Speaker: Michael Filaseta, USC.
Title: A collection of problems to ponder on polynomials.
Abstract: In this talk, we discuss results and problems associated
with polynomials. The topics in our collection of problems will vary, but the
main focus will be on polynomials all of whose coefficients are either 0 or
1 and the problems will typically be ones associated with their factorization.
Although we will have some answers to reveal to some of the questions, there
will be plenty of open problems presented.
Speaker: Matthew Boylan, USC.
Title: Gaussian hypergeometric series.
Abstract: Introduced by Greene in the early 1980s, Gaussian
hypergeometric series are finite field analogs of classical hypergeometric
series. In this talk we will discuss some of their connections with number
theory.
Speaker: Michael Mossinghoff, USC and Davidson College.
Title: Barker sequences: Come on down!
Abstract: A Barker sequence is a finite sequence of integers
a_0,...a_{n-1}, each +1 or -1, for which the absolute value of the sum over j
of a_j times a_{j+k} <= 1 for k not equal to 0. It has long been conjectured
that long Barker sequences do not exist. We describe some recent work
connecting this problem to several open questions posed by Littlewood,
Mahler, Erdos, Newman, Golay, and others about the existence of polynomials
with +1 or -1 coefficients that are ``flat'' in some sense over the unit
circle. If time permits, we will also describe some related work concerning
mean values of L_p norms and Mahler's measure for certain families of
polynomials.
Speaker: Ethan Smith, Clemson University.
Title: A Barban-Davenport-Halberstam asymptotic for number fields.
Abstract: Let a and q be coprime. Dirichlet's Theorem on
primes in arithmetic progressions is a well-known result giving information
about the distribution of primes congruent to a modulo q. That is,
the theorem tells us approximately how many such primes are in the
interval [1,x].
In the mid 1960s, Barban, and independently, Davenport and Halberstam, began
a study of the mean square error for the approximation given in Dirichlet's
Theorem. The so-called Barban-Davenport-Halberstam Theorem essentially says
that the square of the error in Dirichlet's approximation is small on
average. Later, Montgomery, Hooley, and others sharpened their theorem by
giving an asymptotic formula for the mean square error. In this talk, we
will discuss a natural generalization of these ideas to the setting
of number fields.
Speaker: Michael Filaseta, USC.
Title: The density of square-free 0, 1-polynomials.
Abstract: This talk will be based on work with Sergei Konyagin from
several years ago concerning the density of polynomials that are not divisible
by the square of a non-constant polynomial among all polynomials having
coefficients either 0 or 1. We will examine how this density result is
connected to squarefree numbers missing digits in a given base. The talk
makes a good introduction to some analytic number theoretic ideas.
Speaker: Matt Boylan, USC.
Title: S_4-modular forms.
Abstract: Let f be an integer weight modular form with integer
coefficients and let l be prime. By work of Serre and Deligne, one can
associate to f a Galois representation whose image lies in GL_2(Z/lZ), the
group of 2x2 invertible matrices with entries in Z/lZ. For given f, let
G(f,l) be its image. In this talk we will discuss properties of modular
forms f for which G(f,l) modulo scalar matrices is isomorphic to S_4, the
permutation group on 4 elements. It turns out that the Fourier coefficients
of such forms satisfy striking congruence properties modulo l. Moreover,
the study of such forms is related to the famous Artin Conjecture on
the analytic continuation of L-functions constructed from Galois
representations.
Speaker: Tim Flowers, Clemson University.
Title: Asymptotics of Bernoulli, Euler, and Strodt Polynomials.
Abstract:
It is well known that both Bernoulli polynomials and Euler polynomials on
a fixed interval are asymptotically sinusoidal. A recent paper by
Borwein, Calkin, and Manna uses an idea of Strodt to generalize Bernoulli
and Euler polynomials and view them as members of a family of polynomials.
We used these ideas to study the asymptotics of non-uniform Strodt
polynomials. We will describe the experimental process which led to
several conjectures. In addition, we will show how experiments suggested
the methods used to prove some of these results.
Speaker: Dan Baczkowski, USC.
Title: Applications of the Hardy-Littlewood Circle Method.
Abstract: Goldbach's Conjecture states that every even integer > 2 can
be written as the sum of two primes. The ternary Goldbach conjecture
states that every odd integer > 5 is the sum of three primes.
Vinogradov made monumental progress on the latter problem by showing
that it was valid for sufficiently large odd integers. The proof
utilizes ubiquitous techniques from combinatorics, complex analysis,
Fourier analysis, number theory, and much more. We will
introduce the powerful tool referred to as the Hardy-Littlewood
circle method and mention some of its applications. In particular,
we will discuss its relevance when sketching the proof of
Vinogradov's Theorem. Be ready for a short historical overview, an
intoduction to analytic number theory, the recent progress of the
Goldbach conjectures, and enormous fun!
Speaker: Jim Brown, Clemson University. Note that this date is different than usual, but the time and location are as usual.
Title: The Eisenstein ideal and generalizations
Abstract: In a short paper in 1976 Ken Ribet gave a revolutionary proof
of the converse of Herbrand's theorem, a result relating certain
Bernoulli numbers to sizes of pieces of ideal class groups. Ribet was
able to prove this result by studying congruences between modular forms
and the information these congruences give in terms of Galois
representations. We will briefly outline Ribet's argument. From there
we will give a different (more general) formulation of Ribet's result in
terms of the Eisenstein ideal. Time permitting we will then define a
generalization of the Eisenstein ideal that can be used in more general
settings than the one pursued by Ribet. All necessary definitions will
be recalled.
Speaker: John Webb, USC.
Title: Some examples of theorems of Waldspurger and Kohnen-Zagier
with an application to partitions.
Abstract: In this talk we give an introduction to theorems of
Waldspurger and Kohnen-Zagier. Roughly speaking, these theorems assert
that Fourier coefficients of half-integral weight modular forms are square
roots of central critical values of modular L-functions, up to explicitly
identifiable factors. We will discuss Tunnell's solution (which relies
crucially on Waldspurger's work) to the ancient
congruent number problem. The congruent number problem asks for necessary
and sufficient conditions for a positive integer to be the area of a right
triangle with rational side lengths. Time permitting, we will also give
a famous example of Kohnen and Zagier and discuss current work of the
speaker which connects values of the ordinary partition function to
values of modular L-functions via Waldspurger's work.
Speaker: Pradipto Banerjee, USC.
Title: On a polynomial conjecture of Pal Turan.
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