Abstracts of 2010 LSU Math REU

s of 2010 LSU Math REU talks • Name: Suzanne Carter Title: Calculating Picard Fuchs Differential Equations Abstract: Families of algebraic varieties have an associated Picard Fuchs differential equation, with a solution space spanned by the periods of the variety. These periods are integrals of differentials taken over chains in the primitive cohomology of the varieties. The, often unique, holomorphic solution to this differential equation has interesting properties when looking at the coeffiFamilies of algebraic varieties have an associated Picard Fuchs differential equation, with a solution space spanned by the periods of the variety. These periods are integrals of differentials taken over chains in the primitive cohomology of the varieties. The, often unique, holomorphic solution to this differential equation has interesting properties when looking at the coefficients; they satisfy congruences modulo powers of primes. We will discuss two methods for calculating these differential equations, a method outlined by Griffiths and a method that works for families of elliptic curves. Using the results of these techniques, we will discuss properties of these differential equations and the congruences that their solutions satisfy in certain examples. • Name: Steffen Docken Title: Symmetric Square Differential Equations of the Beauville Families of Elliptic Curves Abstract: Each of the Beauville Families of elliptic curves has a corresponding differential equation called a Picard Fuchs differential equation. These Picard Fuchs differential equations are satisfied by modular forms. This talk will illustrate how the Symmetric Square differential equation for a family of elliptic curves can be derived from the Picard Fuchs differential equation. It will also show that the Symmetric Square differential equation is also satisfied by modular forms and that the coefficients of the solutions have interesting properties. • Name: Sarah Loeb Title: Permissible Plane Embeddings of Dessin Blow-ups. Abstract: Given a ribbon graph embedded on a surface, we consider the blowup, that is the three-valent partially oriented graph constructed by replacing each vertex by an oriented circle and attaching the edges around the circle according to the rotation system. A characterization in terms of two forbidden configurations is then given as to which blow-ups have permissible planar embeddings, i.e. those with embeddings where the orientation of the circles is determined by their nesting level. The proof requires Kuratowski’s Theorem as well as an argument by cases based on connectivity. An application of this result is a characterization of which ribbon graphs arise as a state smoothing of a link diagram. • Name: Natasha Potashnik Title: Abstract: Dessins d’enfants provide a surprising connection between group theory and topology. The absolute Galois group acts on dessins, and the orbits of this action are the main subject of our study. We will discuss invariants of these Galois orbits and methods for investigating them. In particular, we will explain the notion of a Belyi-extending function and demonstrate their utility in distinguishing Galois orbits. The second Chebyshev polynomial is known