On the Application of Combinatorial Analysis to Mumber Theory Geometry and Analysis

In this lecture I will discuss the application of some well known and less well known theorems in combinatorial analysis to various other branches of mathematics. In other words I will not mention combinatorial types of reasoning the use of which is of course very wide-spread (e.g. the classical proof of Carleson on the almost everywhere convergence of Fourier series of functions in L2 is full of combinatorial reasoning), but will restrict myself to cases where definite quotable theorems are used. My paper in no ways claim to give a complete survey of all the applications of combinatorial theorems and is certainly heavily biased towards my own work. Though combinatorics has been successfully applied to many branches of mathematics these can not be compared neither in importance nor in depth to the applications of analysis in number theory or algebra to topology, but I hope that time and the ingenuity of the younger generation will change this.