Spring 2017 – Algebra and Discrete Math (ADM) Seminar

Combinatorics of MacWilliams identities In coding theory, a MacWilliams identity expresses a linear relation between the weight distribution of a code and the weight distribution of the dual code. The transformation is characterized by its "Krawtchouk coefficients".When studying additive codes in finite abelian groups, code and dual code are subsets of different ambient spaces, and their weight enumerators refer in general to different weight functions. Invertible MacWilliams identities hold when the weights are mutually compatible. A major problem in this area is the construction of mutually compatible weights, and the computation of the associated Krawtchouk coefficients.Using a combinatorial method, we construct a family of mutually compatible weight functions on finite abelian groups that automatically yield invertible MacWilliams identities for additive codes. The weights are obtained composing a suitable support map with the rank function of a graded (poset-) lattice having certain regularity properties. We express the corresponding Krawtchouk coefficients in terms of the combinatorial invariants of the underlying lattice, giving a closed formula for them.The most important weight functions studied in coding theory (including the Hamming weight, the rank weight and the Lee weight) belong, up to equivalence, to the class that we introduce. This allows in particular to compute classical and new Krawtchouk coefficients employing a unified simple combinatorial method.We conclude discussing some applications of the MacWilliams identities for the rank metric to enumerative problems of matrices.

Symbolic powers Given an ideal I in a polynomial ring, its n-th symbolic power consists of the functions that vanish to order n at each point in the variety defined by I. There is a strong relationship between the ordinary and symbolic powers of an ideal. In this talk, we will introduce symbolic powers, discuss the Containment problem -- which consists of determining which symbolic powers are contained in each fixed power of a given ideal -- and how using characteristic p methods we can prove that a conjecture of Harbourne holds under certain assumptions on the singularities of R/I.No prior knowledge of either symbolic powers nor characteristic p methods will be assumed. The new results in this talk are joint work with Craig Huneke.

A Tour Through Frobenius Distributions of Elliptic Curves Each rational elliptic curve E comes equipped with two important integer sequences indexed by primes: a sequence {N_p} denoting the number of points over the reduced curve E modulo the prime p and a sequence {a_p} denoting the trace of the Frobenius endomorphism of E at the prime p. Of interest is to investigate the statistical distribution of these sequences, either for a single curve or for a family of curves. For instance, one could ask "how often is a_p equal to a fixed integer r?" or "is the distribution of N_p for curves with torsion different than the distribution of N_p for curves without torsion?".In this talk we will begin by discussing a number of the rich and varied problems in this area and conclude with more intricate descriptions of the contributions made by myself and Kevin James. In particular we will give answers on average to questions akin to "how often is N_p as large as possible?" and "how often is a_p prime?".

The Magic of Error Correcting Codes Error correcting codes are used in many technologies that we use every day including CDs, DVDs, and smartphone QR images. This talk will feature a series of magic tricks based on simple error correcting codes. The secrets behind each trick will be revealed so that you can amaze friends and family. By learning how the tricks work, you will also learn the mathematics which explain how error correcting codes work. The talk will conclude with an explanation of the error correcting codes used in QR images.

On some topics of \tau(n)-Number Theory The notion of a \tau-factorization or \tau-products in the general theory of (non-atomic) factorization on integral domains was defined by Anderson and Frazier, in 2006. The idea is basically to study factorizations by just considering a restriction into the element of interest. As the factorizations into primes, irreducible elements, primary elements, primals, rigids, etc. This was easy to considering the relation \tau = SXS, then they called x=\lambda*x_1*...* x_n a \tau-factorization of x if \lambda is a unit and each x_i is in a set S, of desired nonzero nonunit elements. Later, they extended the idea to any symmetric relation \tau on the set of nonzero nonunit elements of an integral domain, which opens the doors to consider many types of factorizations with probably unique representations. Not many people had work with this theory. Maybe is due to too many technical definitions, but it can open new ways of studying graph theory. This talk will present some basic structural factorization properties and how this properties can be inherited into this new theory of factorizations. Then will provide some examples to familiarized with the concept. But the main focus of this talk is to provide a flavor of all the work done when considering the equivalence relation modulo n on the integers, denoted by \tau(n). Among the results, an introduction to the notions of \tau(n)-divisibility, the number of \tau(n)-factors, \tau(n)-GCD and \tau(n)-primes, when \phi(n)<5 and some more general cases.