Mathematics

[KC]

At the end of this school year I will be done with calculus, and I really want to keep learning math, but there's so many ways to go from here that I'm not sure exactly where to start. I guess I'm just looking for some direction as to what to read up on next. Some book ideas would be really helpful also.

[Chrysalis]

Lazar:

This is probably not what you have in mind, but how about taking up philosophy of mathematics. They examine proofs, theory and practice, induction. You'll investigate, for example, Goedel's incompleteness theorem. Other examples to investigate are Pythagorean theorem, Euler's analogy, set theory. I'm throwing these out like I know them, but really I don't. But I have read only lightly on the subject, and this is probably a gloss over of math philosophy.

[ComradeRed]

Before I begin, download this link (http://us.geocities.com/alex_stef/mylist.html) because it has a link for every free book on math that I consider important and then some. Just right click, 'save as...', then save it as whatever.html. It's run by geocities, which is why I suggest saving it offline...it's much faster!

It really depends what you want to do! I don't know what calculus you learned (if you learned vector calc or multivariable, etc.) so I'll list off a few things that may be of interest:

Tensor Analysis - this is one of the most fascinating things I've seen, if you are going to get into tensors, first read up on vectors (or at least know what one is). Also a closely related geometric entity is the "Spinor" which is sort of like a "Quantum vector/tensor". Some books you want to read for tensors are: This thing by NASA (a .pdf file) (http://www.grc.nasa.gov/WWW/K-12/Numbers/Math/documents/Tensors_TM2002211716.pdf), a quick introduction (http://arxiv.org/abs/math.HO/0403252) to tensors, and there are some other books that I do not have at the moment, but I'll go to the library and get back to you on it. You may also like Misner, Thorne, and Wheeler's classic Gravitation although this is more of the application of tensors.

Riemannian and Non-Euclidean Geometry - Remember the postulate that a parallel line never touches the line it is parallel to? What if it were on a curved surface? Think about it, draw a triangle on the globe and its internal angles (if it is equilateral) sums to 270 degrees! Things change with curvature! That is what non-euclidean geometry is about. Hyperbolic geometry is dealing with the surface (where the internal angles of a triangle sums to be greater than 180 degrees) and elliptic geometry deals with the opposite (drawing on the inside of a ball, where a triangle's internal angles sums to less than 180 degrees). Some books you may look for are Riemannian Geometry and Geometric Analysis by Jurgen Jost, and Riemannian Geometry by Peter Peterson.

Topology - this field screws around with shapes and surfaces without tearing the surface or gluing it. Some good links I recommend are Topology: A First Course (http://www.math.uu.se/~oleg/educ-texts.html), or General Topology by Kelley, J.L. I highly recommend learning this sometime before you are thirty!

Algebraic Geometry - It is essentially a combination of Abstract Algebra, Topology, and much much more! Algebraic Geometry: A Total Hypertext Online System (http://odin.mdacc.tmc.edu/~krc/agathos/) is one of the introductory texts that I could find. I read (but didn't completely understand) Algebraic Geometry by Hartshorne, Robin...it is a rather "pure" book.

Field: Discrete Mathematics

This is completely new to you if you haven't studied too much math outside of school, and it is my personal favorite. I'm including abstract algebra for the hell of it too. This is perhaps the hardest math I've dealt with, but not in the sense "Calculus is hard!" More in the sense of Hegel is hard, or Heidegger is hard.

Graph Theory This does not deal with the "x and y axes" or anything of the sort. Instead it deals with - well - anything! It shows the relationship between objects and processes, it's one of the best math tools I've ever used. Some books are free grad level book (http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/) or undergrad, I forgot, but it is nice and easy if you know set theory. A website (http://www.ecp6.jussieu.fr/pageperso/bondy/books/gtwa/gtwa.html) that has a fantastic out of date book. This field really isn't hard to learn and easy to use. It extends to include networks, directed graphs, cyclic and acyclic graphs, bipartite graphs, etc.

Abstract Algebra - I don't really know how to explain this, it has to do with the algebraic structure of strange things. This is very powerful as a tool, and very interesting. It begins with things as simple as sets, to groups, to...much more! A good book that I learned it from was Rings, Fields, Vector Spaces, and Group Theory: An Introduction to Abstract Algebra via Geometric Constructibility although admitedly it is a higher level text book so it may be intimidating at first but don't let this lose you! It was written by a fellow called Sethuraman, B. A. A few good websites includes a sort of archive (http://www.math.niu.edu/~beachy/aaol/contents.html) of theorems, and applied examples (http://www.math.uchicago.edu/~mileti/museum/algebra.html) of a very pure field.

Categories - I've always viewed it as a misleading thing. This is, in practice, just an extension of graphs to be more precise (defining the objects you work with, then the arrows or lines connecting them depict functions, etc.). But most mathematicians call this canonically "abstract nonsense". It's hard to describe. Some good stuff on this is Abstract and Concrete Categories (http://katmat.math.uni-bremen.de/acc/acc.pdf) by Adámek, Jiří, Herrlich, Horst, & Strecker, George E. Or maybe Categories, Types and Structures (ftp://ftp.di.ens.fr/pub/users/longo/CategTypesStructures/book.pdf) by Asperti, Andrea, & Longo, Giuseppe.

Topoi - this I personally consider seperate from a category but it extends the same concept to logic and geometry. Really fascinating stuff, here's a copy of Toposes, Triples, and Theories (http://www.cwru.edu/artsci/math/wells/pub/ttt.html), by Michael Barr and Charles Wells. Or Topos in a Nutshell (http://math.ucr.edu/home/baez/topos.html), or Topoi, the Categorical Analysis of Logic (http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010&id=3), The development of categorical logic (http://publish.uwo.ca/~jbell/catlogprime.pdf). You may want to read up on some topology or algebraic geometry first ;)

Other things which I don't have the time to talk about: functional analysis, complex analysis, real analysis, Laplace and Fourier transforms, combinatorics, mathematical logic, Linear algebra, and some other stuff!

[KC]

Yeah I was considering venturing into set theory or topology or something, and I figured you'd have a lot more info on this than me. Thanks!

[ComradeRed]

Set Theory is rather dull and boring, to be honest. Math as a whole is like a set of tools, but set theory is like a meta-meta-tool.

There is no room for growth in it. But it is an important tool to learn for just that reason!

Topology is interesting but I am not impressed with it (as a physicist :P). You may want to look into algebraic geometry, it combines topology and set theory into a rather interesting field.

[Floyce White]

You're 18 and you're "done with calculus?"

[Janus]

High school calculus, probably, like Calc. 1 and Calc. 2. Not the more advanced stuff in college, I'm guessing.

[ComradeRed]

You're 18 and you're "done with calculus?" I was beginning tensors when I was 17, what's the big deal here? People work at different rates.

I suspect that Lazar sees something in it unique to him, something neither you nor I could see; just as I see something in tensors that no one else can really see.

Theoretically, one could learn calculus by the age of 16 if one really wanted to![b] (Hell, Einstein learned it in fourth grade!).

The high school calc actually gets a firm grounding on vector calculus, and differential geometry. If one gets into say "D-E" (fourth-fifth year), that would be the time to learn things like Schrodigner's equation, applications, Hamiltonian/Lagrangian systems, etc.

But even someone who [b]wanted to learn and just even knows basic derivatives and integrals could learn that stuff quite easily. I daresay that if LaZaR were driven enough and wanted to, he could learn tensors within a week.