I have an undergraduate degree in Probability and Statistics and am contemplating joining a graduate program for PhD. I believe I am reasonably strong in probability theory and would like to do research in the area.
However, I am a bit confused about where the current research in probability is going. I went through some reputed journals in Probability like the Annals of Probability and Journal of Theoretical Probability. I was a bit disappointed to see that currently research in probability constitutes either combinatorics or differential equations. There are lots of models, primarily from Statistical Physics literature.
I like probability and mathematical statistics. I do not really want to be in combinatorics -- never really liked the field. But from what probabilists are currently interested in, it seems that finding expected cluster sizes in random graphs and doing triangulations and stuff are all that are happening.
Where did the research similar to that of Stein go? In my university I am doing courses like Brownian motion, Empirical Processes, Martingale theory, etc. However I don't find any article on these. Are these too outdated? Has the market been completely occupied by triangles and graphs? Is probability too immersed in "models"? One of my professors said, "Everything is a model!" Then where is the identity of a subject? Everybody who seems to do probability and is sufficiently famous seems to be doing these!
I think it is time to really understand where the research in probability is going at present. I have been long disappointed to see no big program in probability, like Perfectoid spaces or Langland's program in Number Theory. There is no big theory which people try to understand.
How does a prospective graduate student understand what is the aim of probability? What are the non-combinatorial areas of probability? How active are they? What are some resources or influential papers on these?
What should one learn in order to attack modern problems in probability? As I said, in other subjects there are things to learn. There are interconnections between Algebraic Geometry and Algebraic K Theory. I understand such theories are not there in probability since it is too young. Eve if they exist, are they really active areas?