Speaker: Matt Boylan, USC.
Title: Powers of Euler's infinite product.
Speaker: Michael Filaseta, USC.
Title: A connection between the factorization of sparse polynomials
and an elementary number theoretic problem.
Speaker:
Kevin James, Clemson University.
Title: Average Frobenius distributions of elliptic curves.
Speaker:
Elizabeth Jurisich, College of Charleston.
2:30 pm, LC 405.
Title: Modular functions and moonshine phenomena.
Speaker:
Frank Thorne,
University of Wisconsin at Madison.
Title: Maier matrices beyond Z.
Speaker:
Scott Ahlgren, University of Illinois at Urbana-Champaign.
Title: Rank generating functions as weakly holomorphic modular forms.
NO SEMINAR
Integers
Conference 2007, Oct. 24-27, Carrollton, Ga.
Speaker: Ognian Trifonov, USC.
Title: Covering Systems of the Integers, II.
I. If n_1 is fixed, what is the smallest possible value of n_l?
II. What is known in the case when all moduli n_1, n_2,...,n_l are
squarefree integers?
Speaker: Samuel S. Gross, USC.
11:00 am, LC 312.
Title: Elementary proofs of the prime number theorem.
Speaker:
Amanda Folsom,
University of Wisconsin at Madison.
Title: Half-integral weight Maass form correspondences and vector
valued forms.
Department colloquium,
LC 412, 3:30 pm.
Speaker:
Jeff Lagarias, University of Michigan.
Title: Hilbert's 18th Problem.
Abstract: Let f(q) be the infinite product whose nth factor is
1-q^n. Fix an integer r, and let p_r(n) be the nth power series coefficient
of f(q)^r. The functions f(q)^r play an important role in number
theory. Note, for example, that f(q)^(-1) is the generating function for
the ordinary partition function. In this expository talk, I will describe
a classical result of Serre which says that if r is even and positive,
then in a precise sense, almost all coefficients p_r(n) = 0 if and only if
r is 2, 4, 6, 8, 10, 14, 26. I will also discuss related open questions,
time permitting.
Palmetto Number Theory
Series (PANTS) 3,
College of Charleston,
Charleston, South Carolina.
Abstract: We discuss some motivation for looking at the irreducibility
of polynomials from the sequence f(x) x^n + g(x) where f(x) and g(x)
are fixed polynomials with integer coefficients and n varies over the
positive integers. We will then turn to the main focus of the talk which
will be to connect the irreducibility of f(x) x^n + g(x) to an elementary
number theory question involving integers modulo n (and, in particular, not
involving polynomials). The latter problem can be resolved using a
pigeon-hole type argument leading to our main result on the factorization of
f(x) x^n + g(x). The talk is based on joint work a few years ago with Kevin
Ford and Sergei Konyagin but motivated largely
by prior work of Andrzej Schinzel.
Abstract:
Let E: y^2 =x^3 +Ax +B be an elliptic curve defined over the rationals.
Then the trace of the Frobenius morphism of E for the prime p is denoted
a_E(p) and satisfies the equation a_E(p) = p+1 - #E(F_p), where F_p denotes
the finite field of p elements. By theorems of Hasse and Deuring,
we know that for a fixed prime p, a_E(p) takes on every integral
value in (-2*sqrt{p}, 2*sqrt{p}) as E varies over all elliptic curves defined
over F_p. It is quite natural to consider the complementary question of how
often a_E(p) takes on a given integral value as p varies and E is fixed.
The Sato-Tate conjecture asserts that if E does not have complex
multiplication, then
#{p < X : 2*alpha*sqrt{p} < a_E(p) < 2*beta*sqrt{p}}
~ (2X/pi*log X)*integral{alpha}^{beta} sqrt{1-t^2}dt.
Lang and Trotter have made the more precise conjecture that if r is an
integer and E does not have complex multiplication or if r is not zero, then
#{p < X : a_E(p) = r } ~ C_{E,r}*(sqrt{X}/log X),
where C_{E,r} is an explicit constant depending only on E and r.
Richard Taylor has recently proved the Sato-Tate conjecture for elliptic
curves defined over the rationals. In fact, he has proved a generalization
of the conjecture for elliptic curves defined over any real number field
which satisfies a mild hypothesis. The more precise conjecture of
Lang and Trotter remains open.
In this talk, we will introduce these conjectures and some generalizations.
We will also discuss some averaging results related to these more general
conjectures.
Note that this date and time are different than
usual.
Abstract: Certain modular functions "hauptmoduls" well known to
number theorists arise as graded characters of the moonshine vertex operator
representation of the Monster simple group. I will give an overview of
the moonshine correspondence, and discuss the case of the generalized
moonshine conjecture.
Abstract: A "Maier matrix" is a combinatorial device used to prove
the existence of irregular and unusual behavior in the distribution of the
primes and related arithmetic sequences. After giving a general overview of
the method, I will discuss my work in extending the method to the polynomial
ring F_q[t] and to rings of integers of certain imaginary quadratic fields
(here, F_q is the finite field with q elements). In particular, we will
obtain natural analogues of 'irregular distribution' results proved by
Maier, Shiu, and Granville-Soundararajan.
Abstract: In the 1950s Atkin and Swinnerton Dyer produced many
complicated identities involving the "ranks" of partitions (some of these
confirmed conjectures made by Dyson). Recent work of Ono, Bringman, and
Rhoades has explained some of these identities in the context of weakly
holomorphic Maass forms. We produce infinite families of phenomena which
explain the remaining identities as special cases. All necessary background
will be given in the talk (which is joint work with S. Treneer).
Abstract: Let x congruent to r_i modulo n_i (where i=1,...,l) be a
covering of the integers with
1 < n_1 < n_2 <...< n_l.
The two main questions we will consider are:
Note that this date is different than
usual.
Abstract: In this lecture, I will discuss a few of the fundamental
notions on which the first elementary proofs are based and how these notions
allow for no less than two elementary proofs, the first coming from Selberg
and the second coming from Wright in 1952. These two proofs are analogous and
I will present the latter as it has been reproduced in great detail. If
time permits, I will also briefly discuss a third and completely
different elementary proof presented by Daboussi in 1984.
Abstract: Recent celebrated works of Zwegers and Bringmann-Ono
have placed the mock $\Theta$-functions and their generalizations in the
context of weight $\frac{1}{2}$ harmonic weak Maass forms. In light of
this, one expects similar correspondences to hold between other spaces of
half-integral weight Maass forms, however missing are natural candidates to
serve as analogues to the mock $\Theta$-functions. In separate works with
Bringmann-Ono and Bruinier-Bringmann-Ono, we make such correspondences
precise by constructing half-integral weight vector valued harmonic weak
Maass forms on the full modular group $\SL_2(\mathbb Z)$ whose
transformation properties are dictated by the Weil representation arising
from elementary theta series. We show that these vector valued Maass forms
give rise to certain Borcherds products and also families of hypergeometric
series. We establish correspondences between spaces of half-integral weight
Maass forms and classical spaces of half-integral weight modular forms.
Abstract: In 1900 David Hilbert presented a famous list of 23 problems
at the International Mathematical Congress in Paris. This talk is about
the 18-th of these problems, which was motivated by problems in
materials science. The 18-th problem concerns crystallographic
groups, tilings of space by identical polyhedra, and packing of space
by identical convex bodies, such as spheres (Kepler problem). This talk
describes the history and results found on this problem. This includes
some recent results found in 2006 and 2007.
Palmetto Number Theory Series (PANTS) 4,
University of South Carolina,
Columbia, South Carolina.
Department of Mathematics main page