Title: Galois Theory 101 and Galois Groups of Laguerre Polynomials

Speaker: Michael Filaseta

Abstract: Following an introduction to basic Galois Theory, a survey will be given of a variety of work done with students and others on the Galois group associated with Laguerre polynomials over the rationals. A particular focus will be on determining Laguerre polynomials with Galois group the alternating group.

Title: Four Seemingly Unrelated Problems

Speaker: Michael Filaseta

Abstract: Four different problems that are number theoretic or combinatorial in nature will be discussed. Two of these problems remain open and the other two have known solutions. We explain how these seemingly unrelated problems are connected to each other. To disclose a little more information, one of the problems with a known solution (by Schinzel) is the following: Is it possible to find an irrational number q such that the infinite geometric sequence 1, q, q^2, ... has 4 terms in arithmetic progression?

Title: Covering Thin Subsets of the Integers

Speaker: Wilson Harvey

Abstract: We consider two famous questions about covering systems of the integers and the analogous questions for covering subsets of the integers. In particular, we show that the Fibonacci numbers can be covered by congruences with the moduli arbitrarily large and with arbitrarily large prime divisors. One consequence of this is the existence of an odd covering of the Fibonacci numbers. If there is time, we will also show that there are arbitrarily thin sets that, if covered, yield a covering of the entire set of integers.

Title: Asymptotics on the Partition Function

Speaker: Harsh Mehta

Abstract: In 1918 Ramanujan and Hardy developed the circle method in order to obtain astonishingly good asymptotic results on the partition function. We discuss the ideas they used and some developments in the circle method since then. There are many applications of the circle method in Waring's problem which hopefully we have time to address.

Title: Asymptotics on the Partition Function, Part II, and Values of Cubic Polynomials Modulo a Prime

Speakesr: Harsh Mehta and Ognian Trifonov

Abstract: The first part of the seminar will be a continuation of last week's seminar. For the second part, let P(x) be a cubic polynomial with integer coefficients, and let p be a prime. We obtain a formula for the number of distinct values modulo p that the polynomial P(x) takes as x runs through the integers.

Title: Some Facts from Algebraic Number Theory, the Cohen-Lenstra Heuristics and Soundararajan's Thesis

Speaker: Kevin Sheng

Abstract: We give an exposition of Kannan Soundararajan's Ph.D. thesis, which is related to the Cohen-Lenstra heuristics, along with some elementary facts from algebraic number theory. His main theorem gives lower bounds on the number of torsion elements of the ideal class group CL(K) for imaginary quadratic fields K = Q(sqrt{-d}). The proof relies on counting the number of square free d satisfying certain Diophantine conditions.

Title: r-Gaps Between Zeros of the Riemann Zeta-Function

Speaker: Caroline Turnage-Butterbaugh (Duke)

Abstract: Denote by 0 < gamma_1 < gamma_2 < ... the imaginary parts of the zeros of the Riemann zeta-function on the critical line. Selberg showed that for all positive integers r there exists an absolute constant c such that

lambda_r := limsup_{n goes to infinity} (gamma_{n+r}/((log gamma_n)/(2 pi r))) > 1 + c sqrt{r}

and

mu_r : = liminf_{n goes to infinity} (gamma_{n+r}/((log gamma_n)/(2 pi r))) < 1 - c sqrt{r}.

Since Selberg's work, there has been much attention given to finding strong qualitative bounds for r = 1. In this talk, I will present some of the methods employed in these investigations and discuss ongoing work with Brian Conrey concerning qualitative bounds on lambda_r and mu_r for r > 0.

Title: Mean Values of Bounded Multiplicative Functions

Speaker: Harsh Mehta

Abstract: This will be a survey on methods used to estimate the mean value of multiplicative functions, primarily focusing on techniques used by Kaisa Matomaki and Maksym Radziwill in their recent paper that talks about the relation between averages in short and long intervals.

Title: Some Novel Applications of Algebra

Speaker: Nigel Boston

Abstract: I shall introduce entropy regions, motivated by network coding, and then note how they are equivalent to basic objects in finite group theory, allowing us to address fundamental questions regarding these regions. Then I shall introduce the Belgian Chocolate Problem, a question from robust control theory amenable to algebraic attack and mysteriously connected to the abc conjecture.