Courses Topics for undergraduate mathematics and physics research

[Faiq]

I will be attending a university in this fall, and I have decided to double major in physics and mathematics. However what haunts me most is the very idea of research. I am supposed to be publishing something new in 4 years both in mathematics and physics for completion of my course. But I can't seem to get any idea on what to research on.

Can somebody point on what can be some potential research topics for physics and mathematics for undergraduates?

Moreover is it possible to fulfill the condition by researching and publishing on a topic that combines both physics and maths? (Example:
Hodge theory and Electromagnetism. (Algebraic topology/Physics)

Electromagnetic theory since the time of Maxwell has been an important source of new mathematics. This is particularly true for topology, specially for what is called "algebraic topology". One fundamental topic in algebraic topology with strong ties to electromagnetism is the so called "Hodge-de Rham theory". Although in its general form this is a difficult and technical topic, it is possible to go a long way into the subject with only Math 233. The article "Vector Calculus and the Topology of Domains in 3-Space", by Cantarella, DeTurck and Gluck (The American Mathematical Monthly, V. 109, N. 5, 409-442) is the ideal reference for a project in this area. (It has as well some inspiring pictures.)

Another direction to explore is the theory of direct current electric circuits (remember Kirkhoff's laws?). In fact, an electric circuit may be regarded as electric and magnetic field over a region in 3-space that is very nearly one dimensional, typically with very complicated topology (a graph). Solving circuit problems implicitly involve the kind of algebraic topology related to Hodge theory. (Hermann Weyl may have been the first to look into electric circuits from this point of view.) The simplification here is that the mathematics involved reduces to finite dimensional linear algebra. A nice reference for this is appendix B of The Geometry of Physics (T. Frankel), as well as A Course in Mathematics for Studentsof Physics vol. 2, by Bamberg and Sternberg.)

 

[berkeman]

You can look through the links to "Similar Discussions" at the bottom of the page, and this Insights article may be of interest: https://www.physicsforums.com/insights/niches-publishable-undergraduate-research/

 

[Faiq]

Yes, I actually consulted that but most of the research topics are engineering-related, no?

 

[berkeman]

Yes, I actually consulted that but most of the research topics are engineering-related, no?

True, but I was hoping that they might spark parallel ideas in your areas of interest. Congrats on heading off to university!

 

[berkeman]

BTW, which university will you be attending, and what are the current research projects being conducted in the Physics and Math departments? Have you read the Bios of all of the professors to see what their research interests are? That might also help to spark some ideas.

 

[Faiq]

I am not sure I can answer the question to its full potential so here's the website http://nyuad.nyu.edu/en/

 

[Dr. Courtney]

Usually, you gain the confidence of a prof and get him to add you to his research group.

Then you select a topic from the things his group is working on. No need to come up with an original idea at this point.

Right now your focus should be on impressing your profs with hard work and great grades so that you are the one student they most want to add to their research group.

Oh, and a lot of the ideas in the "Niches" article are more in the areas of applied physics and applied mathematics. But these have usually been acceptable for fulfilling undergraduate research requirements. "Pure" math and "pure" physics are a bit harder for an undergrad, especially for a new undergrad to see when they are just starting out.

 

[berkeman]

Thanks Dr. C