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Career and Education Questions: May 26, 2022

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I'm a software engineer with a masters in mathematics. My bachleors was almost exclusively pure mathematics, but my masters thesis was on theorical computer science. I'm interviewing for a job, and they are considering me for a quantative developer role (which I'm very excited about). The only problem is I have never had a grasp on differential equations, and I imagine they aren't going to care about my grasp on group theory. They never made sense to me (outside of one class where we briefly used numerical analysis to solve an ODE).

Does anyone have any resources for DE/numerical analysis that could give me a crash course? Ideally something that would go over implementing some numerical techniques to solve some DE.
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Anybody here whose primary interests are probability theory and statistics who studies in Bonn or TU Munich (preferrably in a master's degree)? I have some questions, as I'm currently deciding on where I'm going to do my master's degree in germany
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Why does Operator Theory not feature at all as a research area in US graduate programs? Under Analysis only PDEs and Harmonic Analysis is mentioned.

Is Operator Theory not really an area of interest in the US? Or does it only come up while working on PDEs?

Considering all my Master's projects have been on Operator Theory it feels a mis step now if it'll disadvantage me while applying for PhD. I know US PhDs still have some leeway for taking courses and then starting with the thesis work, but yeah.
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I'm in a math degree and there's a group project for a certain class where we address an unsolved problem. This is going to sound delusional but please accept the premise. I think I have a shot at solving it, but frustratingly the professor is setting us on a stupid anti-productive direction. (I feel horrible saying that, I'm so sorry, it's just true) My 3 group partners don't want to go against it so zero help from them. I'm incredibly thankful for the prof introducing this problem to me, I wouldn't have known about it ever without him. In the unlikelihood that I manage to solve it would it be okay for me to just wait until this class project is done and then publish it with any other professor? Maybe I'm being emotional but it's frustrating for someone to take credit when they're doing the opposite of helping. Any advice be it moral/ethical/academic would be very appreciated. Thanks. And I'm perfectly aware I sound delusional.
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I'm an undergraduate student who's interested in possibly pursuing graduate school for mathematical optimization or something adjacent to that.

I'm going to participate in an REU this coming summer, and I'm wrapping up the year-long undergrad analysis and undergrad optimization sequences this quarter. Since I've more or less completed my graduation requirements, I was thinking of petitioning into some graduate courses next year, ideally in order to get to know some faculty whose areas of research are close to what I'm aiming for. The instructors that I'm closest to are unfortunately more focused on the teaching side of things rather than research, and I've been advised that for graduate admission purposes, it's ideal to have a letter written by a tenured professor.

Some of the grad classes I'm considering are:

1. convex optimization with algorithms emphasis (CS department)
2. convex optimization with modeling emphasis (EE department)
3. convex analysis and nonsmooth optimization (Math department)
4. mathematics of data science (year-long sequence under Math department)

The 2nd one follows the textbook by Stephan Boyd. I think the 4th one will most likely follow the high-dimensional probablity textbook by Roman Vershynin and the high-dimensional statistics textbook by Martin J. Wainwright.

My main questions are:

1. Is trying to foster a relationship with a grad course instructor to get a letter of recommendation or for the distant possibility of having them as an advisor doable or ill-advised?
2. would it be reasonable to ask any of them if I could get involved in their research in some form, or do I lack the experience to be of any real use to them at this point?
3. Which of these courses among those listed should I prioritize? (I don't think it'd be a good idea to take any more than 2 classes a quarter, and that's already probably pushing my luck)
4. Should I focus on classes directly related to optimization, or would it be better to take more foundational classes? (e.g. measure theory or functional analysis)
5. For the Mathematics of Data Science class, do I have the appropriate background to take it? I've taken analysis at the level of Baby Rudin and the first 6 chapters of Royden. I took half of the probability sequence, so while I've seen stuff like Chernoff bounds, transformations, and MGFs, I haven't formally seen stuff in the multivariate setting or things like conditional distributions. I also haven't taken an actual machine learning or data science course yet.
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I have a bit of an issue where I have a paper ready to submit to a journal, but I don't know what journal to submit it to. I wrote a paper last year on using a program to answer some questions about an old card game and showed it to a professor of mine. I was still an undergrad at the time, but when my professor briefly looked over it, he said the results seemed significant enough to be published outside of an undergrad research journal, though he wasn't sure what kind of journal to submit it to (the results are basically just asking a standard question about the likelihood of something, using a program to find the answer, and seeing how the answer changes based on some other factors). He's been telling me he'll look for a journal, but it's been over a year now and I think I'm just gonna try to submit it myself (I assume he's just too busy with other things). I recently submitted another paper in topology, so I understand how the submission process works, I just don't know where to submit this paper. I don't think my results are mind blowing or anything, it's just something that I know other people that play this card game wonder about.
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I'll be taking Real Analysis in the fall and I'd like to prepare over the summer. I've got a solid foundation in computational calculus and some exposure to rigorous calculus, I also took an intro to proofs course last semester, and my intro to linear algebra course was partially proof-based (not much, it's mainly for engineers and CS people). I'm definitely far better at reading and understanding proofs now, but writing my own is still a bit of a struggle. Should I spend the summer working through a transition book like Spivak or polishing my proof-writing abilities? In addition to Spivak, I have The Art of Proof (this was the book I used for class), Solow, Velleman, and Polya on hand as well as Lay's Analysis with an Intro to Proof. If I should do something else entirely I'm open to suggestions! Thank you.
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Is the Math GRE useful/used for master's applications? Or only PhD ones? And is it used anywhere outside of the US?
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I'm double majoring in math and physics, I'll be a senior for the 2022-23 year. For a very long time I've had the goal of getting a PhD in physics and doing theoretical physics research. Recently though, I've really began to fall in love with pure math, things like topology and group theory. I'm having a lot of trouble (and anxiety as I need to make this decision quickly) deciding between pursuing physics or math for a PhD.
  

  
Doing research in either physics or math seems right up my alley, but the job prospects for both don't seem the greatest. Its hard to find things online about what research is really like in the industry, so I'm looking for anyone's experience in industry (or academia, I'd love to hear about it) and if they make use of any high level/abstract math (and how do you use it) or if they use/work on new physics.
  

  
Also, if anyone has had to pick between focusing on theoretical physics or pure math I'd love to hear how you made that decision. I'm happy to add more details if needed.
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