Fourier coefficients of automorphic forms are the building blocks for automorphic L-functions. While these coefficients are often quite mysterious, there is one family of automorphic forms whose Fourier coefficients do have an explicit and rather uniform description -- Eisenstein series. In fact, Langlands' initial study of Eisenstein series' coefficients in the 1960's led him to make conjectures about equalities of L-functions which inform much of modern number theory. I'll discuss two new explicit descriptions for Fourier coefficients of Eisenstein series which hold in great generality and hint at undiscovered connections among automorphic forms, representation theory, and physics. One description makes use of Kashiwara crystal graphs and the other uses the 6-vertex model in statistical mechanics. Both objects possess beautiful combinatorial structure that deserves to be more widely known, though we do not assume familiarity with either and all concepts mentioned above will be defined over the course of the talk.

We examine methods to produce triples of integers which are the sides of a right triangle (i.e., (3,4,5), (5,12,13), or (8,15,17)). Multiplication of complex numbers will make an appearance, the first example of the interplay between algebra and geometry. We will then learn about elliptic curves, which provide a similar, but much more intricate, synthesis of algebra and geometry. Elliptic curves are a current area of intense research in mathematics and computer science, playing a central role in modern cryptology and in the recent proof of Fermat's Last Theorem.

I will start with the definition of the metastable states as dynamical (temporal) states of the system, relaxing to equilibrium, before the equilibrium is reached. I will explain then that in general one should not expect these states to be Gibbs states.