What are some hot topics that mathematical statistics researchers are studying now?
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Persi Diaconis wrote an article on "Mathematical Statistics" in the Princeton Companion of Mathematics. He discusses several ongoing research areas in mathematical statistics, including the search for high-dimensional admissible estimators (Stein's paradox).
Hopefully, modelling the dynamics of tumor progression qualifies for this:
Anderson & Quaranta. Integrative mathematical oncology. Nature Reviews Cancer, 2008.