The talk will be in the general area of birational geometry. Can we find singular representatives of birational equivalence classes of algebraic varieties, with the simplest possible singularities? In particular, can we find the smallest class of singularities that necessarily persist after birational mappings that preserve smooth points and transverse self-intersections of the target spaces? Many of the questions considered were raised by Janos Kollar.
Many branches of mathematics are used to develop algorithms for modern molecular medicine and visualization of its data. I will discuss some examples, including high-resolution DNA melting analysis to determine transplant compatibility, and identifying genes associated with tumor progression that led to a therapy. I will also describe some surprising mathematical connections discovered in the course of this work.
Mathematicians have made a lot of progress in the last 350 years, but not in writing proofs. The proofs they write today are just like the ones written by Newton. In a talk presented at a workshop celebrating Dick Palais' 60th birthday, I explained how to do better. This is a new version of that talk, reflecting 20 more years of experience writing better proofs.