Title:
An Assortment of Problems on PolynomialsSpeaker: Michael Filaseta
Abstract: A number of results and problems associated with polynomials will be discussed. This will be based largely on a prior talk that I gave a few years ago in a seminar on a list of interesting polynomials connected to Number Theory. Some updated material will be included.
Title:
PANTS Rehearsal TalksSpeaker: Wilson Harvey and Robert Wilcox
Abstract: The speakers will present trial runs of their talks that are planned for the Palmetto Number Theory Series XXIV at Emory on September 12-13. Wilson will talk on covering systems for subsets of the integers, and Robert will talk on Universal Hilbert Sets.
Title:
Newton PolygonsSpeaker: Maria Markovich
Abstract: The talk will be an introduction to Newton polygons for obtaining information about the factorization of polynomials with integer coefficients. A selection of nice examples will be given.
Title:
An Application of Newton PolygonsSpeaker: Michael Filaseta
Abstract: This is joint work with Brady Rocks. We establish that more than 2/3 of the polynomials in a certain sequence of polynomials arising from the Strong Factorial Conjecture are irreducible over the rationals.
Title:
The Strong Factorial Conjecture, Part ISpeaker: Brady Rocks
Abstract: In this talk I will introduce the Strong Factorial Conjecture, the subject of my dissertation at Washington University in St. Louis. In particular, I will motivate its study by connecting it to several open problems such as the Jacobian Conjecture and the Rigidity Conjecture of Jean Philippe-Furter. I will also discuss how the problem can be studied via systems of multivariate integer polynomials. Finally, I will discuss some of my results (including partial results) which show that the conjecture holds in several special cases.
Title:
The Strong Factorial Conjecture, Part IISpeaker: Brady Rocks
Abstract: This talk will continue the discussion from Part I given at the last seminar.
Title:
Connections between the Distribution of the Farey Fractions and RHSpeaker: Harsh Mehta
Abstract: This talk will discuss some current results about connections between the farey fractions and RH. I intend to prove that the farey fractions are uniformly distributed and show that the distribution of their power moments is connected to RH.
Title:
Connections between the Distribution of the Farey Fractions and RH, Part IISpeaker: Harsh Mehta
Abstract: This talk will continue the material from last week on the uniform distribution of farey fractions and the connection to RH.
Title:
Mystery TalkSpeaker: Matt Boylan
Abstract: The title says it all, at least all you are going to find here.
Title:
Oscillations in sums involving the Liouville functionSpeaker: Michael Mossinghoff (Davidson College)
Abstract: The Liouville function λ(n) is the completely multiplicative arithmetic function defined by λ(p) = −1 for each prime p. Pólya investigated its summatory function L(x) = Σn≤x λ(n), and showed for instance that the Riemann hypothesis would follow if L(x) never changed sign for large x. While it has been known since the work of Haselgrove in 1958 that L(x) changes sign infinitely often, oscillations in L(x) and related functions remain of interest in analytic number theory. We review some connections between oscillations in this function and its relatives with the Riemann hypothesis and other problems in number theory, and describe some recent work on this topic. In particular, we describe a method involving substantial computation that establishes new bounds on the magnitude of the oscillations of L(x). This is joint work with T. Trudgian.