Serre proved in 1972 that the image of the adelic Galois representation
associated to an elliptic curve E without complex multiplication has open
image; moreover, he also proved that for an elliptic curve over Q the
index of the image is always divisible by 2 (and in particular never
surjective). More recently, Greicius in his thesis gave criteria for
surjectivity and gave an explicit example of an elliptic curve E over a
number field K with surjective adelic representation. Soon after, Zywina,
building on earlier work of Duke, Jones, and others, proved that the
adelic image `random' elliptic curve is maximal.

In this talk I will explain recent joint work with David Zywina in which
we generalize these theorems and prove that a random abelian variety in a
family with big monodromy has maximal image of Galois. I'll explain what
big monodromy and maximal mean an explain the analytic and geometric
techniques used in previous work and the new geometric ideas -- in
particular, Nori's method of semistable approximation-- needed to
generalized to higher dimension.

For an integer n > 2, the unit group modulo n has an even number
of elements, with half of them having representatives in (0,n/2)
and the other half having representatives in (n/2,n). It is
"balanced". Say a subgroup H of this unit group is "balanced"
if each coset of H is evenly split between the bottom half and
the top half. Suppose g>1 is a fixed integer. We are concerned with
the distribution of numbers n coprime to g for which the
cyclic subgroup in the unit group mod n is balanced.
This has an application to the statistical study of the rank
of the Legendre curve over function fields. (Joint work with
Douglas Ulmer.)

Among the most important complexity problems in
coding theory are the maximun likelihood decoding
and the computation of the minimun distance.
In this talk, we explain a self-contained, quick and
transparent proof of the NP-hardness of these
problems based on the subset sum problem over
finite fields.

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.