The McKay correspondence gives a bijection between finite subgroups of SU(2) and affine A,D,E Dynkin diagrams. There is a quantum version of this statement (due to Kirillov Jr and Ostrik) which relates "finite subgroups" of quantum sl(2) and finite A,D,E Dynkin diagrams. We use this correspondence to construct the category of "equivariant" coherent sheaves on the quantum projective line. This is done by defining analogues of the symmetric algebra and the structure sheaf, and using them to define a triangulated category which is a natural analogue of the derived category of equivariant sheaves on the projective line. We then produce natural objects in this triangulated category, and relate our category to the derived category of representations of the corresponding A,D,E quiver. This can be thought of as a quantum analogue of the projective McKay correspondence of Kirillov Jr. We will first review the classical constructions, then discuss the "quantum" analogues.
The Macaulay-Matlis duality is the basis for many duality results in algebra and geometry. It takes a very specific form for ideals in a polynomial ring by interpreting polynomials as differential operators. For example, it explains certain symmetry properties of irreducible ideals aka Gorenstein ideals.
A cellular resolution is a way of representing the free resolution of a monomial ideal by associating it with a cell complex. In this talk we will discuss a cellular resolution of the Stanley-Reisner ring of the n-gon. This is an interesting example because this ring is Gorenstein (symmetric Betti table) and the cellular resolution is a self-dual polytope (symmetric f-vector). The talk will focus on the primary ideas used in the construction of the polytope.
MacWilliams identities play a prominent role in algebraic coding theory because they tell how certain information about a code, encoded in enumerators, can be used to deduce information about the dual code. We will provide a unified approach to MacWilliams identities for various weight enumerators of linear block codes over Frobenius rings. Such enumerators count the number of codewords having a pre-specified property, and MacWilliams identities yield a transformation between such an enumerator and the corresponding enumerator of the dual code. All identities can be derived from a MacWilliams identity for the full weight enumerator using the concept of an F-partition, as introduced by Zinoviev and Ericson (1996).
The talk will be self-contained, and only basic knowledge of linear algebra and introductory ring theory will be required.
One goal of algebraic coding theory is to find linear error-correcting codes with maximum error-correcting capacity. Correlated to the error-correcting capacity of a code C are the generalized Hamming weights of C. Thus, using Hochster's formula and techniques of commutative algebra, we will derive the generalized Hamming weights for one class of BCH codes and partially extend these results to special cases of another class of BCH codes.
Given an ideal I of a local ring (R,m) we are interested in determining when the socle I:m, and in more generality the quasi socle ideals I:m^s where s is a positive integer, is integral over the ideal I. In the first part of the talk we will review what integrality of ideals means and what is known about the problem in the special cases when I is a complete intersection, height two perfect, Gorenstein ideal. The focus will be in looking for an explicit way to determine the generators of these quasi-socle ideals in terms of a presentation matrix of the ideal and in a characteristic free fashion. Time permitting, I will describe some of the new result obtained in an ongoing work joint with Shiro Goto, Craig Huneke, Claudia Polini and Bernd Ulrich.
