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Currently I’m interested in a couple of fields, namely dynamical systems, stochastic processes, and additive combinatorics. I was wondering if it’s feasible to keep pursuing all 3, and whether I can expect any overlap between the fields. I have not really narrowed down my focus too much yet since I’m just starting out, but to give an idea of the specific areas of interest, I’m currently using, or planning to use following books:

Dynamical systems:

  • Katok & Hasselblatt - An introduction to the modern theory of dynamical systems

  • Einsiedler & Ward - Ergodic Theory with a view towards number theory

  • Eisner et. al. - Operator theoretic aspects of Ergodic theory

  • Furstenberg - Recurrence in Combinatorics and Number Theory (Planned)

Stochastic processes:

  • Le Gall - Brownian motion, Martingales and Stochastic Calculus

Additive combinatorics:

  • Tao & Vu - Additive combinatorics

For reference I have a little under a year and a half before I start my PhD. Is there any overlap between these fields where I can combine some of my interests? Or should I specialise at some point after going through these general books?

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  • $\begingroup$ Just a small remark concerning your reference "Operator Theoretic Aspects of Ergodic Theory": The surname of the first author is "Eisner", not Tanja; Tanja is her first name. Besides, the book has actually four authors: Eisner, Farkas, Haase and Nagel ;-). $\endgroup$ – Jochen Glueck Feb 24 at 10:17
  • $\begingroup$ That said, there are certainly a lot of connections between dynamical systems and stochastic processes. I personally tend to look on both from the perspective of operator semigroup theory - so I'd say one connection is that both are related to the theory of positive operator semigroups. I'm sure that much more can be said in response to your question, but this should probably be done by somebody who has a stronger background knowlegde of all three fields than I do. $\endgroup$ – Jochen Glueck Feb 24 at 10:18
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I'd strongly encourage you to read Terry Tao's blog posts on dynamics and relations to additive combinatorics. These are beautiful descriptions of the main ideas and intuitions behind them, and are much more accessible than, say, his book with Vu. To get started, read his course posts for ergodic theory, just search for Math 254A (his 2008 course).

While you're there, also browse other parts of his blog, particularly entries giving advice and perspectives on doing research, writing papers, what to do when you're stuck, and so on. All very interesting and very valuable, especially for people just starting out.

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