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.

Given a Lie group G with Lie algebra g, and a manifold M^n of dimension n >0, what invariants determine whether there is an effective action A of G, or g, on M^n? If A exists what can be said about fixed points? Can A be analytic? If not, what can be said about its kernel?

Typical results:

1. The identity component of the group of upper triangular n-by-n real matrices has effective smooth actions on every M^n. But such an action cannot be analytic, because the fixed point set of some 1-dimensional central subgroup has nonempty interior.

2. Assume for some X in g that ad X has m>0 eigenvalues whose imaginary parts are linearly independent over the rationals. Let A be
an effective analytic action of g on M^n.

(a) If n < 2m then A(X) has no fixed points.

(b) Assume n=2m and M^n is compact. Then the number of fixed points of A(X) equals the Euler characteristic Char(M), which is therefore nonnegative.

EXAMPLES: If M^n is compact and Char(M)

The sequencing of the human genome and the development of new methods for acquiring biological data have changed the face of biomedical research. The use of mathematical and computational approaches is taking advantage of the availability of these data to develop new methods with the promise of improved understanding and treatment of disease.

I will describe some of these approaches as well as our recent work on a Bayesian method for integrating high-level clinical and genomic features to stratify pediatric brain tumor patients into groups with high and low risk of relapse after treatment. The approach provides a more comprehensive, accurate, and biologically interpretable model than the currently used clinical schema, and highlights possible future drug targets.