Undergraduate Math Course Selection Question

[Vbc]

Hello,

I'm a college undergraduate sophomore interested in eventually getting a Ph.D. in biophysics or computational physics. I have the chance to take several graduate math classes in the next two years, and I was wondering which ones would be the most useful for me in the future. They are

• graduate analysis class
• graduate algebra course
• Geomtric/Algebraic topology (two semesters)
• functional analysis
• graduate complex analysis course

 

[scottdave]

Any more description than just the course titles? Perhaps talk with somebody who is in one of those programs to see what could be applicable.

 

[Vbc]

Any more description than just the course titles? Perhaps talk with somebody who is in one of those programs to see what could be applicable.

• The graduate analysis class is over Lebesgue theory of integration
• The Algebra course covers advanced group theory, galois theory, rings, and various theorems in algebra
• Complex Analysis: Complex numbers; analyticity/holomorphicity; Cauchy Riemann equations;line integrals; Cauchy's theorem over 0 -homologous loops; Cauchy's integral formula and its
  consequences (Taylor expansion, Liouville's theorem, and the fundamental theorem of algebra);
  the argument principle and Rouché's theorem; classication of isolated singularities; modulus of
  entire functions; canonical innite products and Hadamard's factorisation theorem.
• Functional Analysis:Banach spaces: review of L^p spaces, linear operators, dual space, Hahn-Banach theorem, weak topologies, Banach-Alaoglu theorem, compact and bounded operators, closed graph theorem; Hilbert spaces: self-adjoint and unitary operators (including spectral theorem), symmetric operators and self-adjoint extensions; if time allows, distributions and Sobolev spaces.

 

[S.G. Janssens]

1. Sorry if this is a silly question, but did you complete the necessary prerequisites? To me, it looks like all of these courses presume the completion of at least one proof-based undergraduate-level course on the same topic, as well as some degree of "mathematical maturity". For example:

for graduate analysis: single-variable analysis
for geometric/algebraic topology: point-set topology, undergraduate algebra
for functional analysis: linear algebra, single-variable analysis, some real analysis
for graduate complex analysis: single-variable analysis, some point-set topology

In case of doubts, send the prof. a mail to ask if your background is sufficient.

2. I like all these courses, and the analysis courses most, but you and I have different interests. Given your interests, it could be more relevant for you to take a rigorous and application-oriented numerical analysis course, maybe a similar course on probability and statistics. Functional analysis could be relevant if there is enough time to discuss its underpinning of both quantum mechanics and numerical analysis.

(Topology has relatively recently found applications in the analysis of biophysical structures, but I think this may not be enough motivation for you to go in that direction: The initial investment is big and you may not get to see those kinds of applications.)

 

[Vbc]

1. Sorry if this is a silly question, but did you complete the necessary prerequisites? To me, it looks like all of these courses presume the completion of at least one proof-based undergraduate-level course on the same topic. For example:

for graduate analysis: single-variable analysis
for geometric/algebraic topology: point-set topology, undergraduate algebra
for functional analysis: linear algebra, single-variable analysis, some real analysis
for graduate complex analysis: single-variable analysis, some point-set topology

In case of doubts, send the prof. a mail to ask if your background is sufficient.

2. I like all these courses, and the analysis courses most, but you and I have different interests. Given your interests, it could be more relevant for you to take a rigorous and application-oriented numerical analysis course, maybe a similar course on probability and statistics. (Topology has relatively recently found applications in the analysis of biophysical structures, but I think this may not be enough motivation for you to go in that direction: The initial investment is big and you may not get to see those kinds of applications.)

Hello, Yes I do have all the pre-reqs for these classes (multiple proof based undergraduate versions of these courses), I understand that a graduate class is a big commitment so I'm just wondering what math others who have gone into the fields I'm interested have taken/used.

 

[S.G. Janssens]

Hello, Yes I do have all the pre-reqs for these classes (multiple proof based undergraduate versions of these courses), I understand that a graduate class is a big commitment so I'm just wondering what math others who have gone into the fields I'm interested have taken/used.

Ah ok, good, then I will leave it up to those who have entered your fields of interest to continue the discussion. In a nutshell, my impression is that the courses you list err a bit on the pure side of mathematics, which is beautiful but not necessarily the most relevant for what you want to reach. (Even then, you could of course always take a course just out of curiosity.)

 

[Dr Transport]

As a computational physicist, I see none of those courses as necessary.

 

[Vanadium 50]

I don't think any of those course will be really advantageous. Doesn't mean it won't be worthwhile for personal development - so take whichever you are most interested in.

 

[Dr Transport]

I'd head towards courses of a more applied nature. Numerical methods, more applied linear algebra, maybe computational chemistry or fluid dynamics come to mind, these would be more useful than topology, analysis etc...

 

[Vbc]

I'd head towards courses of a more applied nature. Numerical methods, more applied linear algebra, maybe computational chemistry or fluid dynamics come to mind, these would be more useful than topology, analysis etc...

Hello, excuse my ignorance, but would functional analysis not be useful as linear algebra knowledge? Or is it the topic too abstract to be useful in your work?

 

[Dr Transport]

Hello, excuse my ignorance, but would functional analysis not be useful as linear algebra knowledge? Or is it the topic too abstract to be useful in your work?

Ok, I am biased. In a 20+ year career in computational electromagnetics and 30+ year computational infrared materials, functional analysis has never been needed to do my work. I've used much more linear algebra and differential equations than anything else.

 

[member 587159]

As someone in pure math, I don't think these courses will be directly useful for your interests.

Maybe functional analysis is the most useful of them, as techniques in functional analysis can be used in digital mathematics/information theory. I think about models of signals etc.