Arithmetic Statistics of Modular Symbols

We will discuss recent progress and resolution of a series of conjectures by Mazur, Rubin, Stein on the statistical behavior of modular symbols in order to understand the central values of twists of L-functions of elliptic curves.  These values have been the object of extensive studies in the last 50 years.  On the other hand, modular symbols have been a useful tool in studying the space of holomorphic cusp forms of weight 2, as well as the homology of modular curves; they have been the object of extensive investigations by many mathematicians including Birch, Manin, and Cremona.