Title: Personal Perspectives on m-ary Partitions
Speaker: James Sellers (from Penn State)
Abstract: A great deal of my research journey has involved the study of m-ary partitions. These are integer partitions wherein each part must be a power of a fixed integer m > 1. Beginning in the late 1960s, numerous mathematicians (including Churchhouse, Andrews, Gupta, and Rodseth) studied divisibility properties of m-ary partitions. In this talk, I will discuss work I completed with Rodseth which generalizes the results of Andrews and Gupta from the 1970s. Time permitting, I will then discuss several problems related to m-ary partitions, including my work with Neil Sloane on non-squashing stacks of boxes, an application of m-ary partitions to objects known as "unique path partitions" (which are motivated from representation theory of the symmetric group), as well as very recent work with George Andrews and Aviezri Fraenkel on the characterization of the number of m-ary partitions of n modulo m. Throughout the talk, I will attempt to highlight various aspects of the research related to symbolic computation. The talk will be self-contained and geared for a general mathematical audience.
Title: The distance to a squarefree polynomial
Speaker: Michael Filaseta
Abstract: Turan conjectured that every polynomial f(x) in Z[x] is close to an irreducible polynomial in the following sense: there is a constant C, independent of f(x), such that there is an irreducible polynomial g(x) over Q of degree less than or equal to deg(f) such that the sum of the absolute values of f(x) - g(x) is less than C. This problem remains open, and we will discuss its history. Recent work of Dubickas and Sha considers the analogous problem with irreducibility replaced by squarefree. After a bit of background, we turn to joint work with Richard Moy on this topic.
Title: The distance to a squarefree polynomial, Part II
Speaker: Michael Filaseta
Abstract: We continue the discussion from the previous week on joint work with Richard Moy.
NO SEMINAR THIS WEEK
SPRING BREAK - NO SEMINAR
Title: An assortment of problems on polynomials
Speaker: Michael Filaseta
Abstract: This is a recurring talk of mine (last updated and given on 09/03/15) which began as a top ten list of polynomials and a way to introduce graduate students and others to some interesting questions on polynomials. However, the top ten list grew, and I ended up wanting to discuss more than one interesting polynomial per topic discussed. So now it's just cool polynomials all the way with the interesting topics that surround them.
Title: Applications of Pade approximations to Number Theory
Speaker: Michael Filaseta
Abstract: Pade approximations provide a powerful tool for applications in Number Theory, in particular with regard to diophantine approximation and transcendence results but also in a number of other contexts. This is the first talk of a two week survey on Pade approximations and there many applications in Number Theory. We will begin with a combinatorial problem (a lattice path counting problem) known as the Ballot problem and go on to irrationality measures, Diophantine equations, and Waring's problem for this lecture.
There is no seminar this week.
You are encouraged to attend SERMON 2019 this weekend for a number of interesting talks. There is no seminar this week.
Title: Applications of Pade approximations to Number Theory II
Speaker: Michael Filaseta
Abstract: We continue our discussion of various applications of Pade approximations to Number Theory. This week we focus on applications to the factorization of n(n+1), prime divisors of binomial coefficients, Galois groups of classical polynomials, the Ramanujan-Nagell equation, the abc-conjecture, k-free numbers in short intervals and k-free values of polynomials and binary forms.
Title: Modular forms modulo 2
Speaker: Matt Boylan
Abstract: Work of Serre and Tate in the 1970s shows that the F_2-algebra of Hecke operators acts locally nilpotently on the space of modular forms with integer coefficients reduced modulo 2. Serre, Nicolas, Bellaiche, and others recently studied this algebra in great detail. We will survey their work and give applications, some of which are related to past work of the speaker.