Dr. Michael DiPasquale, Colorado State University.
Title: Extending Wilf’s Conjecture
Abstract: Suppose $a_1,a_2,\ldots,a_n$ is a set of positive integers which are relatively prime. The set of all integers which can be written as a non-negative integer combination of $a_1,\ldots,a_n$ is called a numerical semigroup. The non-negative integers which are not in the numerical semigroup is called the set of holes of the semigroup. It is known that the set of holes of a numerical semigroup is finite, and the largest hole is called the conductor. In a four-page note in the American Mathematical Monthly in 1978, Herbert Wilf asked a question about the density of the set of holes of a numerical semigroup in the interval from zero through the conductor. This question is still widely open and has become known as Wilf’s conjecture. We will discuss this conjecture and a recent extension of it to higher dimensions which is joint work with C. Cisto, G. Failla, Z. Flores, C. Peterson, and R. Utano.