2015-2016

Fall Semester Schedule (see below for abstracts)

Spring Semester Schedule

Abstracts:

  • Speaker: Jonathan Bayless
  • Title: The prime factors of an integer
  • Abstract: Since Euclid's time, we have known that the factorization of any integer n into its prime factors is unique, up to ordering, so in many ways, the prime factors dividing n tell us everything we need to know about n. For this reason, it is natural to ask many questions about these prime divisors, including their number or their size relative to n. In this talk, we will discuss the history related to these two questions and some interesting sums/integrals whose estimates are tied directly to the answers to these and other questions.
  • Speaker: Paul Kinlaw
  • Title: Explicit Bounds on Several Sums and Functions Occurring in Elementary Analytic Number Theory
  • Abstract: It is well known that the sum of reciprocals of prime numbers is a divergent infinite series. In 2013 Bayless, Klyve and Oliveira e Silva determined explicit numerical bounds on the counting function of prime-index primes (that is, primes whose order in the list of primes is itself a prime number). They also proved that the sum of reciprocals of prime-index primes is convergent, and that the value of the sum lies in the interval [1.04299,1.04365]. In this talk we will consider a generalization to almost primes. A number is called k-almost prime if it has exactly k prime factors (counting repeated factors). Similarly, we may define a number to be k-nearly prime if it has at most k prime factors. We will prove that for every pair of positive integers j and k, the sum of reciprocals of j-nearly primes of k-nearly prime index is convergent. We will look at explicit bounds on the sum of reciprocals of such a sets, a problem which requires determining explicit bounds on the counting function of almost primes up to a given magnitude. Time permitting, we will look at some other problems of a similar nature.
  • Speaker: Elliot Benjamin
  • Title: The 2-Class Field Tower Conjecture: Part 1 (a, b, and c)
  • Abstract: In this first part of my talk, I will review the basics of algebraic number theory and class field theory that are fundamental for understanding the conjecture that the 2-class field tower of an imaginary quadratic number fields with 2-class group of rank 4 is infinite. These algebraic number theory basics include the recovery of unique prime factorization in algebraic number fields through the use of ideals in the ring of algebraic integers, the class group of an algebraic number field, ramification indices, the Hilbert class field, the essential isomorphisms between class groups and Hilbert class fields, including the generalization to p-class groups and Hilbert p-class fields for primes p, and the formulation of the class field tower and p-class field tower. We will also see an algebraic number theory proof of the sum of two squares theorem.
  • Speaker: Elliot Benjamin
  • Title: The 2-Class Field Tower Conjecture: Part 2 (a and b)
  • Abstract: In this second part of my talk, I will discuss the conjecture that the 2-class field tower of imaginary quadratic number fields with 2-class group of rank 4 is infinite. The existence of the infinitude of the class field tower of an algebraic number field, as well as the infinitude of the 2-class field tower of a quadratic number field, was first established in the 1960's by Golod & Shafarevich. By an improvement in Golod & Shafarevich's technique, it was soon established that imaginary quadratic (resp. real quadratic) number fields with 2-class group greater than or equal to 5 (resp. greater than or equal to 6) have infinite 2-class field tower. It was well known that algebraic number fields with 2-class group of rank 1 have finite 2-class field tower that ends at the first layer, and that imaginary quadratic number fields with 2-class group of rank 2 or 3 could have finite or infinite 2-class field towers. The conjecture of the infinitude of the 2-class field tower for imaginary quadratic number fields with 2-class group of rank 4 was stimulated by Martinet asking the question in the 1970s about the infinitude of the 2-class field tower for fields of this type. Through the work of Hajir in the 1990s, my own work in the early 2000s, and the work of Mouhib and Sueyoshi later in the first decade of the 2000s, various cases have been established for when this 2-class field tower is infinite. My recent work resulted in a 2015 paper published in the Journal of Number Theory about the conjecture, and a subsequent paper that makes further progress on my work will be soon coming out in this same journal. Thus this 2-class field tower conjecture problem is an enduring and enticing problem.
  • Speaker: Ken Lane
    • Title: An undergraduate major in data analytics: context and opportunity
      • Abstract: The Husson University Board of Trustees has approved a proposal for the development and delivery of an undergraduate degree in data analytics to be hosted in its School of Sciences and Humanities. Data analytics is identified as a subset of the emerging field of data science. In this talk we consider the emergence of data science as a field of study and explore potential opportunities created by the Husson data analytics initiative.
    • Speaker: David Grinstein
    • Title: The Number-theoretic Basis for the "ac-method" of Factoring Quadratic Trinomials
      • Abstract: One of the small mysteries of elementary algebra is that the method of factoring quadratic trinomials with integer coefficients "by grouping" (also sometimes know as the "ac method" or other names) works so well.
      • This note describes that technique and provides a proof that the technique always succeeds in factoring a trinomial whenever a factorization with integer coefficients exists. It relies on the number-theoretical result that the integers have unique factorization into primes.
      • Speaker: Aitbala Sargent
        • Title: On stability of two level finite difference schemes of hydrodynamics in Lagrangian coordinates.
        • Abstract: Many fluid dynamics problems modeled by Euler equations involve large changes of volume or size of computational domain or moving boundaries and thus have to be treated in moving Lagrangian coordinates. For convergence of the numerical solution, it is important for a scheme to be consistent and stable. As a part of my dissertation, I have explored stability of two level finite difference schemes of hydrodynamics in Lagrangian coordinates. The necessary and sufficient conditions for stability of the linearized schemes have been obtained using Fourier and the energy methods respectively. In this talk, I will first demonstrate the techniques on the example of the 1-D advection equation and then present the stability conditions for two level schemes of hydrodynamics.
      • Speaker: Paul Kinlaw
      • Title: On the number of numbers up to x with k prime factors
        • Abstract: In 1900 Landau proved an asymptotic formula for the count tau_k(x) of numbers up to a given bound x which have exactly k prime factors (including multiplicity). We'll define asymptotic equivalence so that we may state Landau's result formally. We will discuss recent progress in determining explicit upper and lower bounds on tau_k(x). This research is joint work with Jonathan Bayless.
    • Speaker: Jake Emerson
    • Title: Data Analytics Discussion
    • Abstract: Jake Emerson, a data scientist with the Bangor Daily News, will talk with us about the field of data science, employment opportunities, educational goals, etc.
      • Speaker: Jonathan Bayless
      • Title: On sums and products
      • Abstract: So you think you know everything about addition and multiplication tables? Since third grade? Think again! We will talk about what we know and what we don't know about these simple tables.
    • Speaker: David Grinstein
    • Michon talks about the apparent paradoxes involved if one insists on assigning a single value to sqrt(negative number). Further exploration examines sqrt of complex numbers. Try "plot z = sqrt(x + iy)" in wolframalpha. One way of addressing this a two-sheet Riemann surface for sqrt, where one can identify a point on the surface as (r, theta), where theta is modulo 4pi, and sqrt(r, theta) is (sqrt(r), theta/2).
    • Speaker: David Grinstein and Scott Lambert
    • Title: Wonderings on a Riemann surface
    • Abstract: This talk will be a further exploration of branch cuts stemming from our last math seminar talk.
      • Speaker: Ken Lane
      • Title: Reproducible Research with R
      • Abstract: We consider the evolving relationship between science and data and provide an example of the consequence. We present a brief history of the R programming environment and provide elementary examples of its use. A demonstration of the use of RStudio for reproducible research will be given.
      • Speaker: Paul Kinlaw
      • Title: Dickman's Rho Function and Smooth Numbers.
        • Abstract: We will introduce Dickman's rho function, defined as the unique continuous function rho on [0,infinity) such that rho(u)=1 for u in the interval [0,1] and such that rho satisfies the differential-difference equation u*rho'(u)+rho(u-1)=0 for all u>1. We'll prove several basic properties satisfied by rho as well as an asymptotic formula for rho(u) and explicit numerical bounds.
        • We'll look at applications to number theory, including bounds on de Bruijn's function Psi(x,y) counting the numbers up to x that are free of prime factors exceeding y. (Such numbers are called y-smooth). Time permitting, we will look at a proof of the theorem that Psi(x,y) is asymptotic to x*rho(u) as x approaches infinity, where u=(log x)/(log y). We will then discuss attempts to obtain an explicit version of this result and its application to putting explicit bounds on the sum of reciprocals of Carmichael numbers.