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How to get a good undergrad math education outside academia?
Hey everyone. I went to college and got a degree in math, but definitely missed out on a lot of subjects. Just to give you an idea: I hadn't taken any classes in probability, number theory, topology, or functional analysis, and the highest level of calculus-related stuff I took was complex variables or multivariable calc. I had never heard of a tangent space, differential form, homology or cohomology, didn't know what a tensor was...some of this was stuff I didn't even know I didn't know, others was stuff that I knew was out there but didn't look into because I figured I was only interested in discrete math. When I graduated I had a strong enough background in graph theory, abstract algebra, and theory of computation, but little else. (At this point I've also forgotten a lot of that). When I went to grad school, these gaps in my knowledge became much more apparent. But I stuck with it and got my Ph.D. Along the way, I filled in some gaps in my knowledge, especially in the first year. But after finding an advisor I kept my head down, focused on my research, and didn't go much outside of my comfort zone subject-wise. This was partly because I would try some more advanced topics such as Lie groups or Riemannian geometry, get really confused and drop the class because (I suspect) I didn't get a good enough foundation earlier. My lack of knowledge of other subjects also sucked the motivation out of me even in my own research, because I couldn't see the purpose of my research within the bigger picture of math. I'm thinking of reading a textbook on history of mathematics just to understand how different math subjects are related and why people doing modern mathematics study the wildly abstract stuff they do today, but I digress.
My first plan was to continue in academia, but for various life reasons I now work in industry, and I enjoy it. My current job involves some elliptic curve cryptography and fiddling around with finite fields, and I'm glad I was able to find work that involves interesting math. However, I really feel like I did not get a good undergrad math education (not blaming my undergrad institution, that was entirely my responsibility) and would like to "do it over again" and really make myself well-rounded. Plus, I don't know much algebraic geometry or number theory, and feel like this will inhibit my ability at work in the long run. I want to develop a plan and execute it while I'm still young-ish, before I have kids and all study time goes away forever.
So, my questions are:
1. **What would a well-rounded undergraduate mathematics education contain?** One where you could, in theory, specialize in anything in grad school and still be OK.
2. **What are the dependencies between subjects?** I want to know ahead of time so that one subject leads into another and I don't try to start with something that's way over my head.
3. **What is a realistic expectation/goal/schedule to set for myself**, given that I have a job and chores at home to take care of? I started a 600-ish page book on probability and statistics about a month ago. Going through it deliberately and doing all exercises, I'm still only on page 20 or so. At this rate my program of study will take me approximately forever.
4. **What's the appropriate balance between quick and thorough?** That is, between trying to understand everything, writing out all the proofs, coming up with your own examples, and doing all the exercises; and skipping problems and proofs or using the internet, just to be able to move on to the next topic? I have this problem where I want to understand one thing thoroughly, but then get stuck at a certain point, switch to thinking about something else, and then forget the first thing. What I would like to cultivate is consistency and understanding.
5. **If your job involves math, is it appropriate to spend time at work studying math** that's not necessarily directly related to work, but could help you be better at your job in the long run?
Sorry for the long post: I know it's a lot of questions, some of which are about math and some of which are more about time management. Also sorry if this has been asked before, but I *did* make a good-faith effort at Googling and didn't see this question answered.
**tl;dr** How do you get a well-rounded undergraduate mathematics education while adulting with a full-time job?
Hey everyone. I went to college and got a degree in math, but definitely missed out on a lot of subjects. Just to give you an idea: I hadn't taken any classes in probability, number theory, topology, or functional analysis, and the highest level of calculus-related stuff I took was complex variables or multivariable calc. I had never heard of a tangent space, differential form, homology or cohomology, didn't know what a tensor was...some of this was stuff I didn't even know I didn't know, others was stuff that I knew was out there but didn't look into because I figured I was only interested in discrete math. When I graduated I had a strong enough background in graph theory, abstract algebra, and theory of computation, but little else. (At this point I've also forgotten a lot of that). When I went to grad school, these gaps in my knowledge became much more apparent. But I stuck with it and got my Ph.D. Along the way, I filled in some gaps in my knowledge, especially in the first year. But after finding an advisor I kept my head down, focused on my research, and didn't go much outside of my comfort zone subject-wise. This was partly because I would try some more advanced topics such as Lie groups or Riemannian geometry, get really confused and drop the class because (I suspect) I didn't get a good enough foundation earlier. My lack of knowledge of other subjects also sucked the motivation out of me even in my own research, because I couldn't see the purpose of my research within the bigger picture of math. I'm thinking of reading a textbook on history of mathematics just to understand how different math subjects are related and why people doing modern mathematics study the wildly abstract stuff they do today, but I digress.
My first plan was to continue in academia, but for various life reasons I now work in industry, and I enjoy it. My current job involves some elliptic curve cryptography and fiddling around with finite fields, and I'm glad I was able to find work that involves interesting math. However, I really feel like I did not get a good undergrad math education (not blaming my undergrad institution, that was entirely my responsibility) and would like to "do it over again" and really make myself well-rounded. Plus, I don't know much algebraic geometry or number theory, and feel like this will inhibit my ability at work in the long run. I want to develop a plan and execute it while I'm still young-ish, before I have kids and all study time goes away forever.
So, my questions are:
1. **What would a well-rounded undergraduate mathematics education contain?** One where you could, in theory, specialize in anything in grad school and still be OK.
2. **What are the dependencies between subjects?** I want to know ahead of time so that one subject leads into another and I don't try to start with something that's way over my head.
3. **What is a realistic expectation/goal/schedule to set for myself**, given that I have a job and chores at home to take care of? I started a 600-ish page book on probability and statistics about a month ago. Going through it deliberately and doing all exercises, I'm still only on page 20 or so. At this rate my program of study will take me approximately forever.
4. **What's the appropriate balance between quick and thorough?** That is, between trying to understand everything, writing out all the proofs, coming up with your own examples, and doing all the exercises; and skipping problems and proofs or using the internet, just to be able to move on to the next topic? I have this problem where I want to understand one thing thoroughly, but then get stuck at a certain point, switch to thinking about something else, and then forget the first thing. What I would like to cultivate is consistency and understanding.
5. **If your job involves math, is it appropriate to spend time at work studying math** that's not necessarily directly related to work, but could help you be better at your job in the long run?
Sorry for the long post: I know it's a lot of questions, some of which are about math and some of which are more about time management. Also sorry if this has been asked before, but I *did* make a good-faith effort at Googling and didn't see this question answered.
**tl;dr** How do you get a well-rounded undergraduate mathematics education while adulting with a full-time job?
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