take a look at the variational principles of mechanics by him as well.

I was in the same position as you in high school (and am finishing my math major this semester). Calculus is not "math" in the sense you're referring to it, which is pure mathematics, without application, just theory and logic. Calculus, as it is taught in high school, is taught as a tool, not as a theory. It is boring, tedious, and has no aesthetic appeal because it is largely taught as rote memorization.

Don't let this bad experience kill your enthusiasm. I'm not sure what specifically to recommend to you to perk your enthusiasm, but what I did in high school was just click around Wikipedia entries. A lot of them are written in layman enough terms to give you a glimpse and you inspire your interest. For example, I remember being intrigued by the Fibonacci series and how, regardless of the starting terms, the ratio between the (n-1)th and nth terms approaches the golden ratio; maybe look at the proof of that to get an idea of what math is beyond high school calculus. I remember the Riemann hypothesis was something that intrigued me, as well as Fermat's Last Theorem, which was finally proved in the 90s by Andrew Wiles (~350 years after Fermat suggested the theorem). (Note: you won't be able to understand the math behind either, but, again, you can get a glimpse of what math is and find a direction you'd like to work in).

Another thing that I wish someone had told me when I was in your position is that there is a lot of legwork to do before you start reaching the level of mathematics that is truly aesthetically appealing. Mathematics, being purely based on logic, requires very stringent fundamental definitions and techniques to be developed first, and early. Take a look at axiomatic set theory as an example of this. Axiomatic set theory may bore you, or it may become one of your interests. The concept and definition of a set is the foundation for mathematics, but even something that seems as simple as this (at first glance) is difficult to do. Take a look at Russell's paradox. Incidentally, that is another subject that captured my interest before college. (Another is Godel's incompleteness theorem, again, beyond your or my understanding at the moment, but so interesting!)

In brief, accept that math is taught terribly in high school, grunt through the semester, and try to read farther ahead, on your own time, to kindle further interest.

As an undergrad, I don't believe I yet have the hindsight to recommend good books for an aspiring math major (there are plenty of more knowledgeable and experienced Redditors who could do that for you), but here is a list of topics that are required for my undergrad math degree, with links to the books that my school uses:

• elementary real analysis
  
• linear algebra
  
• differential equations
  
• abstract algebra
  
  And a couple electives:
  
  
• topology
  
• graph theory
  
  And a couple books I invested in that are more advanced than the undergrad level, which I am working through and enjoy:
  
  
• abstract algebra
  
• topology
  
  Lastly, if you don't want to spend hundreds of dollars on books that you might not end up using in college, take a look at Dover publications (just search "Dover" on Amazon). They tend to publish good books in paperback for very cheap ($5-$20, sometimes up to $40 but not often) that I read on my own time while trying to bear high school calculus. They are still on my shelf and still get use.

I’m not sure precisely what you mean by “contemporary” or “geometric algebra” or “basic number elements and algebra”. What did you feel was missing from Lang’s book? (I’m not familiar with its contents.)

If you want something in line with the standard high school curriculum, but maybe a bit more rigorous than most, this book by Kiselev was the standard Russian school text for generations (review)

Or you could try the Art of Problem Solving geometry book (site).

There’s a lot of good stuff in Coxeter and Greitzer’s book Geometry Revisited, but I’d say it probably assumes a standard high school geometry course as a prerequisite.

Not really limited to plane geometry, but I really like Hilbert and Cohn-Vossen’s book Geometry and the Imagination (review). I’d recommend getting a used copy of the original printing; the recent ones are printed on demand and not as nice.

Also let me recommend Apostol and Mamikon’s lovely book New Horizons in Geometry (review), though it’s more about calculus than algebra per se.

If you want to study plane curves from a complex number perspective, you could try Zwikker’s 1963 The advanced geometry of plane curves and their applications

If by geometric algebra you mean Grassmann/Clifford/Hestenes style algebra, check out the stuff Jim Smith has been doing, or you could take a look at this thing (I haven’t read it), or try these papers.

They probably aren’t what you’re looking for, but I think Farouki’s Pythagorean Hodograph Curves are pretty neat (that book also has a lot of other interesting material in it). Also neat for formalistic theorizing about algebras for spline curves is Ramshaw’s monograph On Multiplying Points: The Paired Algebras of Forms and Sites (probably a bit abstract for what you want here).

What are your goals? Do you want to design lenses and mirrors for cameras? Model classical mechanics systems? Construct arbitrary shapes out of polynomial curves so you can draw fonts or animate characters on a computer screen? Design cut paths for CNC machines? Approximate transcendental functions by some type of function that you can more easily compute with? Find the prettiest proofs of thousand-year-old theorems about circles? Prepare yourself to study differential geometry or algebraic topology? ...

We need to make a few definitions.

A group is a set G together with a pair of functions: composition GxG -> G and inverse G -> G, satisfying certain properties, as I'm sure you know.

A topological group is a group G which is also a topological space and such that the composition and inverse functions are continuous. It makes sense to ask if a topological group for example is connected. Every group is a topological group with the discrete topology, but in general there is no way to assign an interesting (whatever that means) topology to a group. The topology is extra information that comes with a topological group.

A Lie group is more than a topological group. A Lie group is a group G that is also a smooth manifold and such that the composition and inverse are smooth functions (between manifolds).

In the same way that O(n) is the set of matrices which fix the standard Euclidean metric on R^n, the Lorentz group O(3,1) is the set of invertible 4x4 matrices which fix the Minkowski metric on R^4. The Lorentz group inherits a natural topology from the set of all 4x4 matrices which is homeomorphic to R^16. It is some more work to show that the Lorentz group in fact smooth, that is, a Lie group.

It is easy to see the Lorentz group is not connected: it contains orientation preserving (det 1) matrices and orientation reversing (det -1) matrices. All elements are invertible (det nonzero), so the preimage of R+ and R- under the determinant are disjoint connected components of the Lorentz group.

There are lots of references. Munkres Topology has a section on topological groups. Stillwell's Naive Lie Theory seems like a great undergraduate introduction to basic Lie groups, although he restricts to matrix Lie groups and does not discuss manifolds. To really make sense of Lie theory, you also need to understand smooth manifolds. Lee's excellent Introduction to Smooth Manifolds is an outstanding introduction to both. There are lots of other good books out there, but this should be enough to get you started.

Anti-disclaimer: I do have personal experience with all the below books.

I really enjoyed Lee for Riemannian geometry, which is highly related to the Lorentzian geometry of GR. I've also heard good things about Do Carmo.

It might be advantageous to look at differential topology before differential geometry (though for your goal, it is probably not necessary). I really really liked Guillemin and Pollack. Another book by Lee is also very good.

If you really want to dig into the fundamentals, it might be worthwhile to look at a topology textbook too. Munkres is the standard. I also enjoyed Gamelin and Greene, a Dover book (cheap!). I though that the introduction to the topology of R^n in the beginning of Bartle was good to have gone through first.

I'm concerned that I don't see linear algebra in your course list. There's a saying "Linear algebra is what separates Mathematicians from everyone else" or something like that. Differential geometry is, in large part, about tensor fields on manifolds, and these are studied by looking at them as elements of a vector space, so I'd say that linear algebra is something you should get comfortable with before proceeding. (It's also great to study it before taking quantum.) I can't really recommend a great book from personal experience here; I learned from poor ones :( .

Also, there are physics GR books that contain semi-rigorous introductions to differential geometry, even if these sections are skipped over in the actual class. Carroll is such a book. If you read the introductory chapter and appendices, you'll know a lot. On the differential topology side of things, there's Schutz, which is a great book for breadth but is pretty material dense. Schwarz and Schwarz is a really good higher level intro to special relativity that introduces the mathematical machinery of GR, but sticks to flat spaces.

Finally, once you have reached the mountain top, there's Hawking and Ellis, the ultimate pinnacle of gravity textbooks. This one doesn't really fall under the anti-disclaimer from above; it sits on my shelf to impress people.

I think these days it's really important to make it to the generalized stokes theorem, not just for an honors crowd but in general. This means covering differential forms. Hubbard and Hubbard has been mentioned.

Not a book but in my mind a very nice update on H&H is Ghrist's video lecture on multivariable calculus which covered traditional integral theorems (Green, Gauss and Stokes) while showing their full relationship to generalized stokes in a very natural way. I really think this is a kind of template how modern courses on multivariable/vector calculus should be taught these days. it's not just the content but also the order of presentation that is very neat and maximizes clarity.

There are a bunch of books that had treaded this path over the years. Loomis & Sternberg, and Harold Edwards are books worth considering, though H&H is in some sense most detailed while also having a nice pace.

I actually believe that there is a dearth of really good updated and polished books in the area, and that there are so few really good options calls for some effort to develop lecture notes into books on the topic.

Okay, I'm going to go against the mold here and say Lee's Topological Manifolds is your best bet. This book seemed to be the clearest, most thorough treatment of the topic of the topic at an introductory level. I think it's definitely a lot more verbose than Munkres, but I see it as an advantage as a lot of details are spelled out explicitly (which I something I like to see in textbooks that introduce a new topic to a unfamiliar audience). Plus the extra emphasis on manifolds is a very nice feature of the text.

That is just my recommendation, though. I think Munkres is a great book as well, but Lee's book seemed to present things in a way that clicked with me a bit better.

You need some grounding in foundational topics like Propositional Logic, Proofs, Sets and Functions for higher math. If you've seen some of that in your Discrete Math class, you can jump straight into Abstract Algebra, Rigorous Linear Algebra (if you know some LA) and even Real Analysis. If thats not the case, the most expository and clearly written book on the above topics I have ever seen is Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.

Some user friendly books on Real Analysis:

1. Understanding Analysis by Steve Abbot
   
   
2. Yet Another Introduction to Analysis by Victor Bryant
   
   
3. Elementary Analysis: The Theory of Calculus by Kenneth Ross
   
   
4. Real Mathematical Analysis by Charles Pugh
   
   
5. A Primer of Real Functions by Ralph Boas
   
   
6. A Radical Approach to Real Analysis by David Bressoud
   
   
7. The Way of Analysis by Robert Strichartz
   
   
8. Foundations of Analysis by Edmund Landau
   
   
9. A Problem Book in Real Analysis by Asuman Aksoy and Mohamed Khamzi
   
   
10. Calculus by Spivak
    
    
11. Real Analysis: A Constructive Approach by Mark Bridger
    
    
12. Differential and Integral Calculus by Richard Courant, Edward McShane, Sam Sloan and Marvin Greenberg
    
    
13. You can find tons more if you search the internet. There are more superstars of advanced Calculus like Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra by Tom Apostol, Advanced Calculus by Shlomo Sternberg and Lynn Loomis... there are also more down to earth titles like Limits, Limits Everywhere:The Tools of Mathematical Analysis by david Appelbaum, Analysis: A Gateway to Understanding Mathematics by Sean Dineen...I just dont have time to list them all.
    
    Some user friendly books on Linear/Abstract Algebra:
    
    
14. A Book of Abstract Algebra by Charles Pinter
    
    
15. Matrix Analysis and Applied Linear Algebra Book and Solutions Manual by Carl Meyer
    
    
16. Groups and Their Graphs by Israel Grossman and Wilhelm Magnus
    
    
17. Linear Algebra Done Wrong by Sergei Treil-FREE
    
    
18. Elements of Algebra: Geometry, Numbers, Equations by John Stilwell
    
    Topology(even high school students can manage the first two titles):
    
    
19. Intuitive Topology by V.V. Prasolov
    
    
20. First Concepts of Topology by William G. Chinn, N. E. Steenrod and George H. Buehler
    
    
21. Topology Without Tears by Sydney Morris- FREE
    
    
22. Elementary Topology by O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev and and V. M. Kharlamov
    
    Some transitional books:
    
    
23. Tools of the Trade by Paul Sally
    
    
24. A Concise Introduction to Pure Mathematics by Martin Liebeck
    
    
25. How to Think Like a Mathematician: A Companion to Undergraduate Mathematics by Kevin Houston
    
    
26. Introductory Mathematics: Algebra and Analysis by Geoffrey Smith
    
    
27. Elements of Logic via Numbers and Sets by D.L Johnson
    
    Plus many more- just scour your local library and the internet.
    
    Good Luck, Dude/Dudette.

Ghrist's book makes a great overview of not only a bunch of topics in algebraic and differential topology, but also has a bunch of applications. I don't think it would be very good as a first introduction to topology, but it's certainly good for browsing and getting a general idea of things.

For a textbook, you might be best getting Munkres and working through that. Another book I really like that is shorter than Munkres is Armstrong's topology book.

There's really no easy way to do it without getting yourself "in the shit", in my opinion. Take a course on multivariate calculus/analysis, or else teach yourself. Work through the proofs in the exercises.

For a somewhat grounded and practical introduction I recommend Multivariable Mathematics: Linear Algebra, Calculus and Manifolds by Theo Shifrin. It's a great reference as well. If you want to dig in to the theoretical beauty, James Munkres' Analysis on Manifolds is a bit of an easier read than the classic Spivak text. Munkres also wrote a book on topology which is full of elegant stuff; topology is one of my favourite subjects in mathematics,

By the way, I also came to mathematics through the study of things like neural networks and probabilistic models. I finally took an advanced calculus course in my last two semesters of undergrad and realized what I'd been missing; I doubt I'd have been intellectually mature enough to tackle it much earlier, though.

>I didn’t mean to offend anyone.

None taken; I just meant that math is huge, and none of it is inherently boring, but I understand that it can seem that way if you're not used to thinking about it.

It looks like there are already some good suggestions regarding how math relates to other subjects, so let me propose something purely mathematical: knot theory. It may not seem like "math", depending on what you've been exposed to, but this is what might make it a great topic for you. Here's the kind of thing a knot theorist thinks about:

Take a piece of string, tie a knot in it, and then take the two ends and tape them together. There are many ways of doing this. Here's something you might get; and here's another possibility. But maybe secretly, those are actually the same, in the sense that you can get from one to the other by just adding twists or moving bits of the string around. How can you tell if they're different or not? This is what knot theory focuses on.

If this interests you, here's a really great book written for laypeople about knot theory. It has lots and lots of pictures that you can learn a tremendous amount from just by staring for a while. It will probably be a challenge to read through, and a lot of it might go over your head, but that's fine; that's what reading math is like for everyone. There's also a large chance that reading this will feel nothing like math at all, and in this sense there's no need to be afraid.

If you want a general sense for what math is like, I recommend Elements of Advanced Mathematics by Steven G. Krantz. (pdf)

As the title suggests, it does cover some higher level topics and consequently, will be a bit of a slow read. Getting through it is rewarding though, as it will give some good context for the many different areas of math.

I've also been told that Thinking Mathematically by John Mason is good for learning general problem solving skills in Mathematics.

Good luck!

For me, a "good read" in mathematics should be 1) clear, 2) interestingly written, and 3) unique. I dislike recommending books that have, essentially, the same topics in pretty much the same order as 4-5 other books.

I guess I also just disagree with a lot of people about the
"best" way to learn topology. In my opinion, knowing all the point-set stuff isn't really that important when you're just starting out. Having said that, if you want to read one good book on topology, I'd recommend taking a look at Kinsey's excellent text Topology of Surfaces.

If you're interested in a sequence of books...keep reading.

If you are confident with calculus (I'm assuming through multivariable or vector calculus) and linear algebra, then I'd suggest picking up a copy of Edwards' Advanced Calculus: A Differential Forms Approach. Read that at about the same time as Spivak's Calculus on Manifolds. Next up is Milnor Topology from a Differentiable Viewpoint, Kinsey's book, and then Fulton's Algebraic Topology. At this point, you might have to supplement with some point-set topology nonsense, but there are decent Dover books that you can reference for that. You also might be needing some more algebra, maybe pick up a copy of Axler's already-mentioned-and-excellent Linear Algebra Done Right and, maybe, one of those big, dumb algebra books like Dummit and Foote.

Finally, the books I really want to recommend. Spivak's A Comprehensive Introduction to Differential Geometry, Guillemin and Pollack Differential Topology (which is a fucking steal at 30 bucks...the last printing cost at least $80) and Bott & Tu Differential Forms in Algebraic Topology. I like to think of Bott & Tu as "calculus for grown-ups". You will have to supplement these books with others of the cookie-cutter variety in order to really understand them. Oh, and it's going to take years to read and fully understand them, as well :) My advisor once claimed that she learned something new every time she re-read Bott & Tu...and I'm starting to agree with her. It's a deep book. But when you're done reading these three books, you'll have a real education in topology.

Usual hierarchy of what comes after what is simply artificial. They like to teach Linear Algebra before Abstract Algebra, but it doesn't mean that it is all there's to Linear Algebra especially because Linear Algebra is a part of Abstract Algebra.

Example,

Linear Algebra for freshmen: some books that talk about manipulating matrices at length.

Linear Algebra for 2nd/3rd year undergrads: Linear Algebra Done Right by Axler

Linear Algebra for grad students(aka overkill): Advanced Linear Algebra by Roman

Basically, math is all interconnected and it doesn't matter where exactly you enter it.

Coming in cold might be a bit of a shocker, so studying up on foundational stuff before plunging into modern math is probably great.

Books you might like:

Discrete Mathematics with Applications by Susanna Epp

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

Building Proofs: A Practical Guide by Oliveira/Stewart

Book Of Proof by Hammack

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al

How to Prove It: A Structured Approach by Velleman

The Nuts and Bolts of Proofs by Antonella Cupillary

How To Think About Analysis by Alcock

Principles and Techniques in Combinatorics by Khee-Meng Koh , Chuan Chong Chen

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!) by Carol Ash

Problems and Proofs in Numbers and Algebra by Millman et al

Theorems, Corollaries, Lemmas, and Methods of Proof by Rossi

Mathematical Concepts by Jost - can't wait to start reading this

Proof Patterns by Joshi

...and about a billion other books like that I can't remember right now.

Good Luck.

In addition to the ones mentioned already, another excellent book is Topology by James Dugundji. I know a lot of older mathematicians who prefer that book over both Munkres' and Kelley's, because it covers more material and has very clear and concise explanations, plus some more challenging exercises (including some esoteric material not normally found at a book at this level).

For a simpler introduction, I think that Basic Topology by M.A. Armstrong is pretty good. It starts out with point-set topology then goes into algebraic topology. It takes an intuitive and geometric approach, and has a good conversational style that's well-suited for an elementary course at the undergrad level.

Study what you find the most interesting!

Does your linear algebra include the spectral theorem or Jordan canonical form? IMHO, a pure math subject that is relatively the easiest to learn and is useful no matter what you do is linear algebra.

Group theory (representation theory) has also served me well so far.

If you want to learn GR and Hamiltonian mechanics in-depth, learning smooth manifolds would be a must. Smooth manifolds are basically spaces that locally look like Euclidean spaces and we can do calculus on. GR is on a pseudo-Riemannian manifold with changing metric (because of massive stuffs). Hamiltonian mechanics is on a cotangent bundle, which is a symplectic manifold (whereas Lagrangian mechanics is on a tangent bundle.) John Lee's book is a gentle starting point.

Edit: If you feel like the review of topology in the appendix is not enough, Lee also wrote a book on topological manifolds.

I'm currently working through Munkres' book on Topology, and I am using the video lectures found here. I know these are in an annoying form factor, but, trust me, these are the only videos that go into any depth you will find on the internet. They use Munkres, too, which is a plus.

On the same site are video lectures for Algebraic Topology. For this, I definitely recommend buying Artin's "Algebra" (1st edition can be found cheaply, and I don't think there's really any significant difference from 2nd), and watch these video lectures by Harvard. Then, you can finally move on to the Algebraic Topology video lectures which uses the free textbook "Algebraic Topology" by Allen Hatcher.

Hope this helps.

I'll be honest: most explanations of differential forms suck, and it makes you wonder if people really know what they're talking about. Of course, the answer, in general, is that they do; they just don't know how to communicate it. One issue is the absolute proliferation of different viewpoints regarding these things. Sometimes you get the impression that an author's intuition follows one viewpoint and his formal definitions follow another, resulting in a presentation that is, to say the least, a mess.

A fairly good treatment is the topology book Differential Forms in Algebraic Topology. Check the preview on there. You'll be able to see the basic introduction to forms in it and see if it makes sense to you (or potentially could make sense), that way you don't have to waste money on the book if it doesn't seem up your alley. That said, that book is pretty much as good as it's ever going to get when it comes to exposition on this subject.

Also, just in case it's not clear, when the authors there present the symbols dx_1, dx_2, ..., in the first paragraph, they mean those to be just formal symbols satisfying those two relations. They are not referencing some previous definition that they assume you know.

Knot Theory

It's simple and accessible great for casual learning.
If you've already had much topology, you'll want a more in-depth book after you finish this one. This book is accessible to undergrad and even high school learners. But it's a great jumping off point even if you're more mathematically sophisticated.

Here are some great books that I believe you may find helpful :)

• Elements of Algebra by Leonhard Euler
  
• Foundations of Mathematics by Ian Stewart
  
• Geometry Revisited by H. S. M. Coxeter
  
• Excursions in Calculus by Robert M. Young
  
• New Horizons in Geometry by Tom M. Apostol and Mamikon A. Mnatsakanian
  
• The Calculus Gallery: Masterpieces From Newton to Lebesgue by William Dunham
  
• A Primer of Real Functions by Ralph P. Boas
  
• Understanding Analysis by Stephen Abbot
  
• Chaos: Making a New Science by James Gleick
  
• Logic: An Introduction to Elementary Logic by Wilfred Hodges
  
  and last but definitely not least:
  
  
• A Mathematical Gift I, II, III by Kenji Ueno et al.
  
  Later on:
  
  
• Conceptual Mathematics: A First Introduction to Categories by F. William Lawvere
  
• Twelve Landmarks of Twentieth-Century Analysis by D. Choimet & H. Queffélec
  
• Theory of Complex Functions by Reinhold Remmert
  
• Algebraic Combinatorics. Walks, Trees, Tableaux & More by Richard P. Stanley

I think your background is definitely sufficient. The prerequisites for topology are very minimal, though analysis gives motivation to a lot concepts encountered.

The main idea of topology is generalizing the notion of neighbourhoods from metric spaces to more general spaces where there may not be a canonical choice of distance function, or even any notion of distance at all.

I'm a big fan of Ronald Brown's Topology book. It's under $10 for the ebook, and under $30 for the paperback on Amazon. Brown starts off with topology on the real line, and gives good motivation and intuition for the axioms. I personally prefer it to the often-recommended Munkres.

The book Ideals, Varieties and Algorithms by Cox, Litle and O'Shea is a very good undergraduate level algebraic geometry book. It has the benefit of teaching you the commutative algebra you need along the way instead of assuming you know it.

I'm not really aware of any algebraic topology books I'd consider undergraduate, but most of them are accessible to first year grad students anyway, which isn't too far away from senior undergrad. Some of my favorite sources for that are Munkres' book and Fulton's Book.

For knot theory, I haven't really studied it myself, but I've heard that The Knot Book is quite good and quite accessible.

The answer is "virtually all of mathematics." :D

Although lots of math degrees are fairly linear, calculus is really the first big branch point for your learning. Broadly speaking, the three main pillars of contemporary mathematics are:

1. Analysis
   
2. Algebra
   
3. Topology
   
   You might also think of these as the three main "mathematical mindsets" — mathematicians often talk about "thinking like an algebraist" and so on.
   
   Calculus is the first tiny sliver of analysis and Spivak's Calculus is IMO the best introduction to calculus-as-analysis out there. If you thought Spivak's textbook was amazing, well, that's bread-n-butter analysis. I always thought of Spivak as "one-dimensional analysis" rather than calculus.
   
   Spivak also introduces a bit of algebra, BTW. The first few chapters are really about abstract algebra and you might notice they feel very different from the latter chapters, especially after he introduces the least-upper-bound property. Spivak's "properties of numbers" (P1-P9) are actually the 9 axioms which define an algebraic object called a field. So if you thought those first few chapters were a lot of fun, well, that's algebra!
   
   There isn't that much topology in Spivak, although I'm sure he hides some topology exercises throughout the book. Topology is sometimes called the study of "shape" and is where our most general notions of "continuous function" and "open set" live.
   
   Here are my recommendations.
   
   Analysis If you want to keep learning analysis, check out Introductory Real Analysis by Kolmogorov & Fomin, Principles of Mathematical Analysis by Rudin, and/or Advanced Calculus of Several Variables by Edwards.
   
   Algebra If you want to check out abstract algebra, check out Dummit & Foote's Abstract Algebra and/or Pinter's A Book of Abstract Algebra.
   
   Topology There's really only one thing to recommend here and that's Topology by Munkres.
   
   If you're a high-school student who has read through Spivak in your own, you should be fine with any of these books. These are exactly the books you'd get in a more advanced undergraduate mathematics degree.
   
   I might also check out the Chicago undergraduate mathematics bibliography, which contains all my recommendations above and more. I disagree with their elementary/intermediate/advanced categorization in many cases, e.g., Rudin's Principles of Mathematical Analysis is categorized as "elementary" but it's only "elementary" if your idea of doing math is pursuing a PhD. Baby Rudin (as it's called) is to first-year graduate analysis as Spivak is to first-year undergraduate calculus — Rudin says as much right in the introduction.

I don't know if it's ever used in research, but working with groupoids can be beneficial in basic algebraic topology because of the freedom allowed by not fixing a base point. Ronald Brown wrote a book that goes through a first course on algebraic topology using groupoids for homotopy theory, covering spaces, etc.

Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.

Amazon search for Dover Books on mathematics

There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm

Pre-Calculus / Problem-Solving

• The Art of Problem Solving - Lehoczky/Ruczyk - in two volumes, both have full solutions available separately
  
  Calculus
  
  
• Calculus - Stewart - ignore the bad reviews, it's very clear and covers all of basic calculus well - it's just the book that most intro students use and as a consequence gets a bad rap
  
  
• Calculus - Spivak - this will take your understanding of calculus to the next level
  
  
• Calculus - Apostol - there are two volumes and they are very comprehensive, albeit difficult
  
  Linear Algebra
  
  
• Linear Algebra - Lay
  
  
• Linear Algebra Done Right - Axler
  
  Differential Equations
  
  
• Elementary Differential Equations - Boyce/DePrima
  
  
• Ordinary Differential Equations - Tenenbaum
  
  Number Theory
  
  
• Number Theory - Pommersheim for a change of pace and proof skills
  
  Proof-Writing
  
  
• Transition to Higher Mathematics - Dumas/McCarthy - helps to bridge the gap between "lower" and higher math
  
  Analysis
  
  
• The Way of Analysis - Strichartz
  
• Principles of Mathematical Analysis - Rudin
  
• Real and Complex Analysis - Rudin
  
  Complex Analysis
  
  
• Complex Analysis - Ahlfors
  
  Functional Analysis
  
  
• Functional Analysis - Rudin
  
  
• Introductory Functional Analysis with Applications - Kreyszig
  
  Partial Differential Equations
  
  
• Partial Differential Equations for Scientists and Engineers - Farlow
  
  Higher-dimensional Calculus and Differential Geometry
  
  
• Calculus on Manifolds - Spivak - for a modern take on multivariable calculus
  
  
• Differential Geometry - Kreyszig
  
  Abstract Algebra
  
  
• A First Course in Abstract Algebra - Fraleigh
  
• Abtract Algebra - Dummit and Foote
  
  Geometry
  
  
• The Elements - Euclid - for culture and for understanding where math has come from, plus it's still an awesome book after all these years; don't need to go through all of it
  
• Euclidean and Non-Euclidean Geometries - Greenberg
  
  Topology
  
  
• Introduction to Topology - Mendelson
  
  
• Topology - Munkres
  
  Set Theory and Logic
  
  
• Introduction to Set Theory - Hrbacek/Jech
  
  
• Elements of Set Theory - Enderton
  
  
• Mathematical Logic - Kleene
  
  
• Set Theory and the Continuum Hypothesis - Cohen
  
  Combinatorics / Discrete Math
  
  
• Combinatorics - Cameron
  
  
• Concrete Math - Graham/Knuth/Patashnik
  
  
• Discrete Math - Biggs
  
  Graph Theory
  
  
• A First Course in Graph Theory - Chartrand/Zhang
  
  P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
  

Thanks for the detailed reply. I think it's probably best to give Abbott's a shot then. Right now I'm working through Hubbard's multivariate text alongside Spivak's Calculus on Manifolds. I'm having a lot of difficulty with Spivak because I just haven't done enough work with single variable analysis to be comfortable doing it all in n dimensions, as you've said. It took my until the end of chapt. 2 to realize I'm not really getting it and I need to take a step back and figure out the simpler stuff first.

As a side question, what do you think about a side-text for topology/metric spaces like: https://www.amazon.com/Introduction-Topology-Third-Dover-Mathematics/dp/0486663523 ? The only exposure I have to it is from a linear algebra text and the beginning of Spivak (in other words, not much at all).

If you want to do more math in the same flavor as Apostol, you could move up to analysis with Tao's book or Rudin. Topology's slightly similar and you could use Munkres, the classic book for the subject. There's also abstract algebra, which is not at all like analysis. For that, Dummit and Foote is the standard. Pinter's book is a more gentle alternative. I can't really recommend more books since I'm not that far into math myself, but the Chicago math bibliography is a good resource for finding math books.

Edit: I should also mention Evan Chen's Infinite Napkin. It's a very condensed, free book that includes a lot of the topics I've mentioned above.

These two are great in that they kind of take a CS approach to proofs by using the idea of a pattern:

Proof Patterns https://www.amazon.com/dp/3319162497/ref=cm_sw_r_cp_apa_i_0Y44BbDC5HWYB

How to Prove It: A Structured Approach, 2nd Edition https://www.amazon.com/dp/0521675995/ref=cm_sw_r_cp_apa_i_C244Bb8QY01F2


This is great for understanding limitations and the history of the development of proof techniques:

Reverse Mathematics: Proofs from the Inside Out https://www.amazon.com/dp/0691177171/ref=cm_sw_r_cp_apa_i_M044BbM0BMAE3


As has been suggested, you should look into mathematical logic, modern symbolic logic, which includes propositional and quantificational logic, relational logic, and perhaps even modal logic. Copi is great for a classical treatment of modern and Aristotelian logic. Gensler is great for learning translations, and Smullyan is great for mathematical logic.

You might also look into set theory and hott or type theory, but only after you've approached some of the other stuff. I can't emphasize logic enough if you want to really understand proof theory. Perhaps even check out computability and complexity theory. A lot of these topics in theoretical computer science run into the limitations and possibilities of these and other ideas. You'll then realize as others have suggested, that yes there are some fundamentals taken for granted, but real math is difficult because of the intuition needed in approaching a proof, and the possibility that you could be working on a problem that is unanswerable.

The standard/classic intro undergrad textbook is Munkres.

I actually never took a proper Topology course, I've just been forced to pick up a lot of it along the way. This book has been helpful for that. It's very friendly for reading/self-study.

If you don't want to buy a $60 book, I'm sure you can find it online somewhere, though I learn a lot better when trying to teach myself from a book I can easily flip through rather than a pdf in any form.

Though not an intro to topology book in the strictest sense (it doesn't go into too much detail on seperation axioms, metrization theorems, or Tychnonoff's theorem) I found going through Lee's Introduction to Topological Manifolds to be a fantastic book to learn from, especially if you're interested in going into differential geometry afterwards. I find that many a topology book commits the sin of focusing way too much on curious pathological counterexamples and the beginning students find themselves awash in a sea of formalities without any intuition to guide them. Having many geometrically motivated examples will make the subject highly approachable and eases the transition into thinking about abstract topological spaces.

The classic textbook for a first course in topology is Topology by Munkres. It's a very good book.

Michael Starbird offers his topology "book" free of charge on his website. Here's the link. It's really closer to lecture notes for the course, and it's intended for an inquiry-based learning (IBL) course. What this means is that all of the proofs are omitted. The reader is expected to prove each result themselves. This obviously works much better in a group setting.

If you see any book titled "algebraic topology," I would recommend you ignore it for now. Algebraic topology courses assume you've at least had the one semester course in point-set topology (i.e. the books I linked) and one or two semesters in abstract algebra.

"General Topology" by Stephen Willard, has always been my sentimental favorite. The sections and exercises I would recommend are

1. Chapter 6, Section 19, Compactification (Exercises 19H, 19J, and 19K; then if you are up for it, 19L and 19M.)
   
   
2. Chapter 7, Section 32, Metrization (Exercises 23B, 23C, 23D are well-wroth anybodies time.)
   
   The exercises from (1) anticipate the exercises form (2); at end of both sections and exercises I've suggested, you'll have seen some cool machinery from more than a couple of different perspectives.
   

Check out this list. It includes most if not all books mentioned here.

From more to less advanced, generally:

• Geometry by Gray, Esplen, and Brannan
  
  
• The Infinite in the Finite
  
  
• A Geometrical Picture Book
  
  
• 99 Points of Intersection: Examples-Pictures-Proofs
  
  
• Lines and Curves: A Practical Geometry Handbook
  
  
• The Heart of Mathematics
  
  
• The Symmetries of Things
  
  
• A Mathematical Gift, 1: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World) (v. 1)
  
  A Mathematical Gift, II
  
  A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World) (v. 3)
  
  Note the first three volumes are cheaper as a set.
  
  Intuitive Topology (Mathematical World, Vol 4)
  
  Groups and Symmetry: A Guide to Discovering Mathematics (Mathematical World, Vol. 5)
  
  Knots and Surfaces: A Guide to Discovering Mathematics (Mathematical World, Vol. 6)
  
  
• A Beginner's Guide to Constructing the Universe
  
  and 5 activity books in the same style
  
  
• Going beyond books, but visual, manipulative, and tactile, the Zome Tools

As far as I know, there's no "standard" book for the rite of passage, but obviously Munkres is an alright intro to Point-Set [and that's all you should do from the book] and Hatcher is a wonderful introduction to Algebraic Topology.

Hatcher has this tendency to ramble a bit and to not be exceptionally clear, though. Moreover, there is a bit lacking in Hatcher's text [no doubt done on purpose!]. Because of this, I usually recommend Bredon's Topology and Geometry.

I'll note also that AT is currently a pretty hot-topic because of its application to Data Analysis; you may want to read Topology and Data by Carlsson and the "What is...Persistent Homology?" paper, just so you can see some of the things people are doing with AT.

If you are enjoying your Calc 3 book, I highly recommend reading Topology, which provides the foundations of analysis and calculus. Two other books I would highly recommend to you would be Abstract Algebra and Introduction to Algorithms, though I suspect you're well aware of the latter.

"Topology and Groupoids" is a great one, think it would fit well in your list.

Just saw in my place to the most of Felix Klein](https://www.amazon.com/Indras-Pearls-Vision-Felix-Klein/dp/1107564743) . . Lastly 15+ years because it was nice. Something different note, if I wouldn't say about your heart while gyrating our own choice. **A MAN's choice.

Once you've gotten started with Munkres, I highly recommend Bredon's Topology and Geometry for algebraic topology. It's very well written, covers a ton of material, and has some nice pictures.

My favorite two books in the whole world are Differential Forms in Algebraic Topology by R. Bott and L. Tu, and Characteristic Classes by J. Milnor and J. Stasheff. Every time I read them I learn something new.

Lee is still the easiest and best for self study. https://www.amazon.com/Introduction-Topological-Manifolds-Graduate-Mathematics/dp/1441979395/ref=sr_1_3?keywords=lee+manifolds&qid=1558265795&s=gateway&sr=8-3

followed by:

https://www.amazon.com/Introduction-Smooth-Manifolds-Graduate-Mathematics/dp/1441999817/ref=sr_1_2?keywords=lee+manifolds&qid=1558265795&s=gateway&sr=8-2

​

It's long, taking almost two volumes to get to Stokes theorem. If conciseness is important, you can just read Warner:

https://www.amazon.com/Foundations-Differentiable-Manifolds-Graduate-Mathematics/dp/0387908943/ref=sr_1_1?keywords=Warner+manifolds&qid=1558265966&s=gateway&sr=8-1

​

Tu looks good, but I haven't read it carefully.

It depends. Consider Advanced Calculus by Sternberg/Loomis. You need to be comfortable with something like baby Rudin to tackle this book.

These were the most enlightening for me on their subjects:

• Differential and Integral Calculus, Vol. 1&2 by Richard Courant
  
• Linear Algebra by Georgi Shilov
  
• Differential Equations and Their Applications by Martin Braun
  
• Introduction to Geometry by H.S.M. Coxeter
  
• A Course in Mathematical Analysis, Vol. 1&2 by Edouard Goursat
  
• Introduction to Topology and Modern Analysis by George Simmons
  
• The Theory of Functions of a Complex Variable by A.I. Markushevich
  
• Basic Topology by M.A. Armstrong
  
• Topology by James Dugundji
  
• Ordinary Differential Equations by V.I. Arnold
  
• Partial Differential Equations by Fritz John
  
• Introduction to Probability Theory by Paul Hoel, Sidney Port and Charles Stone
  
• Linear Programming by Katta Murty
  
• Nonlinear Programming: Theory and Algorithms by M. Bazaraa, H. Sherali and C. Shetty
  
• A First Course in Numerical Analysis by Anthony Ralston and Philip Rabinowitz
  
• A First Course in Stochastic Processes by Samuel Karlin and Howard Taylor
  
• Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields by John Guckenheimer and Philip Holmes
  
• Homology Theory by James Vick

I love Jack Lee's series on manifolds:

Introduction to Topological Manifolds

Introduction to Smooth Manifolds

I've heard Munkres' Topology is fantastic as an introduction to general topology, but never read it myself.

I've heard good things about Adam's book The Knot Book if knot theory is your thing.

One thing that you run into in math is a distinct lack of books and articles that explore ideas without proving them. They certainly exist, but are not the norm. We also suffer from a wealth of very poorly written books.

Survey articles and introductory tracks aimed at introducing young mathematicians to advanced topics do exist in some volume and they tend to be better written and in a more conversational tone. They want to familiarize you with a body of work that proving all in one place would be a monumental task so they just don't do it. If I were you I'd look in that direction, but you wont learn very much about the nuts and bolts of what's going on.

You will enjoy this book -Indra's Pearls: The Vision of Felix Klein .

The wiki entry for this book

I don't think you'll "spoil" what you'll learn later. If anything, seeing the material before will help you understand cooler stuff during the class next year. There's a lot of remarks and subtle examples I missed the first time I went through the standard undergrad math topics, that I only learned later.

But if you still want to avoid the topics you'll see in class, you could try some point-set topology (e.g. Munkres Topology). It would be beneficial for the real analysis class too. For differential geometry, I'd recommend Jänich Vector Analysis, which says it only needs calculus and linear algebra as prereqs.

I own this book
https://www.amazon.com/Introduction-Topology-Modern-Analysis-Simmons/dp/1575242389/ref=tmm_hrd_swatch_0?_encoding=UTF8&qid=&sr=

Which seems to be free here.

http://susanka.org/HSforQM/[Simmons]_Introduction_to_Topology_and_Modern_Analysis.pdf

Munkres is the canonical text, but I would strongly recommend Gamelin's Introduction to Topology. It's cheap and it's short, and the exercises actually have answers, which is priceless for self-study.

Thanks a lot! I'm currently trying to get through Bert Mendelson's Intro to Topology book. Would you say this is a good enough start in your opinion?

Set Theory:

Naive Set Theory

Number Theory:

Elementary Number Theory

Introduction to Analytic Number Theory

A Classical Introduction to Modern Number Theory

Topology:

Topology

Introduction to Topological Manifolds

The standard textbook, which doesn't require much background (just calculus and a bit of set theory) is Topology by James R. Munkres.
Topology stands at the base of many mathematical subjects, but I don't know of many real world applications of general topology per se. Algebraic topology and knot theory have applications in biology, astronomy and I'm sure plenty else.

Introduction to Topology by Mendelson is good.

http://www.amazon.com/Introduction-Topology-Third-Dover-Mathematics/dp/0486663523 I liked this little guy. Very introductory, if that's what you are looking for.

I liked this one by Munkres

http://www.amazon.co.uk/Topology-Featured-Titles-James-Munkres/dp/0131816292

I wish I was only taking those two. I've also got Abstract Algebra II (Ring Theory), and teaching the one class on top of that. This is my "tough" semester. The next two I'll probably only be taking 2 classes each semester, plus teaching.

What book are you using for Topo? We're using Munkres.

And what are you using for Real Analysis? I know Baby Rudin is sort of the standard, but we're using Ross.

Yes that's the book I ended up doing catch up with when I took graduate differential topology with this book.

From the ground up, I dunno. But I looked through my amazon order history for the past 10 years and I can say that I personally enjoyed reading the following math books:

An Introduction to Graph Theory

Introduction to Topology

Coding the Matrix: Linear Algebra through Applications to Computer Science

A Book of Abstract Algebra

An Introduction to Information Theory

Don't know your background, but I'd look at Pollack's https://www.amazon.com/Differential-Topology-AMS-Chelsea-Publishing/dp/0821851934 and, of course, Spivak's https://www.amazon.com/Comprehensive-Introduction-Differential-Geometry-Vol/dp/0914098705

MIT has lectures on OCW, as well.

Any graduate student should know basic topology already, just to be clear. Also this could be another very cheap and accessible textbook.

We're using Armstrong, which has gotten mixed reviews but I think that it's alright, its informal style suits me. But I have a copy of Munkres just in case. Which book are you using?

Armstrong and Lee are both worth checking out.

I prefer terse books for some reason, so Armstrong is my personal favorite, but other people may prefer Lee, which includes lots and lots of explanations and examples.

You may be interested in Gamelin and Greene, Introduction to Topology, which is also an inexpensive Dover book. It is recommended by Allen Hatcher (PDF), no less. The math stackexchange has more suggestions here, and here.

If you want to earn credits towards an engineering degree, not that there's anything wrong with that: probability, statistics, multivariable calculus, differential equations, linear algebra

If you want to have fun and broaden your horizons: point-set topology (Munkres!), abstract algebra.

Find out which teacher(s) at your high school have mathematics degrees, and ask them for advice. Even if you want to study by yourself, see if you can work out an arrangement where they check your problem sets and give regular feedback. They may also be able to set up a seminar with like-minded students. And they will know what the local community colleges have to offer.

Indra's Pearls is pretty unique among math textbooks. Through programming exercises and visualizations it teaches about the theory of Kleinian groups and their limit sets. The emphasis is not so much on proofs but on getting a feel for the subject.

You are missing Abstract Algebra that usually comes before or after Real Analysis. As for that 4chan post, Rudin's book will hand anyone their ass if they havent seen proofs and dont have a proper foundation (Logic/Proofs/Sets/Functions). Transition to Higher Math courses usually cover such matters. Covering Rudin in 4 months is a stretch. It has to be the toughest intro to Real Analysis. There are tons of easier going alternatives:

Real Mathematical Analysis by Charles Pugh

Understanding Analysis by Stephen Abbot

A Primer of Real Functions by Ralph Boas

Yet Another Introduction to Analysis

Elementary Analysis: The Theory of Calculus

Real Analysis: A Constructive Approach

Introduction to Topology and Modern Analysis by George F. Simmons

...and tons more.

Do you know anything about this book from Dover Books on Mathematics? I bought it the other day cause it's so damn cheap and I've been looking to do some self-study in topology cause my school doesn't have a course.

I've used Dover before for other subjects and never had any complaints, just figured I'd ask since the question of books came up.

Bredon's book is an algebraic topology book also it has something about manifolds and something about smooth manifolds. Honestly the book is rind of ridiculous. Just look at the table of contents.

My buddy (phd student) told me that if I were to do a reading course, or just want to do self study that I should use Munkres. I think you can find international editions for much cheaper than that. We were using Rudin for our analysis class and spent a lot of time on ch.2. These are my only suggestions because I haven't done much with topology or analysis.

Munkres' book is the standard intro to topology. If you have no experience in it at all, it has a good intro to most everything you'll need to know in Point-Set Topology and the second part is a fairly intuitive intro to Algebraic Topology. Once you are familiar with Point-Set Topology, you can also learn from Hatcher.

The most important thing is to do the problems, you'll just be another buzzword-filled physics student if you don't prove anything.

That's why it's general topology and not just topology.
https://www.amazon.com/dp/0486434796/ref=rdr_ext_tmb

My topology textbooks were Munkres, Hatcher, and Bredon.

I am in a knot theory class and we are using "The Knot Book" by Colin Adams. I think the book is very readable and informative and doesn't really expect much any prior knowledge.

Topology by James Munkres

Read The Knot Book by Colin Adams.

Topology by Munkres

I don't see why you couldn't start with the standard graduate math text on topology, Munkres. If you have the formal maturity to do proofs, you can just start here. Analysis and abstract algebra not necessary.

There's also Zomorodian, which I wouldn't consider a complete introduction to topology in a mathematical sense, but the intended audience here is exactly you. YMMV.

Basic Algebra by Nathan Jacobson

Advanced Calculus by Shlomo Zvi Sternberg, Lynn Harold Loomis

Last two problems in The Knot Book:

> 10.10 Figure out how to represent a four-sphere in six-space. "Draw" an unknotted four-sphere in six-space.

> 10.11 Draw a knotted four-sphere in six-space.

Though maybe that's not as challenging as it sounds (you know, drawing objects with four spatial dimensions on two-dimensional paper, twisted in six dimensions). I haven't read that far yet.