Speaker: A.V. Geramita (Queen's and Genoa)
Title: SUMS OF SQUARES: EVOLUTION OF AN IDEA
Date and Location: Jan. 21, 2011 at 2:30 in RB 2047
Abstract: Beginning with Fermat’s characterization of primes which are the sum of two squares and Lagrange’s 4-squares Theorem (with Gauss’ addendum) I will discuss how these two theorems naturally lead to the famous Waring Problems for integers. With that as a backdrop, we will see that there is ananalogous formulation of Waring’s Problems for homogeneous polynomials.
These Polynomial Waring Problems have a beautiful geometric reformulation and once that reformulationis apparent, other similar problems become very apparent and are the target of much current research. Problems, for example, which generalize the notion of rank for matrices. Moreover, these problems have important applications in Phylogenetics and Communication Theory as well as in Statistics.
Speaker: Ben Babcock (Undergraduate Student, Lakehead)
Title: Revisiting the Spreading and Covering Numbers
Date and Location: September 28, 2010 at 11:30 in RB 1045
Abstract: Let R be a polynomial ring in n variables and S denote the monomials of degree d in R. Take any subset of S and multiply it by the indeterminates of R. If the resulting monomials are distinct, we say they are ”spread out” among the monomials of degree d+1. The spreading number, alpha_n(d), is the cardinality of the largest subset of S that satisfies this property. The covering number, rho_n(d + 1), is the cardinality of the smallest subset of S that generates all the monomials of degree d + 1. This talk is an overview of research done during a summer NSERC USRA. We examine two methods of computing n(d): constructing a simplicial complex and computing its dimension, and using the symmetry of a graph. Additionally, we present an algorithm for computing upper bounds of rho_n(d + 1) that improves upon the explicit formula for general n, d.