# Triple Products of Eisenstein Series

The main result of this thesis is the construction of Massey triple products of Eisenstein series. Massey triple products are a generalization of the ordinary notion of multiplication; instead of multiplying two objects together, the Massey triple product is a way of combining three different objects at once. The objects that are multiplied in this way are called cocycles, which can be thought of as representing the mathematical properties of a certain geometric space. The most familiar application of Massey triple products is in the study of the Borromean Rings.

This configuration of three rings has the surprising property that if any one of the rings is removed, the remaining two rings are unlinked. Despite this, the three rings cannot be pulled apart. One may ask: how can the Borromean ring configuration be distinguished mathematically from a set of three unlinked rings? Massey triple products provide an answer to this question. The geometric space occupied by three unlinked rings has only a trivial Massey product (i.e. all products are 0), but the geometric space occupied by the Borromean configuration supports a non-trivial Massey triple product (i.e. some products are nonzero).

In this work, the objects of study are cocycles for a more complicated space than the Borromean ring configuration. The space of study is called the modular curve, and its cocycles are given by functions called Eisenstein series. Eisenstein series are a kind of modular form, which are mathematical objects that have many deep connections to number theory.

In particular, there is number-theoretic motivation for studying products of Eisenstein series. Recent work by Brown has established that the ordinary products of Eisenstein series predict algebraic relations between multiple zeta values, which are transcendental numbers arising from the Riemann zeta function. This dissertation extends Brown's work by construction Massey triple products of Eisenstein series, which conjecturally predict new relations between multiple zeta values as well.

Anil Venkatesh graduated in September 2015. His thesis, advised by Professor Richard Hain, is described above.