Choice between two grad math classes, DG vs. PDE

[mordechai9]

I am trying to decide between two different graduate math classes in the fall. These are the only classes that sound really interesting to me, as well as fitting in with the rest of my schedule. My other courses will consist of several physics type courses, notably, Jackson's electrodynamics, intro to plasma physics, and intro to nuclear physics.

One is called "Differentiable manifolds", using the textbook "Comprehensive introduction to differential geometry, vol. 1", with a description like the following: "This course is the first introduction to differentiable manifolds. We will cover the basics: differentiable manifolds, vector bundles, implicit function theorem, submersions and immersions, vector fields and flows, foliations and Frobenius theorem, differential forms and exterior calculus, integration and Stokes' theorem, De Rham theory, etc. If time allows it, we will branch into Riemannian manifolds."

The second course is "Partial differential equations", using the textbook "Partial Differential Equations" by L. C. Evans. The description is: "This is an introductory course in partial differential equations. We will follow the textbook by L. C. Evans. The course will consist of the following parts: 1. Basic properties of solutions of Laplace's equation and the heat and wave equations. 2. Second order linear elliptic and parabolic equations: existence, regularity, maximum principles. 3. First order nonlinear PDE: introduction to Hamilton-Jacobi equations and conservation laws."

Both classes sound interesting and useful, but I'm a little bit unsure which one I'm actually capable of handling. Both of them sound pretty tough and I am a little bit worried about my background being sufficient. Both subjects are interesting and I am unsure which class would be better for my overall mathematical maturity at this point in time. Eventually I will be doing research in physics, but I think both classes would be helpful for my end goals.

My background is just having graduated with a math degree, completing the following math courses: Real analysis I (Intro to analysis), Real analysis II (Intro to Lebesgue integration, measure theory, limit theorems), complex variables, differential equations, topology, abstract algebra, linear algebra, curves and surfaces (intro to differential geometry)... and that's all. I think I may have a little deficiency in terms of finite dimensional vector spaces, or very rigorous calculus of multiple variables.

Any advice would be really appreciated...

 

[ice109]

i guess you're a physics major? pdes will be much more useful but obviously dg will be necessary if you're interested in GR

 

[Mene]

I would recommend carefully considering your goals and interests before making a decision between these two graduate math classes. Both Differential Geometry and Partial Differential Equations are important and useful fields of study, but they have different applications and may require different levels of mathematical maturity.

In terms of your background, it seems like you have a strong foundation in mathematics with courses in analysis, topology, and abstract algebra. However, as you have noted, you may have some deficiencies in finite dimensional vector spaces and rigorous calculus of multiple variables. It may be helpful for you to review these topics before beginning either course, as they are fundamental for both Differential Geometry and Partial Differential Equations.

In terms of choosing between the two classes, it may be helpful to consider your future research interests in physics. If you are interested in topics such as general relativity or differential geometry in physics, then the Differential Geometry course may be a better fit for you. On the other hand, if you are interested in topics such as fluid dynamics or quantum mechanics, then the Partial Differential Equations course may be more relevant.

Additionally, you may want to consider the teaching style and approach of each instructor and the textbooks being used. It may be helpful to speak with students who have taken these courses before or to meet with the instructors to discuss your concerns and determine which course may be a better fit for your mathematical background and goals.

Ultimately, both courses will likely be challenging and require a strong understanding of mathematics. Don't be afraid to challenge yourself and push your limits, but also be mindful of your own abilities and choose a course that will best support your academic and research goals. Good luck with your decision!