Would you mind telling more about your general experiences of grad school and afterwards?
Aside from the issues that I had with my adviser, graduate school was a really great place. When I first started out, however, I was overwhelmed, because I was placed on a very, very large grant project and given a very complex, open-ended mathematical task to complete with little direction. To make matters worse, no one on the project told me what they expected of me, where I should pick up relevant information, what journals I should be reading, who I could possibly collaborate with, how long I had to complete certain objectives, and so on. As well, they assumed I joined the project with a deep knowledge of probability and statistics, which I didn't have (and they should have known, considering that I mentioned it to them several times during the interview process), since all of my undergraduate coursework was on the hardware side of electrical and computer engineering.
In any event after a few months of floundering, banging my head against the wall, bringing myself up to speed, slogging through classes, and just trying to find the time to go out and meet people, I heard that I was accepted into medical school and also was set to receive a fellowship. As if by magic, all of my worries melted away with that news, since I wouldn't have to rely on my position on the grant to pay for tuition or my shoebox-sized apartment, I could afford to go out to eat, and would ultimately have much more freedom with regards to how many classes I took or how much free time I could allot myself. In fact, looking back, I really made the most of those 5 years of funding, as I had enough time off to meet a lot of new friends (something I didn't do when I was an undergraduate, since I spent all of 2 years, after high school, on my S.B. EE), was able to travel the world, see new places, experience new foods and cultures, could afford to take whatever classes I wanted without question (I loaded up on math, biology, neuroscience, and statistics courses during my off-years of medical school), could work on any research projects, and, overall, just enjoy life.
While I did relish all of that freedom, one of the biggest hurdles I ran into was publishing on my own, as I had no collaborators and was fairly isolated due to my fellowship. Instead of sticking it out and spending 2-4 months per journal paper, I decided that it would nice if I could offload some of the work, give up some credit, and churn out journal papers every 3-4 weeks (under a light class load). Initially, I tried doing this with one of my publishing-inclined peers, but I ended up having a huge argument with him, as he tried to usurp ownership of an idea (despite contributing only a small proof that turned out to be wrong and unmendable) and publish it on his own at some conference. Eventually, I thought that it would be good try collaborating on a grant project again, as there would be a bit more oversight and I could avoid having someone trying to claim ownership of my work. Unfortunately, I made a big mistake when I mentioned this desire to my adviser (who had the authority to cut the strings on my fellowship after 4 years) and he effectively strong-armed me into working on his grants if I wanted to keep my fellowship. (The only upshot to that arrangement with him was that he had to pay me by the hour, to get around the fellowship restrictions, so I ended up making almost as much as the assistant professors in the electrical engineering department my final year there.)
In any event, after my entire ordeal with him and the justice system, I had the option of going back, getting another adviser, and paying to finish a Ph.D./M.D. Instead, I opted to head back to my alma mater for a bit, to build up my network of contacts and find people to write letters of recommendation, before transferring to a better school. Fortunately, with my body of coursework and publications, I was able to test out of the classes that were similar to ones I had taken before or was able to write a journal paper with the professor to "demonstrate" my knowledge of the subject matter. As well, I was able to find a younger, MUCH better adviser that time around, who recognized that I was a paper-writing fiend (which helped with building up his case for tenure) and just threw a bunch of money and support at me so that I would keep publishing.
Since graduating, I've had a fair number of opportunities ranging from becoming a tenure-track professor or a research professor to taking a job out in industry doing biostatistics work or designing hardware. However, despite all of my problems I had with academia, I truly enjoy the environment, let alone publishing, researching, and sometimes teaching.
What kind of mathematical work do you do?
A good majority of the mathematical work that I do is relatively simple compared to the stuff you'd find in some of the higher-end SIAM journals, Journal of Differential Geometry, Geometry and Topology, Journal of Algebraic Geometry, etc. For instance, in a typical paper that I send to engineering journals, e.g., various IEEE/ACM Transactions, the most I'll end up doing is: (i) constructing a Gibbs, reversible-jump Markov chain Monte Carlo, or variational Bayes sampler for a generative model (the latter of which just involves taking some derivatives), (ii) proving some simple properties about a statistical model (such as its mean and variance, that one can obtain a predictive formula through marginalization, or that it behaves a certain way, which basically amount to taking some anti-derivatives and recalling basic probability assumptions), or (iii) possibly giving a convergence proof of a method to show that the iterates tend to a local optimum (which relies on modifications of standard nonlinear programming results like from Zangwill, transforming the method into a contractive fixed-point algorithm, etc.). All-in-all, the amount of work usually takes a day or two to complete on my own.
Now, there will be times where I break out of this mold and actually flex my mathematical muscles. A recent example of this is a paper that I just submitted that provided a means of simplifying the algorithmic complexity of a method by casting the underlying computational problem on a very special kind of manifold (which, unfortunately, resisted simple analysis). In order to finish the paper (which was a relatively short 30 pages) I ended up having to translate and rework the concepts in several differential geometry books and papers (which made for a 150 page appendix) to ensure that all of the derivations my work relied on were correct. While I can't remember exactly when I began on the project, I believe it took approximately a year to complete, along with the input and help of a few mathematicians.