When I was a kid I wanted to get a PhD in math and physics and comp sci and etc etc etc. I was really disappointed when I got older and realized I would only have time for one. So I chose math because I wanted to study mathematical logic, differential equations, numerics and etc etc etc. Then I got to grad school and realized I would only have time for one. So I chose numerics because I wanted to study molecular dynamics, computational neuroscience, fluid dynamics, etc etc. Then I had to choose an advisor, and I realized I'd only get in an education in one sub discipline. So I chose fluid mechanics, because I wanted to study polymeric fluids, multiscale dynamics, fluid-structure interactions, etc etc. Then I finally started doing real research and realized I'd be lucky just to make a noticeable contribution to one of those.
There are exceptions, of course.
(I don't mean to sound pessimistic, just realistic. Research has been good to me, and I love doing it)
For example, I'm no researcher, but I tend to try and read a few good books about maths, and sometimes even actual mathematical books, just for the fun. Do you tend to do the same?
One of the reasons I chose applied math was because I thought it was the best centralized position in all of science. It would take me years to switch to another branch of math, but in applied math I can at least switch the focus of application, possibly in under a year or two.
Elon Lindenstrauss is being awarded the 2010 Fields Medal for his results on measure rigidity in ergodic theory, and their applications to number theory.
Fields Medal - Elon Lindenstrauss, Princeton University Lindenstrauss has made far-reaching advances in ergodic theory, the study of measure preserving transformations. His work on a conjecture of Furstenberg and Margulis concerning the measure rigidity of higher rank diagonal actions in homogeneous spaces has led to striking applications. Specifically, jointly with Einsiedler and Katok, he established the conjecture under a further hypothesis of positive entropy. It has impressive applications to the classical Littlewood Conjecture in the theory of diophantine approximation. Developing these as well other powerful ergodic theoretic and arithmetical ideas, Lindenstrauss resolved the arithmetic quantum unique ergodicity conjecture of Rudnick and Sarnak in the theory of modular forms. He and his collaborators have found many other unexpected applications of these ergodic theoretic techniques in problems in classical number theory. His work is exceptionally deep and its impact goes far beyond ergodic theory.