Some of my favorites. 1. The whole theory of elliptic functions and modular forms. 2. The Riemann mapping theorem: The interior of any plane simply conncted region whose boundary consists of more than one point can be mapped conformally onto the interior of the unit disk. 3. The uniformizaton theorem: Let P(x,y) be a polynomial, with x and y varying over the complex plane, including infinity. P(x,y)=0 defines an algebraic function, in general multivalued. Suppose the Riemann surface defined by P(x,y)=0 has genus g. We say that the multivalued function is uniformized by single-valued functions [in the 19th century, they were called "uniform functions"] x(t), y(t) if P(x(t),y(t))=0 identically. Then the algebraic function can be uniformized by means of (a) Rational functions, if g=0, (b) Elliptic functions, if g=1, (c) Fuchsian functions of the first kind, if g>1. As an application consider x(t)^n + y(t)^n = z(t)^n. For n=2, we have a polynomial solution x=(1-t)^2, y=2t, z=(1+t)^2, but not for n>2. [See e.g. Ford, "Automorphic Functions", which is one of my favorite books.] 4. Wedderburn's theorem: A finite (associative) division ring is a field. [Question: is a division ring D always a division algebra? Take the ground field of the algebra to be the set of x in D that associate and commute with all elements of D.] This extends to nonassociative division algebras of characteristic not 2,3,5. [Schaffer, "Nonassociative algebras"] 5. A number theoretic consequence that follows from the theory of modular forms. Let sigma(p,n) be the sum of the p-th powers of the divisors of n. sigma(7,n) = sigma(3,n) + 120 sum(sigma(3,k) sigma(3,n-k),k=1..n-1). 6. The theory of the motion of a rigid body. [See Whittaker "Analytical Dynamics".] 7. The compact-Hausdorff theorem: If a topological space is compact and Hausdorff, then in any stronger topology it fails to be compact, and in any weaker topology it fals to be Hausdorff. 8. The classification of simple Lie algebras over an algebraically closed fields of characteristic zero. [Jacobson, "Lie Algebras", another of my favorite books.] Gene __________________________________________________ Do You Yahoo!? Tired of spam? Yahoo! Mail has the best spam protection around http://mail.yahoo.com