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Princeton Summer School in Analysis and Geometry, 2009 Welcome to the Princeton Summer School in Analysis and Geometry, 2009!
Below please find some supplementary material for the course. If you are interested in typing up some notes for the course and share them with others (may they be a recap of what you have learned in the lecture, some interesting discussion you had with other people in the class or some solutions to interesting problems that you have solved), I'll be collecting and displaying them here. Let A denote the Banach algebra of holomorphic functions on the open unit disc in the (one-dimensional) complex plane that are continuous up to the boundary. In one of the problems handed out in class, we were asked to show that any multiplicative and linear functional E on A must be an evaluation map at some point in the closed unit disc. Part of our discussion went into discussing whether such a map E must be continuous. Thanks to Yakov and Otis, here are two articles that summarizes some of our discussion. A proof that multiplicative linear functionals on A are continuous, by Yakov Shlapentokh-Rothman. A discussion of the original problem about the characterization of the multiplicative linear functionals on A, by Otis Chodosh. In another problem, we were asked to prove that any L^2 function on the real line whose distributional derivative vanishes is almost everywhere a constant. Here is a related discussion which generalizes this. The solution was due to Khoa Nguyen, and written up by Otis Chodosh. Andy Manion has kindly reproduced the solutions to some of the problems we discussed in class. The original problems can be found here. |