Professor Chai on several problems in arithmetic geometry. These include canonical integral models of Shimura varieties and the theory of p-adic symmetric spaces with a view to an analogue of Schmidt's theorem on degenerations of Hodge structures. This project falls into the general area of arithmetic geometry - a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful - having recently solved problems that withstood generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.