Mathematics is a discipline for young people.
Most of the greatest mathematicians in history started very early their work and, often, they had the best results when still very young.
I believe, anyway, that there is still some space for older mathematicians…
After some research, I ended up with this funny list:
- Joan Birman got her PhD when she was 41, she did a great job in several areas among which braid theory;
- Eugène Ehrhart got his PhD when he was… 60;
- Albrecht Fröhlich worked on Galois theory (there is even a prize named after him); he got his PhD when he was 36;
- George Green was an autodidact that did undergrad studies when he was 40; apparently Albert Einstein commented that Green had been 20 years ahead of his time.
But the most striking case is, for me, Preda Mihăilescu: he got his PhD when he was 42 and just a few years later he gave the proof of the 158-year-old Catalan’s conjecture.
A more scientific study examined the correlation between age and scientific creativity but I believe that, after all, it is not only a matter of age.
Check this very touching blog post by Tanya Khovanova to understand what I mean…
Not all the mathematics that I studied until today made a sign in me. Many books and articles come and go and just a few things remain.
I’d like to list here some of the things that somehow had an impact in my personal and professional life:
- some books about recreational mathematics that I read when I was at the elementary school (mostly from Martin Gardner): at the time I was following the suggestions by my uncle Marcello (that was obsessed by prime numbers);
- some classes in high school with relative manuals and exercises: I was bad in analytic geometry but I think I understood well calculus (that we called analysis at the time);
- logic at university [Negri, Elementi di logica (several chapters); Odifreddi, Classical recursion theory (few sections), Chung-Keisler, Model theory (few sections), …]: I applied a sort of pattern-recognition to formulas without really understanding (the problem was that we never asked to do exercises) but this material had a deep impact on me;
- abstract algebra at university: I don’t remember which books I used and I complemented with some tutorials online on group theory; unfortunately I didn’t perform so well at the final exams but group theory is one of the best things I’ve never seen;
- Sergei Treil, Linear Algebra done wrong: for the very first time I tried, with this book, to really understand proofs, without skimming through them; I think I retained several parts of the book and linear algebra is the branch of mathematics that I know better today;
- MIT online courses on calculus and linear algebra: I understood pretty well while following them and I believe that every professor should watch Gilbert Strang to learn how to teach;
- [1999-now] self study in signal processing [some tutorials online, parts of Lyon’s book on DSP, parts of Ripples in mathematics (wavelets), some chapters of Smith’s Mathematics of the DFT, sections of Mallat’s book, …]: I believe I understood some linear algebra, something about filters, something about Fourier, something about wavelets and convolution;
- selected parts in some analysis textbooks.
There are probably several other books or articles that I read about mathematics (especially during my PhD in… mathematical logic or my post-doc at École normale supérieure in Paris) but probably they didn’t mean too much for me.
I believe, anyway, that the part that one needs more when doing mathematics is exactly doing it. Lack of practice is always an issue.