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"There are three different topics related to this project. The first one is an interplay between probability and Morse theory. The main idea is to exploit in the probabilitisc framework a suggestion of Witten and recover the Morse-Smale complex of a Morse function using asymptotics of Wiener functionals. The second topic is the study of transportation cost inequality on the path space of a Riemann ...

This proposal belongs to the area of modular forms and L-functions, and it consists of both algebraic and analytic problems related to the structure of the space of modular forms, of integral and half integral weight. The methods I use involve the theory of periods of modular forms, developed in the 1970s by Eichler-Shimura-Manin, and in the 1980s by Kohnen and Zagier. Part of the proposal is conc ...