Which topics of mathematics should I study?

Question

I'm a first year econometrics student with a great interest in mathematics. I very much enjoy my study, but still I am interested to learn about more topics in mathematics which are not part of my study. Some of the main topics which I am already familar with or which will soon be covered in my study are: calculus, linear algebra, optimization, statistics/probability/combinatorics.

Which topics in mathematics would you advise me to study still (that are not part of this list)? I have a general interest in mathematics, so any advice on interesting and or essential topics in mathematics that are worth studying is appreciated.

If possible, could you also give me advice on books/references which I should study from?

Thanks in advance.

Edit: To be more specific, I am looking for topics on which the general consensus is that they are essential to know of for any mathematician or very interesting to study.

Answer 1

You'll just need to get some universities syllabi an pick some topics to study. The syllabi do have a lot of subjects but they also provide you an order in which they should be studied that will restrain a little of your choices. You won't be lost with a diversity of fields of study because you'll have to cover some fundamentals first.

Cambrige and Oxford have nice materials for guiding your study - It'll also be useful in the case that you know some of the first subjects, you'll be able to pick more advanced stuff. The folowing resources are going to be very useful:

• How to Become a Pure Mathematician

  This website has book recomendations and also information on what order it should be studied.

• All the Mathematics You Missed: But Need to Know for Graduate School

  This book comments on the importance of some math fields for the mathematical study.

• Cambridge Syllabus

• Oxford Syllabus

  I've found both syllabi enlightening and informative, with them you're going to have something similar to the site I mentioned before: fields of study, the order in which they should be studied, some intruction on what you should be able to do after covering the topics and a little about the importance of them.

There are also some all-in-one books and book collections that you should look:

• Mathematics, It's Content Method and Meaning

• Fundamentals of Mathematics

• The World of Mathematics

• Mathematics Form and Function

• What it Mathematics?

• Princeton Companion to Mathematics

For the end, as a personal suggestion: Don't get afraid, just get the books and start reading, when the things start to become dark you can use the torches of our fellow members to lighten your path! Good Luck!

Answer 2

Study what you enjoy!

There is no way I or anyone else can tell you what you might find interesting. Read around on wikipedia and try to get an overview of what some subjects are about and see which one appeals to you. Try to gain mathematical maturity by studying many different branches of mathematics. Some combinatorics, abstract algebra and real analysis are fun subjects and after taking them maybe you will find something that appeals to you more than others.

Try to take classes with friends too, if possible. It will make the reading a lot more fun and you can discuss with eachother.

Answer 3

Real and complex variables, fractal geometry, complex dynamics, partial differential equations, and numerical analysis. Definitely not abstract algebra.

Of course, you might not be me.

Edit

To be crystal clear - I'm joking. I'd go as far to say that anyone who seriously advises you against studying abstract algebra is a bad person. Of course, the language of abstract algebra suffuses so much of modern mathematics that an analyst can scarcely do without it. The point is that Dedalus is correct, you must follow your passion.

As a professor, students frequently ask me for advice on courses, majors, careers, and graduate school. Often, students state that they might want to go to graduate school because it looks like my job is pretty chill. That's an insufficient reason - you've got to have the passion to make it through a Ph.D.

That's my 2 cents, anyway.

Answer 4

Here's a link to a page which in turn links to documentation about the first three years of the Cambridge Mathematics Tripos (look at the Guides to Part IA, Part IB and Part II). This will tell you what one famous course thinks is pretty essential (in the first two years) and then -- as you begin to specialize -- what the next steps might be. Of course, there can be heated arguments about what should go in undergraduate courses when: but these documents, and similar ones from other places with top-ranked mathematics courses, should give you some very useful pointers.

Answer 5

Here is a suggestion for an explicit first step. In my (not the most qualified) opinion, real math is grounded in rigor. Typically the first step is real analysis. You can get a sense what constitutes a proof, etc.

These notes, almost verbatim, of lectures by Fields Medal (~ Noble Prize in math) Vaughan Jones are an outstanding, very readable, self-contained entry point. They are his own treatment and are a master presenting the material:

https://sites.google.com/site/math104sp2011/lecture-notes

They start from the beginning so you can go right in.

Answer 6

This is a link to the Mathematics Programs offered at the University of Toronto (St. George):

http://www.artsandscience.utoronto.ca/ofr/archived/1213calendar/crs_mat.htm

If you scroll down you'll find the course requirements for "Mathematical Applications in Economics and Finance Specialist Program" which includes subjects like Real and Complex Analysis and PDE's which aren't on your list. However, if you'd like to follow the Mathematics Specialist program I could tell you which texts they use/have used for quite a few of them. A course number with a Y indicates a full year course (72 hrs of lecture) and a course number with H indicates a half year course (36 hrs of lecture):

First Year

MAT157Y1 - Analysis I Text: Calculus by Spivak. Used in the past: Principles of Mathematical Analysis by Rudin.

If you have never been exposed to abstract mathematics Spivak is probably better to go with. UofT has been teaching from Spivak's for awhile now.

MAT240H1 & Mat247H1: Linear Algebra I & II Text: Linear Algebra by Friedberg et al. Used in the past: Linear Algebra Done Right by Axler.

Second Year

MAT257Y1 - Analysis II

Text - Analysis on Manifolds by Munkres Used in th past: Calculus on Manifolds by Spivak

Go with Munkres on this one. Spivak is barely a little over 100 pages in length! So you can imagine how terse it is.

MAT267H1 - Advanced Ordinary Differential Equations Text - Differential Equations, Dynamical Systems, & Introduction to Chaos by Hirsch et al. & Elementary Differential Equations by Boyce and DiPrima

Third Year

MAT347Y1 - Groups, Rings, & Fields Text: Abstract Algebra by Dummit and Foote

MAT354H1 - Complex Analysis I Text: Complex Analysis by Stein & Shakarchi. Used in the past: Real and Complex Analysis by Rudin

MAT315H1 - Introduction to Number Theory Text: An Introduction to the Theory of Numbers by Niven. Used in the past: A Friendly Introduction to Number Theory by Silverman.

MAT344H1 - Introduction to Combinatorics Text: Applied Combinatorics by Tucker

MAT327H1 - Introduction to Topology Text: Topology by Munkres.

MAT357H1 - Real Analysis I Text: Real Mathematical Analysis by Pugh. Used in the past: Real and Complex Analysis by Rudin.

MAT363H1 - Introduction to Differential Geometry Text: Elementary Differential Geometry by Pressley.

Fourth Year

A lot of these courses are cross listed so they're actually graduate courses. Check here for texts and references:

http://www.math.toronto.edu/cms/tentative-2012-2013-graduate-courses-descriptions/

Hope this helps!