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Career and Education Questions: May 19, 2022
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I want to start building up a list of PhD universities to look into before finishing my Masters.

I was wondering what are the most active places for Low Dimensional Topology (Knot Theory) or Combinatorics?

Edit: I should've prefaced that I am looking to apply anywhere and not only in the US.
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Anton here studying financial math? What are the potential careers for this major?
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I'm interested if anybody has experience applying to grad school several years after finishing undergrad. I'm in a bit of a weird situation - I have a B.S. in math from a top 10 school and graduated with very high GPA, but I didn't really make strong connections with any professors. I technically have one publication, but it's a very short, obscure paper from a random summer project and unrelated to my actual research interests.

After undergrad, I made the worst decision of my life, which was to get an M.A. in education and become a high school math teacher. I'm re-evaluating my career and considering attempting to apply for math PhD programs, but I feel like there's nobody who could write me a decent recommendation. It's been over three years since I've last spoken to any of my former professors, and despite having very good grades, I mostly kept my head down and didn't stand out from the crowd. I've continued to seriously study math in my own time, and I could probably pass quals at most schools right now, but there's not really anybody who can attest to that ability.

Has anybody experienced a similar situation, and if so, what advice would you give?
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Which areas of mathematics have any special connection with quantum physics?
And with special I mean sth not quite trivial.
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Does being good at the Putnam help in grad school? I don't mean if it helps get into grad school, I mean if the ability to be good at solving Putnam problems actually helps do research.
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I am a recent math grad and plan on going to grad school for mathematics. I learned that graduate school is mainly proof writing and linear algebra, and these are the subjects I have the most wobbly foundation on. So, I was wondering: what are some ways I could be a rockstar at these 2 subjects? I was thinking of taking a linear algebra class over the summer to get a solid understanding but everything is closed. The reason I would like a class is because it forces structure on me (in the form of hw and lectures) which I so desperately need and the assigned problems help a lot as well. Without the structure I feel like I am wandering aimlessly. Same with proof writing. Is there some type of class like setting with assigned problems online that I could follow to get better at these 2 things?
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I'm a first year student majoring in math. I am just itching to get my hands on some experience - but all the internships are generally for the penultimate year. What can I do as a first year student for this? Any experience with math involved will do, I'm not picky.
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Trying to figure out what classes i should take to fulfill my main job goal (math teacher) as well as give me other options if that doesn’t work out.

Classes I’m really interested in:

•algebraic thinking

•business finance & management foundation and intermediate

•calculus and analytical geometry 1, 2 and 3

•web design 1, 2, and 3

•web programming foundation, intermediate, advanced
 
•abnormal psychology

•abstract algebra

•precalculus 1 and 2

•statistics

•descrete math

•linear algebra

•principals of financial and managerial accounting
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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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This is not a question about me but my dad. He is a professor of mathematics at a university in Hong Kong, having been a lecturer there since the late 1990s. He, like lots of other researchers in Hong Kong, are looking to leave and if possible continue their careers overseas. One of his colleagues is in his 30s and recently came to the UK and now has a teaching position in a Statistics Department. However, my dad is in his late 50s and wants to go to the US. Is it possible to get an equivalent position in the world of mathematics once you leave your institution? I have a vague impression that staff with tenure tend to stay where they are and that it's very hard to get into senior teaching positions anyway because of the sheer amount of competition. My dad sounded optimistic but I am worried about it turning out badly.  What would be the best way to approach this? What kind of positions should he be looking for and what websites are good places to look for the US? Is there a recruitment cycle, i.e. a time of year where there tend to be more vacancies posted? He has been on sabbatical to the US before and conferences but I'm not sure how much people connections can help.

Thanks in advance.

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