The letter below is written by my son. I have been sending him text books and looking for answers on the internet to keep his interest up. He has progressed so far on his own and now he needs direction and assistance from a professional in mathematics. Any advice or assistance you can provide is greatly appreciated.
My name is —, I'm 25, I've been in prison for the past 6 years, and I'm self-taught in mathematics. I began with a list of courses required in a standard undergraduate curriculum and studied the required texts from each course. I covered the basics in this way before branching off into my own interests, beginning with partial differential equations and eventually landing in scattering theory.
I began studying mathematics because it was fun and interesting (and passed the time), but it has since become so much more. The progress that I've made, combined with the observation that I am capable of at least understanding research in my fields of interest, has compelled me to take the next step into conducting research of my own, and my current goal is to make advances of publishable value. I am just beginning in this process, yet already I have made progress studying scattering resonances. At the moment, I'm working on a number of problems related to resonance counting. In particular, my primary focus is on "inverse resonance counting": By assuming an asymptotic formula for the resonance counting function (as well as some other results concerning distribution), my goal is to determine properties of the potential. Similarly, in the case of a surface with hyperbolic ends, the goal is to determine properties of the surface from knowledge of an exact asymptotic formula for the counting function. My primary resources at present are Mathematical Theory of Scattering Resonances by Dyatlov and Zworski, and Spectral Theory of Infinite-Area Hyperbolic Surfaces by Borthwick.
I'm not sure what I'm asking for here, I just know that I am ready for the next step and seek some guidance as I enter the world of research mathematics. I encounter many problems when it comes to research, such as staying up to date on current topics, finding open problems which suit my skills and interests, and finding papers on topics I need to study more deeply. For example, right now I am in need of results on how resonances change under smooth, small changes in the potential. One of my texts mentioned the paper of P. D. Stefanov, Stability of Resonances Under Smooth Perturbations of the Boundary (1994), but I need more, and that paper makes no citation to papers of the same content. How do I find papers which are similar, or even cite this one?
In short, without direct access to the internet or fellow researchers, I hit many roadblocks which are not math-related, and that can be frustrating. I'm looking for ways to make my unconventional research process go a little more smoothly. If anyone has any suggestions, please let me know here.
And thanks in advance.
Best Answer
I have received a response back from my son he said. "I took Calculus my first and only year at Michigan State University, prior to my incarceration. That is the highest course I have taken formally. Shortly after beginning my sentence, I asked my father for a multivariable calculus textbook and he sent me one. I studied it deeply, and enjoyed it so much that I asked my dad if he could find anything online about what's after calculus in a standard undergraduate curriculum. He found MIT's opencourseware website, and sent me screenshots of a page listing which courses were required of an undergrad math major at MIT, some sample undergrad course loads, as well as pages listing course titles, with descriptions and prerequisites. From there I would ask my dad for all the info on a given course, including required textbooks, lecture notes, and problem sets. Many courses even gave dates on which the problems were due. He'd order me the book(s), and print out and mail me the problems and notes. Starting with linear algebra and a course on ordinary differential equations, I proceeded this way for a couple of years. Often, I would also study the chapters in the books which weren't required by the course and at least attempt every problem. Here is a sample of some of the books required with the courses:
Strang - Introduction to Linear Algebra
Zill - A First Course in Differential Equations
Pinter - A Book of Abstract Algebra
Rudin - Principles of Mathematical Analysis
Ahlfors - Complex Analysis
do Carmo - Differential Geometry of Curves and Surfaces
Simmons - Introduction to Topology and Modern Analysis
Lee - Introduction to Smooth Manifolds
As I matured mathematically, I stopped using the opencourseware site, and began studying in areas which had interested me. I no longer read textbooks linearly, nor do I try to digest every concept in every book I obtain. But I have many, many books. Among them are about 4 on PDE's in general, a handful on more technical but related topics, like perturbations, scaling, dimensional analysis, waves. I own Hormander's, The Analysis of Linear Partial Differential Operators, Vols 1-4, and Reed and Simon's, Methods of Modern Mathematical Physics, Vols 1-3. I have 2 on semiclassical and microlocal analysis, 2 on analytic number theory. Of course in addition to Volume 3 of the Reed and Simon series, the books mentioned in the original post are my resources on scattering theory. I also have a handful of introductory physics texts. Aside from books, I have a few research articles in scattering related to my recent attempts at research. Scattering is the only field in which I've made a serious attempt at research. Hopefully this answers your question, and if not, please follow up!"