Modular curves are the most famous example of the title. As moduli space towers they exhibit a "Frattini property," based on their monodromy groups as covers of the j-line. Using the goals of Serre's "l-adic representations" book I will treat, in parallel, two cases of general ideas.
Modular curves here derive from the semi-direct product of Z/2 acting through
multiplication by -1 on Z; and
the equally rich case from Z/3 acting irreducibly on Z2.
This view has modular curves as families of sphere covers attached to dihedral groups. In this case we see something familiar their cusps and the monodromy on homology in a fiber in a new way. Then, with analogous methods, we outline the 2nd case to show how the tools extend. To take Serre's Open Image Theorem beyond modular curves, to general moduli of abelian varieties, has failed to master the limiting effect of correspondences read motives of arithmetic monodromy on special tower fibers. Our Z/3 case shows how Frattini data in our Hurwitz space approach helps tame that structure.

Primality testing and finding large prime numbers has significant applications to cryptography. In this talk I will discuss a deterministic, polynomial time algorithm for determining if an integer is prime developed by Agrawal, Kayal, and Saxena. Here polynomial time means that there exists constants c,d such that the number of operations to determine if the given integer is prime is less than c log^d(n) where n is the number we are testing for primality.

Variables Separated Equations and Finite Simple Groups: Davenport's
problem is to figure out the nature of two polynomials over a number
field having the same ranges on almost all residue class fields of the
number field. Solving this problem initiated the monodromy method.
That included two new tools: the B(ranch)C(ycle)L(emma) and the
Hurwitz monodromy group. By walking through Davenport's problem with
hindsight, variables separated equations let us simplify lessons on
using these tools. We attend to these general questions:
1. What allows us to produce branch cycles, and what was their effect
on the Genus 0 Problem (of Guralnick/Thompson)?
2. What is in the kernel of the Chow motive map, and how much is it
captured by using (algebraic) covers?
3. What groups arise in 'nature' (a 'la a paper by R. Solomon)?
Each phrase addresses formulating problems based on equations. We seem
to need explicit algebraic equations. Yet why, and how much do we lose/
gain in using more easily manipulated surrogates for them? To make
this clear we consider the difference in the result for Davenport's
Problem and that for its formulation over finite fields, using a
technique of R. Abhyankar.

Dynamical entropies are measures of the complexity of orbit structures. The topological entropy considers all the orbits, whereas the measure theoretic entropy focuses on those ``relevant" to a given invariant probability measure. The variational principle says that the topological entropy of a continuous self-map of a compact metrizable space is the supremum of the measure theoretic entropy over the set of invariant probability measures for the map.

A well known fact is that every transitive hyperbolic (Anosov) diffeomorphism has a unique invariant probability measure whose entropy equals the topological entropy. We analyze a class of deformations of Anosov diffeomorphisms containing many of the known nonhyperbolic robustly transitive diffeomorphisms. We show that these $C0$-small, but $C1$-macroscopic, deformations leave all the high entropy dynamics of the Anosov system unchanged, and that there is a partial conjugacy identifying all invariant probability measures with entropy close to the maximum for the deformation with those of the original Anosov system.

Additionally, we show that these results apply to a class of nonpartially hyperbolic, robustly transitive diffeomorphisms described by Bonatti and Viana and a class originally described by Mane. In fact these methods apply to several classes of systems which are similarly derived from Anosov, i.e., produced by an isotopy from an Anosov system.