After introducing the basic theory of automorphic forms and L-functions, we will discuss the characterization
of the nonvanishing of the central value of certain Rankin-Selberg L-functions in terms of periods of automorphic
forms. This is part of the global Gan-Gross-Prasad conjecture, which was first announced in early 1990's by
Gross and Prasad and was reformulated by Gan, Gross and Prasad in 2010. Our results were accummulated in
a series of my three papers (2004, 2005, 2009), joint with Ginzburg and Rallis and a more recent paper (2010)joint with Ginzburg and Soudry.

Finding a point on a variety amounts to finding a solution to a system of
polynomials. Finding a "rational point" on a variety amounts to finding a
solution with coordinates in a fixed base field. (Warning: our base field
will not be the field of rational numbers Q.) We will present some
theorems about when it is possible to find such a rational point. We will
state Tsen's theorem and the Chevalley-Warning Theorem. We will also
state some more recent results of Hassett-Tschinkel and
Graber-Harris-Starr, which rely on the notion of a "rationally connected
variety". This notion is an analogue of the notion of "path
connectedness" in topology.

The Weil Conjectures are one of the most beautiful theorems in mathematics. In the number field context zeta and L-functions are transcendental. It is well known, for example, that zeta(2)=pi^2/6. The values of these functions, even at integers, are not well understood. The Weil conjectures state the perhaps shocking result that the function field analogues of these functions are almost as simple as possible: they are rational functions. Further, they include the analogue of the Riemann Hypothesis for function fields. In this talk we will explore what the Weil conjectures say, as well as how they are proven.