There's a lot of work in statistical methods for designing clinical trials and analyzing molecular data. You might not consider statistics to be mathematics, but there are a lot of more core mathematical problems that come up in the execution of the statistics. For example, in the process of implementing Bayesian clinical trial methods, I've had to solve problems in special functions, numerical analysis, probability, optimization, etc.
Another area that comes to mind is optimization problems to determine how to maximize the radiation delivered to a tumor while minimizing the radiation delivered to healthy tissue.