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.

I will address questions about uniqueness of tangent maps of harmonic maps to non-positively curved metric spaces.

I will also address applications to superrigidity of group representations to non-positively curved groups.

The self-dual Yang Mills equations are equations for a connection in a principal bundle on a 4-manifold with a real structure group such as SU(2). They have been a source of immeasurable geometric, analytic and topological interest since they were introduced to the mathematics community in the l960's. It is natural to define a complex connection with the same structure group; Kapustin and Witten have introduced a one parameter family of equations for this complexified connection which are related in a natural way to the Yang-Mills equations. We carefully review some of the geometry and topology of the self-dual Yang Mills connections and describe how the Kapustin-Witten equations are related to these older self-dual and anti self-dual equations. After touching briefly on interesting aspects of complex geometry which arise for these complex equations over a real four manifold, we finish by describing the basic unsolved question of the existence of global estimates.