Deciding What Math Courses to Take Next

[Mondayman]

Hi folks,

I am trying to decide which math topics to study next, I am a bit indecisive at the moment. I just finished my second year in my physics degree and have taken calculus I-IV, linear algebra I and II, and differential equations. On my own I pick up Boas' book every weekend and study or review a few topics, so I have a spattering of knowledge on things like PDE's and Fourier series.

Because of the Covid situation, and me transferring institutions, I am not taking physics courses this year. I plan to load up on math courses, and I'm trying to determine which will be most applicable to physics.

I am for sure taking:

Intro to Partial Differential Equations
Mathematical Methods: Mathematical analysis of linear systems. Fourier and Laplace transforms, applications and numerical methods. Functions of a complex variable and applications.
Numerical Analysis I and II
Complex Analysis I

I have the option to take three of following courses this year, and more next year if I wish.

Discrete Math
Tranformation Geometry
Differential Geometry
Optimization
Linear Spaces with Applications
Wavelets, Signals, and Image Processing
Analysis I and II
Algebra I and II
Calculus on Manifolds (requires Analysis I and II)

My hope is to do more computational or theoretical physics, so I am not sure if I should focus on the application courses, or if I should start taking more pure math and eventually learn things like Lie groups and functional analysis, which my school offers in later years. Any thoughts?

 

[FreeRoger]

I definitely suggest some computer science classes if you're interested in computational physics. If you want more theoretical options, I definitely suggest Differential Geometry and Linear Spaces with Applications.

 

[Vanadium 50]

You are talking about taking ~4 math classes at the same time? Even math majors don't do this.

I'd also encourage you to look at the background of people replying. For some reason this section attracts middle-schoolers who want to advise PhDs.

 

[StatGuy2000]

You are talking about taking ~4 math classes at the same time? Even math majors don't do this.

I'd also encourage you to look at the background of people replying. For some reason this section attracts middle-schoolers who want to advise PhDs.

When I was a specialist in math (equivalent of a math major at the University of Toronto, my alma mater), I had taken 4 math classes at the same time during my 2nd and 3rd years , and I was hardly unusual among my cohort of students.

Also, note that @Mondayman finished his second year of his physics degree.

 

[Vanadium 50]

4 math classes at the same time

Wow.

 

[mathwonk]

Certainly not all math classes are the same level of difficulty. As a 1960's math student at Harvard, I think I know why caution is being urged for this plan of taking 4 at a time. Even today most people taking say math 55 at Harvard would probably be advised to take at most one other math course, and that one should overlap and provide support for 55. It also depends on how well prepared you are in the material, how theoretical the presentation is, and what are the expectations.

From the lists above, I would suspect that Analysis, Calculus on manifolds, differential geometry, and complex analysis, would be quite a heavy load, not to mention pde, although there would be some overlap between diff geom and manifolds. But since we are not there looking at the same courses, the professors and their requirements, I can only say that it sounds like a lot, maybe much too much, and you should probably consult locally about it first. But it really depends on the level and degree of demand in the courses. Different professors can be very different in this regard, even for the same course.

E.g. some first year calculus professors teach mainly the mechanics of finding derivatives and antiderivatives, while in my honors first year calc class we once covered open set topology, Lipschitz continuity of definite integrals, and sup norm convergence in Banach spaces of continuous functions. You need to find out which it will be. But if other students at your level normally do this at your school, you may be fine. I would talk both to other students with experience and the professors. And prepare in advance.

 

[mpresic3]

Because of the Covid situation, and me transferring institutions, I am not taking physics courses this year. I plan to load up on math courses, and I'm trying to determine which will be most applicable to physics.
[/QUOTE

This doesn't seem to follow. Doesn't the institution you are transferring into offer physics courses? Doesn't the same institution that is offering the math courses offer physics courses? It seems if you are most interested in computational or theoretical physics, you would take a few physics courses, over the year.

That being said. You are entering the third year in this most unusual situation we all find ourselves because of the pandemic. I remember entering my third year, I had a good idea of where I wanted to go, and how fast I wanted to get there. I expected to go into particle physics. My current career is about half electrical engineer and half physicist. For me, this was a good change in direction. Let's look at the courses:

Discrete Math (cannot say ; I never took it)

If I had taken the following, I would be more well suited in my current career after graduating. Maybe it would have altered my direction for the better. It certainly would have broadened my outlook:

Optimization
Wavelets, Signals, and Image Processing

Save for graduate study (or later); these two should be paired together

Differential Geometry - Interesting;
Calculus on Manifolds (requires Analysis I and II)

Linear Spaces with Applications - Useful for graduate quantum mechanics possibly save for later.

Tranformation Geometry - Cannot say, I never took it.

I only recommend these for math majors. Or save for (much) later
Analysis I and II (I took Analysis I)
Algebra I and II (I took Algebra I) (knowing proofs of the Sylow theorems did not help me when I took Group Theory in Quantum Mechanics later.) Algebra I out of the math department will not give you a better understanding of Lie groups.
Don't get me wrong there is a lot of interesting material here, but I would consider it elective, and not compulsory for a physics major (graduate) .

All told: Try to see about physics courses you can take. As a entering upper undergraduate, you may have the opportunity now to broaden your focus, so I would recommend optimization, or signal processing. You can save the heavy theory for later unless you know you want to major in mathematics.

This is just what I have found, Take it with a grain of salt. I haven't taken any math courses for 40 years and your school may be different.

 

[Mondayman]

I should clarify. I'm not taking physics courses cause I'm a late applicant to my new university. I am technically in open studies, and they close certain courses to open studies. I do not know their reasoning.

I believe I am capable of handling the course load. I love math for the sake of math, and am okay with fully throwing myself into it. Plus I will no longer be working while studying. I managed my time well with 4 courses and work.

Thank you mpresic3. I lean towards linear spaces and differential geometry for sure. I have not made up my mind on the last one. Perhaps I should consider more computer science courses, as suggested.

As for peoples backgrounds, I've been around PF long enough to know that people here are more knowledgeable than I. I trust that I'll get good advice.

 

[mpresic3]

Thank you. I think you are doing the right thing getting as many points of view as possible. I always worry, when I suggest courses, the poster will follow the path I laid out, which is not relevant today as it (may have been) many years ago. Best

 

[Mondayman]

Wow.

Not sure if this is sarcasm. Sure it's going to be a real challenge, but if you have good time management and passion for the subject it's not as tall an order as it seems.

 

[Dr. Courtney]

Slow down there. Don't take more courses than you need for a full time load, and make sure you actually master each course well. Impress your profs enough to get invited to do research with them. If you're as good as you think you are, that should be straightforward for you.

Don't try to impress others with a long course list. Impress your professors with mastery and work ethic. And then work with them in their research.

 

[MathematicalPhysicist]

What is this course: " Transformation Geometry"?

I heard of the book by Tammo Dieck called "Transformation Groups":
https://www.amazon.com/dp/3110097451/?tag=pfamazon01-20

Though I never had the pleasure to read the book.

 

[Vanadium 50]

Not sure if this is sarcasm.

It is not. 4 math classes at once, when a full load is ~5 (usually 5 classes a term is what you need for graduating in 4 years) is very impressive.

 

[StatGuy2000]

It is not. 4 math classes at once, when a full load is ~5 (usually 5 classes a term is what you need for graduating in 4 years) is very impressive.

Granted that for me I took 4 math classes (as a math major) after my second year of university. I can readily see this being more of an issue for physics majors (or those double-majoring in math and physics).

And it isn't something I would necessarily recommend either.

 

[Mondayman]

I think my issue is I want to take as many courses as I can while I'm in university. Everything looks exciting. And it earns me a math minor. This doesn't really contribute at all to my goal to get into physics grad school though.

I've just finished taking two physics and two math courses and found I handled it quite well, but I think they were watered down courses compared to most universities. I will heed the advice to take it easier as I move into a new school. I'll take linear spaces and differential geometry one semester, and optimization and wavelets the next. Maybe I'll look at a history course to round it out.

In the end these courses are more for fun than anything; I won't graduate any faster.

What is this course: " Transformation Geometry"?

Geometric transformations in the Euclidean plane: Symmetry, Frieze, and Wallpaper groups.

 

[MathematicalPhysicist]

I think my issue is I want to take as many courses as I can while I'm in university. Everything looks exciting. And it earns me a math minor. This doesn't really contribute at all to my goal to get into physics grad school though.

I've just finished taking two physics and two math courses and found I handled it quite well, but I think they were watered down courses compared to most universities. I will heed the advice to take it easier as I move into a new school. I'll take linear spaces and differential geometry one semester, and optimization and wavelets the next. Maybe I'll look at a history course to round it out.

In the end these courses are more for fun than anything; I won't graduate any faster.Geometric transformations in the Euclidean plane: Symmetry, Frieze, and Wallpaper groups.

When I took the course Non-Euclidean Geometries from the maths department I used along it the three books by Marcel Berger. I planned on buying those books but the Corona came and ruin it all, damn you Jehovah!

 

[member 587159]

You are talking about taking ~4 math classes at the same time? Even math majors don't do this.

I'm unfamiliar with the American system but I take 6 pure math courses this semester (but not anything else).

 

[mathwonk]

The requirement for a major in math at Harvard is 12 semester courses, at least 8 of which must be in math, and the other 4 in math or related fields. Thus a typical 4 year program, even for someone with a strong interest and background in math, requires taking at most three math courses per year, one in one semester and two in the other semester; or at most two math courses at a time, and that only in 4 of the 8 semesters of study.

here is a link to their document showing a suggested program:
https://www.math.harvard.edu/media/courses.pdf

 

[gwnorth]

That doesn't sound right - 12 courses out of 40 for a major? Are you sure all the listed courses are just 1 semester long? Just quickly looking at my son's school a math major requires around 24 semester courses:
Year 1 - 4
Year 2 - 5
Year 3 - 6-8
Year 4 - 6-8

 

[jasonRF]

That doesn't sound right - 12 courses out of 40 for a major?

mathwonk attended Harvard and the website is very clear - so I suspect 12 is correct. It looks like Harvard students take 4 classes (16 credits) per semester on average, with a major consisting of 10-14 classes
https://oue.fas.harvard.edu/college-curriculum

Most other US colleges and universities have similar requirements. Just a few minutes on Google tells me: MIT and Princeton require something like 12-13 semesters of math courses for the math major; UMass Amherst (a good but non-elite public university) requires 14; Williams College (a highly selective liberal arts college) requires 9-10.

Most schools will let students take more than the required number of classes, and certainly some math majors do that. However, breadth of education is pretty high on the list of objectives for many schools in the US. This leads some (but certainly not all) schools to limit the maximum number of credits from a single department, and/or insist on a minimum number of credits from outside of the major.

jason

 

[Mondayman]

I personally think 4 is a good number for total classes. Many universities do 5 as a requirement to graduate in 4 years. The engineering program has them taking 6-7 for a few semesters. I think this is ridiculous - there's no real learning being accomplished when you drown students in information with little time to comprehend it.

 

[MidgetDwarf]

You should inquire about how the Complex Analysis Course is taught at your university. It can be a very applied course, a theoretical math course, or a mixture of both. Moreover, to make sense of it, it is easier to know some Analysis on R, which would be an intro Analysis course. Have you taken any math courses (proof based) from the math department at your school? Pure mathematics is different from the mathematics shown in high school.

The ODE course can also be applied, theory, or mixed. So I would ask around. Not to mention teacher specific requirements. Ie., weekly quizzes, project, etc. I personally hate weekly quizzes, and do far better with midterm/final. So in the classes I had weekly quizzes, I tended to struggle more. But did well in midterm/final courses.

I knew someone in college that was a physics major. We both transferred from the same CC to the same university. He tried to take some courses from the Mathematics department (pure). I think it was either intro analysis or introduction to modern algebra that did him in. Not quite sure. He never took pure math courses again from the department, but took instead math methods courses and some applied vector calculus pde/ode.

However, he was very talented in Physics, and was able to transfer to an IVY school for PhD. I think Princeton or Yell from this school.

Not to discourage you. But maybe email an instructor teaching a course you are interested in, or maybe even the math department chair. Although some chairs can see such inquiries as a waste of time and will give rude/nasty responses, or outright not reply. But some chairs can be nice.

Remember Mathematics is a different beast.