A substitution rule is an algorithm for replacing a symbol with a finite string of symbols (for example, replace 0 by 01 and replace 1 by 0) and a substitution sequence is a sequence obtained from repeated applications of the substitution rule. A Sturmian sequence is a non-periodic sequence of minimal complexity. Both of these objects that are central to the study of mathematical models of quasicrystals. We will discuss interesting dynamical, algebraic, and combinatorial properties of these two families of sequences, as well as their relation to each other.

In this talk I will present the idea that by providing a "natural representation" of the solution (e.g. is the solution is C^k, or does the solution have known jumps, does it have a characteristic structure...), one may devise discretizations that are in principle of arbitrary order, and optimally compact. In many cases, the structure required may be hidden, and thus requires a closer look at the geometrical properties of the underlying operator. I will illustrate these concepts by examining 3 popular PDEs: the linear advection, Poisson's equations with jump discontinuities, and the heat equation.

The endomorphism rings of ordinary jacobians of genus two curves defined over finite
fields are orders in quartic CM fields. The conductor gap between two endomorphism rings is
defined as the largest prime number that divides the conductor of one endomorphism ring but not
the other. We call a genus two curve isolated, if its endomorphism ring has large conductor gap
(>=80 bits) with any other possible endomorphism rings. There is no known algorithm to explicitly
construct isogenies from an isolated curve to curves in other endomorphism classes. I will
explain results on criteria for a curve to be isolated, as well as the heuristic asymptotic
distribution of isolated genus two curves.

We want to understand spaces that parameterize projective subvarieties. One way to do this is to look at Algebraic Cycles. An Algebraic Cycle is a formal sum of irreducible closed subvarieties. If we take a family of irreducible subvarieties, its limit may have several irreducible components, i.e. the limit may be a general sycle.
We want to study this phenomenon and the Chow Varietes are a way of doing thins. Simply put, the points of a Chow variety are Algebraic Cycles. We will explain at the Chow - Van der Waerden Theorem that imbeds the variety into projective space. Finally we move on to a specific example, 0-cycles. We can use symmetric polynomials to work with 0-cycles. Using this we will look at the tangent space, and derive a formula for the tangent space of a multiple of smooth point.