# Hey guys! What course should I take? (Pure maths)

1. Aug 8, 2013

### ktheo

So I have a list of "optional electives" for my math degree which I am about to finish. I've held off on these courses and finished up all my mandatory courses; analysis, ODEs, group theory etc. I have some fourth year courses to finish (PDEs, dynamical systems, lie algebras) and I have to pick a course amongst a list keeping in mind right now my GPA is 3.33 and I'm trying to get into grad school, and I know that the 4th year courses will be demanding. These are 2nd year coded courses; they were just optional and not pre-reqs for any of my degree requirements.

The professors in both the courses are pretty good, so I'm essentially looking for whatever will be least demanding and most interesting to someone who really doesn't *love* pure math, although neither of these choices could be designated as applied math. Could anyone throw in some input as to what one might be best for me? I feel like the Foundations course might be easier since I have already done set theory and a boatload of proofs (I don't know why they didn't just make this course mandatory...), but I also hate set theory.

INTRODUCTION TO GEOMETRY

Euclidean and non-Euclidean geometries; affine geometry, projective geometry. Transformations and transformation groups

Or

FOUNDATIONS OF MATHEMATICS

Introduction to proofs, set theory and the foundations of mathematics. Propositional logic, introduction to predicate logic and axiomatic theories. Proof techniques (direct, by contradiction, by cases, constructive and non constructive, induction). Informal set theory (sets, functions, equivalence relations, order relations). Paradoxes. Introduction to axiomatic set theory and to the encoding of mathematics. Axiom of Choice, Zorn's Lemma. Cardinality of sets.

2. Aug 8, 2013

### verty

Why do you hate set theory? It is the one subject I thought no one could hate. Perhaps the foundations course would be good if it gets you to like set theory again. And, I just thought of this, if you are going to grad school, you'll want to know the set theoretic language very well because it'll be used.

I don't have an opinion on the geometry course.

3. Aug 8, 2013

### Theorem.

You should learn to love set theory if you plan on going into graduate school, I mean it is at the heart of the foundation of mathematics and you use it all the time. Maybe a good solid course on set theory could be beneficial to you! at the same time, the geometry course looks pretty fun too. You have to look at other things like what course will help you the most become a more solid and well founded mathematician, and both these things could help a lot- so it's tough. I think for your purposes it might be better to learn more set theory, but I don't really know enough about you.

4. Aug 8, 2013

### Naty1

not exactly an inspiring list of criteria.

suck it up and take both courses.... 'adversity builds character' and taking two will keep you out of 'extra curricular trouble'

or..... get an exemption and take an alternative course you love.

5. Aug 8, 2013

### lurflurf

Why do you like set theory? It is the one subject I thought no one could like. I joke. Anyway the way set theory is used in other areas is not the same as a set theory course. As such courses in set theory while worthwhile tend to not be particularly useful or interesting in service to other subjects.

The course description provided does not actually seem to be a set theory course but some kind of remedial math. So avoid. The geometry course described seems interesting and useful. This is especially true as many students do not get a nice geometry course in high school or college and geometry is an important area.

6. Aug 8, 2013

### nonequilibrium

The set theory course looks like a joke for someone at your stage. And I don't mean that positively, it would just be a waste of your time. Obviously go for the first choice if you care about your education.

7. Aug 8, 2013

### QuantumCurt

I'm still pretty early in my undergrad, but the Foundations of Mathematics course sounds like an "introduction to higher mathematics" type course, which seems a bit below your level. One of the first post-Calculus courses I'm going to be taking is going to be a Fundamentals of Mathematics course, and the description is quite similar. I'd love to take some upper level geometry eventually. It's a fascinating subject.

edit-This is the description for the course I'm referring to- "Fundamental ideas used in many areas of mathematics. Topics will include: techniques of proof, mathematical induction, binomial coefficients, rational and irrational numbers, the least upper bound axiom for real numbers, and a rigorous treatment of convergence of sequences and series. This will be supplemented by the instructor from topics available in the various texts. Students will regularly write proofs emphasizing precise reasoning and clear exposition."

The only prerequisite for it is Calculus II. It sounds fairly similar, though this course doesn't mention Set Theory. It sounds like it kind of implies some Set Theory aspects some. I may be wrong on that part though.

8. Aug 8, 2013

### nonequilibrium

The course mentioned by the OP is a tad more advanced than yours due to it also adressing topics like the axiom of choice. But at the same time the description of the course implies those issues will only be addressed in a cursory fashion, so that's why I said "tad". But so overall you are correct in believing the course mainly concerns issues that are usually found at the beginning of math undergrad, rather than at the end of it.

9. Aug 8, 2013

### Robert1986

Do they offer a topology class? Or perhaps another analysis class (or do you already have to take these?) If it is between these two, I would pick the Geometry class.

10. Aug 9, 2013

### verty

I agree with you that the foundations course looks pretty elementary and may only be about set theory at a superficial level. I know that set theory, once it gets beyond the finite and countable, becomes highly esoteric and not applicable to other areas. I suppose one could argue that what one needs from set theory is taught in algebra and analysis courses, or basic proof courses like this foundation course is. What is left to learn, right?

I just thought it would give an opportunity to fill any gaps, to look again at logic/set theory and get over this hatred of it that he has. If the course is below him which it could be, he could always the time to learn more about mathematical logic and set theory, I don't see how this could ever hurt.

One thing though, I don't think this course is remedial. It is surely not meant for people who are failing their advanced courses. I think it is meant for any second year student who has now done the applied basics and wants to progress to pure math, to learn what they need to know to prove things and understand the set theoretic language used in advanced courses.

Only KTheo knows what his preparation is like. He will know if there is nothing to be gained in this foundations course.

11. Aug 9, 2013

### deluks917

I would take the geometry course. Understanding geometry in these concrete cases is pretty useful intuition for some things in Algebraic geometry which is a huge chunk of modern mathematics.

12. Aug 9, 2013

### ktheo

I'm not sure why I have a disdain for set theory; perhaps it stems from the fact that early in undergrad I just had to go through it SO MANY times, from having to re-cap it in abstract algebra, to re-covering it in logic, then again in consumer theory in economics, and it felt so stale to me. I should also mention that in November I will be applying to graduate school but it will not be in the typical MS math stream; I plan to take applied and industrial mathematics.

As mentioned above I plan to do graduate school from a purely applied approach, so I am not sure how much set theory would be useful to me; ive seen the course offerings for the subject i'm interested in at a few different schools and it's mostly dynamical systems, PDEs, applied algebra, stuff like that.

The list isn't very inspiring because I left out that I have already taken a few of them (my program required 4, I already opted to take both probability and statistics as my two other optionals)... the only other possibility was discrete math, but that isn't offered well in my schedule.

I will be taking fourth year functional analysis:

Introduction to Banach and with emphasis on Hilbert spaces. Fourier series and Fourier transforms. Linear operators on Hilbert spaces. Introduction to spectral theory. Selected topics among: Compact operators, unbounded operators, etc.

I have already taken the basic analysis courses. These are courses that can't be replaced by topology etc. which are 3-4th year courses at my school.

I am leaning towards geometry at this point. The professor is in very good regard at my school. You are right; in a "perfect" course sequence, the foundations course is supposed to serve as that "stepping stone" before higher level proof mathematics takes place, which I essentially did by grinding through abstract algebra without the stepping stone.

I'm leaning this way after the comments.

13. Aug 9, 2013

### ktheo

This is kind of one aspect that holds me back from just jumping into the geometry. I haven't ever taken a strictly "geometry" gauged class in my LIFE. I truly don't really even know what it entails... My math career has pretty much been Linear algebra>Abstract linear>abstract algebra>advanced linear algebra>lie groups on the algebra side and then calculus 1>calculus2>calculus 3>ode>dynamical systems then some standalone maths in probability and statistics, which ironically are my favorite focus and have some heavy courses in it (probability theory, mathematical statistics, regression, applied probability).

I'd love to learn geometry better, I'm just trying to be economical at this point... I am planning on grad school, have some tough 4th year courses, and the last thing I wanna do is be studying as hard for a 2nd year geometry class as I did for nightmarish introductory analysis.

With all that being said, I think i'm still going to choose geometry.