Upper Division ODE vs PDE class

  • Thread starter Ash L
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  • #26
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Want to summarize Student100? My institution does not give me access to the article...

I will reserve full judgment until I have a deeper understanding of the relationship between representation theory and physics.
 
  • #27
WannabeNewton
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It is clear that the level of ignorance here is far too high to have a discussion that will be of any worth. One cannot appreciate the usefulness of a subject unless he/she has actually seen it used. Physics isn't just about providing concrete numerical results and plots, it's also about making sense of these results within a logical-mathematical framework to whatever desired echelon of abstraction.
 
  • #28
jgens
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I will reserve full judgment until I have a deeper understanding of the relationship between representation theory and physics.
The Eightfold Way predicted the existence of an elementary particle before it had been observed. Gell-Mann received a Nobel prize in part for this work. Sounds pretty important to me.
 
  • #29
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The framework is often built in terms of 19th century mathematics, rather than modern mathematics; my limited experience with QFT in research and projects has not led me to an encounter with math surpassing that time period; even when the math gets esoteric (differential geometry in GR, a course I am currently taking), the levels of rigor are often significantly less, and I fail to see why such levels are ever of use in the first place (take L'Hospital's rule, for instance; it was invented long before it was proven at a modern level of rigor, from my understanding)

When was representation theory invented? I am getting conflicting reports.

EDIT: Lie groups were invented in the 19th century, from what I can tell. Does that invalidate the mathematician's contribution? No, the project of studying the reasoning itself, independent of the applications, is not necessarily a doomed project, it just indicates that it ceased to be relevant ages ago.
 
  • #30
jgens
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I fail to see why such levels are ever of use in the first place (take L'Hospital's rule, for instance; it was invented long before it was proven at a modern level of rigor, from my understanding)
Here is an example. Euler is famous for playing fast-and-loose with infinite series, but his results always made sense. When people tried to mimic what Euler did for other infinite series they got nonsense. With the modern framework we can understand why Euler's tricks worked where they did and why they failed elsewhere. That seems pretty useful to me.

When was representation theory invented? I am getting conflicting reports.
Over a long period of time. The representation theory traditionally used in QM and QFT was developed around the 1930s or so.

EDIT: Lie groups were invented in the 19th century, from what I can tell. Does that invalidate the mathematician's contribution? No, the project of studying the reasoning itself, independent of the applications, is not necessarily a doomed project, it just indicates that it ceased to be relevant ages ago.
There is more to the subject than date of invention. Lie theory did not really take off until the 20th Century. But if you want truly modern examples, then results proved in the 1980s by mathematicians about 4-manifolds are now instrumental in understanding Gauge theory. These techniques helped solve long-standing open problems in the field.
 
  • #31
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To make this discussion more relevant to the topic, I'll pose a slightly different question: Given it's apparent usefulness in physics, should a physics student take a course in abstract algebra? I have taken two such courses, and found that the vast majority of the knowledge contained within does not (at least so far) seem to have been worth the effort. Student100's links point out that mathematician's aesthetic tastes and objectives are often quite unrelated to those of applied disciplines; much of my group theory course was spent on the development of tools to classify finite groups, for instance, a project which turns out to be not terribly useful to physicists or applied mathematicians*. Likewise, the course in general topology concentrated almost entirely on questions relevant to analysts; the only time it has cropped up in my graduate course on GR has been as an alternate (and significantly more laborious) method for solving a problem involving a finite dimensional periodic universe (you can solve the problem much more easily with a bit of intuition regarding flashlights ;) ). You've made the case that the mathematical objectives do produce, on occasion, useful results, but my concern is that the OP will be wasting her/his time sitting in a course which concentrates upon such objectives.

*From what I can tell.
 
  • #32
jgens
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Given it's apparent usefulness in physics, should a physics student take a course in abstract algebra?
The answer to this depends on your interests in physics and I can name physics professors with conflicting opinions on the matter. This seems to be a largely personal question that depends highly on individual tastes.

much of my group theory course was spent on the development of tools to classify finite groups, for instance, a project which turns out to be not terribly useful to physicists or applied mathematicians.
To be honest many facts about finite groups are even useless to mathematicians. While a first course in group theory usually focuses on finite groups, it should also emphasize things like recognizing group decompositions and group actions, both of which are useful elsewhere in mathematics and physics.

Likewise, the course in general topology concentrated almost entirely on questions relevant to analysts
General topology is more like a dictionary of terms anyway rather than a real field of study anymore, in my humble opinion at least. Some results in general topology are handy for global methods in GR, but someone like WannabeNewton would have to give you the details about that, since my GR background is comparatively weak.

You've made the case that the mathematical objectives do produce, on occasion, useful results, but my concern is that the OP will be wasting her/his time sitting in a course which concentrates upon such objectives.
This is definitely possible. I doubt that getting a better understanding of the mathematics would hurt anything, but it certainly might not help either.
 
  • #33
MarneMath
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I've taken all the courses you are asking about and I happen to have a masters in Stats and currently working towards a PhD in statistics.

ODE, to me seems like you'll get the most out of, especially if you plan to eventually study SDE. (Stochastic Differential Equations.) A solid understanding of theory tends to be more important in such a course than methods. I've felt that a PDE course that doesn't require complex analysis is underwhelming. While, I think the subject is neat, you (or rather I) spent most of the semester learning to solve PDE that apply to Physics by hand and with some occasional theory thrown in. In the end, I felt I could've learned most of the techniques by simply reading the book on my own. Now removed from that course, I realize how more pointless it was since i've never had to solve a PDE by hand since the final for that course -_-. Nevertheless, I thought it was pretty cool at the time. As for chaos, it was neat to learn some new terms and see new techniques used, but overall, it was mostly a fulfill my curiosity type course.

Overall, go for the ODE, learning how to handle existence, uniqueness and stability is a good skill to have especially if you plan to take more advance courses that depend on ODE's. Once you can handle that, I have no doubt you will be able to read an intro to chaos or PDE book on your own and follow it rather easily.
 

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