Algebra Seminar - Hasan Mahmood & Miguel Pazcka - f-ideals and Quasi f-ideals & Commutative algebra and coding theory
Speaker: Hasan Mahmood (Tulane)
Title: f-ideals and Quasi f-ideals
Abstract: The notion of f-vector has a fundamental importance in algebraic, topological and combinatorial study of simplicial complexes and polytopes. It has been studied since the time of Leonhard Euler. f-Ideals are those square-free monomial ideals in the polynomial ring S=K[x_1,x_2,..., x_n] for which the facet complex of I and the non-face complex of I have the same f-vector. The notion of an f-ideal was introduced in 2012 and has been studied in various papers since then. One importance of these ideals lies in the fact that it helps to compute the Hilbert series of S modulo the ideal through the f-vector of the facet complex, which is more direct and easy complex as compared to the Stanley-Reisner complex. The motivation of defining quasi f-ideals is to extend the class of those ideals for which Hilbert function and Hilbert series can be computed through the f-vector of their facet complex. The idea is to read off the f-vector of the non-face complex with the help of f-vector of its facet complex. In my talk, I intend to describe f-ideals and quasi f-ideals, and to show what has been done in this particular direction, and propose some new problems to see what can be done more.
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Speaker: Miguel Pazcka (Instituto Politécnico Nacional)
Title: Commutative algebra and coding theory
Abstract: We introduce and study the family of affine Reed-Muller type codes using Commutative algebra, Algebraic Geometry and Grobner basis techniques. Given an affine Reed-Muller type code, we define the basic parameters ( length, dimension and minimum distance). The goal is to compute the basic parameters giving explicit formulas. In this talk we focus on a special type of affine codes, parameterized affine codes by odd cycles. Let K be a finite field. Let X be a subset of the affine space K^n, which is parameterized by odd cycles. In this talk we give an explicit Grobner basis for the vanishing ideal, I(X), of X. We give an explicit formula for the regularity of I(X).