Undergraduate Math Course Selection Question

In summary, the individual interested in pursuing a Ph.D. in biophysics or computational physics is seeking advice on which graduate math courses to take. The suggested courses include graduate analysis, algebra, geometric/algebraic topology, complex analysis, and functional analysis. The conversation discusses the necessary prerequisites for these courses and suggests that courses in numerical analysis, probability and statistics, and computational fields may be more relevant for this individual's interests. The usefulness of functional analysis for linear algebra knowledge is also questioned. Ultimately, as someone in pure math, it is suggested that these courses may not directly apply to the individual's interests.
  • #1
Vbc
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0
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
 
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  • #2
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.
 
  • #3
scottdave said:
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.
 
  • #4
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.)
 
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  • #5
S.G. Janssens said:
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.
 
  • #6
Vbc said:
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.)
 
  • #7
As a computational physicist, I see none of those courses as necessary.
 
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  • #8
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.
 
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  • #9
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...
 
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  • #10
Dr Transport said:
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?
 
  • #11
Vbc said:
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.
 
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  • #12
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.
 
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Related to Undergraduate Math Course Selection Question

1. What are the prerequisites for an undergraduate math course?

The prerequisites for an undergraduate math course vary depending on the specific course and university. Generally, students are expected to have a strong foundation in algebra, geometry, and trigonometry. Some courses may also require knowledge of calculus or other higher-level math topics.

2. How many math courses should I take each semester?

The number of math courses you should take each semester depends on your individual academic goals and workload. It is recommended to consult with your academic advisor to determine the appropriate course load for your specific situation.

3. Are there any specific math courses that are required for my major?

Some majors may have specific math course requirements, while others may have a list of recommended courses. It is important to check with your academic advisor or department to ensure that you are taking the necessary courses for your major.

4. Can I take math courses from other departments to fulfill my math requirements?

In some cases, math courses from other departments may count towards your math requirements. However, it is important to check with your academic advisor or department to ensure that the course will fulfill the necessary requirements.

5. How do I know which math courses are right for me?

Choosing the right math courses can be a personal decision based on your interests, career goals, and academic strengths. It is recommended to research the course descriptions and speak with your academic advisor for guidance on which courses align with your academic and career goals.

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