Dimensions of Prym Varieties

Given a tame Galois branched cover of curves pi: X -> Y with any finite Galois group G whose representations are rational, we compute the dimension of the (generalized) Prym variety corresponding to any irreducible representation \rho of G. This formula can be applied to the study of algebraic integrable systems using Lax pairs, in particular systems associated with Seiberg-Witten theory. However, the formula is much more general and its computation and proof are entirely algebraic.Comment: LaTeX, 9 pages, no figures. This work was part of my Ph.D. thesis at U. Pen

Automorphism groups of some AG codes

We show that in many cases, the automorphism group of a curve and the permutation automorphism group of a corresponding AG code are the same. This generalizes a result of Wesemeyer beyond the case of planar curves.Comment: added a reference, fixed error in remark

Suzuki-invariant codes from the Suzuki curve

In this paper we consider the Suzuki curve $y^q + y = x^{q_0}(x^q + x)$ over the field with $q = 2^{2m+1}$ elements. The automorphism group of this curve is known to be the Suzuki group $Sz(q)$ with $q^2(q-1)(q^2+1)$ elements. We construct AG codes over $\mathbb{F}_{q^4}$ from a $Sz(q)$-invariant divisor $D$, giving an explicit basis for the Riemann-Roch space $L(\ell D)$ for $0 < \ell \leq q^2-1$. These codes then have the full Suzuki group $Sz(q)$ as their automorphism group. These families of codes have very good parameters and are explicitly constructed with information rate close to one. The dual codes of these families are of the same kind if $2g-1 \leq \ell \leq q^2-1$

Codes from Riemann-Roch Spaces for Y2 = Xp - X over GF(P)

Let Χ denote the hyperelliptic curve y2 = xp - x over a field F of characteristic p. The automorphism group of Χ is G = PSL(2, p). Let D be a G-invariant divisor on Χ(F). We compute explicit F-bases for the Riemann-Roch space of D in many cases as well as G-module decompositions. AG codes with good parameters and large automorphism group are constructed as a result. Numerical examples using GAP and SAGE are also given

Coordinating IBL and non-IBL Calculus II

Increasing amounts of research support the efficacy of inquiry and projects based learning. However, teaching via inquiry can be challenging for an individual instructor to adopt in a highly coordinated environment where a course is taught by multiple instructors, and all sections are expected to follow a common syllabus and take a common final exam. In this paper, we describe our efforts to make space for an inquiry approach to teaching calculus within this constrained environment where the new approach is not adopted by all instructors. Our efforts started with the collection, adaption and development of materials to cover the topics already defined for the course. We piloted our materials with a small group of instructors in the first semester and then opened up the materials to other instructors in subsequent semesters. We have now implemented this method over the past four semesters. Through this process we have shown that the integration of inquiry methods and projects within the pedagogy of individual instructors can be effective, but efforts should be taken to ensure the timing of instruction and coverage of materials is comparable to the efforts of colleagues teaching via lecture methods

Prym varieties and integrable systems

Given a Galois cover of curves π : X [special characters omitted] Y with any finite Galois group G whose representations are rational, we may consider the Prym variety Prym ρ(X,Y) corresponding to any irreducible representation ρ of G. In chapter two, we will compute the dimension of a Prym variety. In chapter three, we look at the decompositions into Prym varieties of the Jacobians of quotients of X, in the case where G is a Weyl group and the quotient is by a parabolic subgroup P. This corresponds to the decomposition into irreducible representations of the permutation representation [special characters omitted] We look for irreducible components which are common to the permutation representation of all parabolic subgroups of G, and find that for exceptional Weyl groups, there is an “extra” common irreducible component, which does not appear for classical Weyl groups