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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.

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You can follow any syllabus to get some new topics of your interest. –  Vÿska May 17 '13 at 16:01
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During a pre-seminar tea, some professors were discussing what should and should not be included in the brand-spanking new "Mathematics for Economists" masters which my uni has started to offer, and I was listening in. One rather brilliant but quite scatty professor piped up "We need to teach them algebraic topology. Economists need to know about algebraic topology!" There was a long, slightly awkward pause, then I asked if there was a reason why economists should care about algebraic topology. I have no idea what the reply was, but the gist of it was "Yes. There is a very good reason." –  user1729 May 17 '13 at 16:03
    
@GustavoBandeira I know, but since there are so many topics, I am looking for advice on topics for which the general consensus is that they are essential and or interesting to study. –  dreamer May 17 '13 at 16:03
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@cruise Actually not. There are various topics but there is a order in which they should be studied. This will narrow a lot of your choices, for example: With Oxford syllabus, you'll be able to have an introduction to pure mathematics, complex numbers, linear algebra, groups and group actions. Princeton's syllabus is going to have linear algebra, introduction to number theory and real analysis. –  Vÿska May 17 '13 at 16:07

6 Answers 6

up vote 7 down vote accepted

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:

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

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!

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You're welcome. I got impressed Mathematics, It's Content Method and Meaning and Fundamentals of Mathematics. They cover a lot of stuff. –  Vÿska May 17 '13 at 16:49
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With the syllabi, you'll have to read real textbooks for learning those topics. Some of the books I mentioned are written in a more friendly way. If you have the motivation, I suggest you to look the syllabi. –  Vÿska May 17 '13 at 16:57
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I've started my journey in the study of mathematics with mathematics for the nonmathematician - I couldn't simply open a textbook and study it for no reason. This book was very important for giving me motivation to study stuff today. The problem with textobooks is that they are kinda cold, there are no dreams, no speculations, no fantasies, only that black, cold text. - I kinda need a little of that dreams and stuff to proceed. –  Vÿska May 17 '13 at 16:58
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I loved the one by Oxford. I even printed it. –  Vÿska May 17 '13 at 16:59
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Then that seems like the right place for me to start from. Again, thanks a lot for all your help, it will undoubtedly be of great use! –  dreamer May 17 '13 at 17:01

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!

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+1 for Pugh (esp. over Rudin), an unsung hero. –  Andrew May 17 '13 at 20:42
    
Thank you for your help! This seems quite a useful and comprehensive list. –  dreamer May 17 '13 at 21:18
    
You're welcome! –  BU982T May 18 '13 at 10:45

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.

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+1 for the "Of course, you might not be me." Hahahha! –  Albert Renshaw May 17 '13 at 16:45
    
Thank you for your advice. Why definitely not abstract algebra though, if I may ask? –  dreamer May 17 '13 at 17:05
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I believe this band influenced his opinnion. –  Vÿska May 17 '13 at 17:13
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@cruise Look here and here. –  Vÿska May 17 '13 at 17:27
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@cruise If it wasn't obvious enough that I was joking, i've edited my post. –  Mark McClure May 17 '13 at 17:58

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.

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Surely I want to study topics that I find intersting, however, I am just very much curious about which topics most people view as essential to know of and or very interesting to study. –  dreamer May 17 '13 at 16:06
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Then I think that you should try to get a good undergraduate knowledge of real analysis, combinatorics and abstract algebra. From there you can specialize more if you want. –  Dedalus May 17 '13 at 16:09
    
Studying what you find interesting is important, but it is also useful to study what you might need to know later. However, necessity often leads to interest. For example, @Dedalus is probably now very interested in the mathematics behind how aeroplanes fly. –  user1729 May 17 '13 at 16:10
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Also, point-set topology is good to know, some complex analysis. But if you just want to know what mathematicians find essential to study, look at an undergrad curriculum in mathematics. –  Dedalus May 17 '13 at 16:11

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.

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Thank you for the advice. I hadn't actually thought of looking at university curricula yet, but that indeed seems a very useful thing to do. –  dreamer May 17 '13 at 16:50

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.

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Thank you for your advice :). I will definitely take a look into the link you mentioned. –  dreamer May 17 '13 at 17:09
    
From what I read on wikipedia (en.wikipedia.org/wiki/List_of_real_analysis_topics), a lot of topic in real analysis are topics that were covered in my calculus courses. Is real analysis basically the same as calculus or are there specific areas of it which I should still study? –  dreamer May 17 '13 at 17:26
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@cruise Nice question. If you scan trrough the notes, you will see material (not this particular order) on topology, metric spaces, different types of continuity, development of the real numbers as a completion of the rationals to name a few. The most important aspect of this process aside from becoming familiar with the vocabulary, is to learn how proofs are developed. There is a particularly outstanding construction of a major proof - Stone-Weierstrass Th. - which makes use of a lot of the above mentioned material and way of thinking. Then the material goes on to calculus. –  Andrew May 17 '13 at 17:47
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@cruise...but with a rigorous development. You will see proofs of derivatives and integrals and other theorems that enhance your understanding. It's a far cry from what I (at least) studied in two years of calculus in high school (mamy years ago). –  Andrew May 17 '13 at 17:49
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@cruise I would not worry about that. Just master everything you study. That is developing what is called "math maturity." –  Andrew May 17 '13 at 19:16

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