Matt Boylan, USC.
Title: Arithmetic of the partition function, I.
Abstract:
Let p(n) denote the number of partitions of a positive integer n; p(n)
counts the number of non-increasing sequences of positive integers whose sum
is n. The function p(n) satisfies well-known congruences modulo 5, 7, and
11 (for example, p(5n+4) is always a multiple of 5). These congruences have
been the subject of much study in the last eighty years. Here, we prove
that these congruences are, in fact, the only ones of their kind. We also
resolve infinitely many cases of a classical conjectrue of Newman on the
distribution of p(n) modulo primes l.
Matt Boylan, USC.
Title: Arithmetic of the partition function, II.
Abstract:
As in the first talk, let p(n) be the ordinary partition function. We show
that if l is prime and p(ln+a) is divisible by l for all n, then (l,a) is one
of (5,4), (7,5), or (11,6). This is proved using the Serre and
Swinnerton-Dyer theory of modular forms modulo l. An introduction to this
theory will be provided.
Michael Filaseta, USC.
Title: From an Introduction to Galois Theory 101 to a Problem in
Diophantine Analysis.
Abstract:
There are two roots to the quadratic x^2 + 1. In an introductory complex
variables course, one defines i as a root of x^2 + 1. Which one is it?
Having taken advantage of the fact that those attending the seminar will
ponder this question in advance or perhaps wonder why I would ask it, we will
embark on a layman's introduction to Galois theory. This will eventually
lead us in a natural way to a study of questions similar to the following:
If n is a positive integer > 8 and m is the largest factor of n (n+1) that is
relatively prime to 6, then how large must m be? For this particular
problem, we will show in a subsequent seminar that m > n^0.27. This week,
however, will be a light lecture simply introducing the connection between
the topics in the title of the talk.
Link to related material (8.9 mb movie file): GaloisThry101.mov.
Michael Filaseta, USC.
Title: On the factorization of x^2+x revisited, Part I.
Abstract:
Fix two primes p and q. This talk will center around results about the
largest factor of x^2 + D x that is relatively prime to p q. The aim is
to show that for many choices of p and q, if x is an integer that does
not belong to a fixed specified set depending on p and q, then the
largest factor of x^2 + D x that is relatively prime to p q is larger
than x^t where t > 0 is a precise positive number depending on p and
q. This week, we will concentrate on the beginning of the proof.
Michael Filaseta, USC.
Title: The factorization of $x^2+x$ revisited, Part II.
Abstract:
The seminar is a continuation of last week's seminar, and
we will come closer to finishing and perhaps even finish the proof of the
main result which implies that for n >9, the product n (n+1) has
a factor > n^0.27 that is relatively prime to 6. This implication
requires a bit of work and some details on how it follows as a
consequence of the main result will be done later.
Michael Filaseta/Matt Boylan, USC.
Title: The factorization of x^2+x revisited, Part III (Michael).
Title: The modular j-invariant and traces of singular moduli (Matt).
Abstract:
To begin, Michael will continue with the proof of his main result,
namely, that for n >9, the product n (n+1) has a factor > n^0.27 that
is relatively prime to 6.
Matt's abstract is as follows:
The modular j-invariant, j(z) = q^(-1) + 744 + 196884q + 21493760q^2 + ... ,
is a modular form of weight zero on the group SL2(Z) which is holomorphic on
the complex upper half plane, but which has a pole at infinity. In this
talk, I will discuss some of its important connections to other areas of
mathematics.
In particular, values of j at quadratic irrationalities in the upper half
plane like z=(-1+sqrt(-3))/2, called singular moduli, play a fundamental
role in algebraic number theory. They have been studied extensively since
the work of Kronecker and Weber in the 1800s. Remarkably, singular moduli
are algebraic numbers. Recently, Zagier initiated a study of their traces.
We will discuss developments stemming from Zagier's work, including some of
the speaker's results.
Gang Yu, USC.
Title: Upper bounds for general Sidon sets.
Abstract:
Given positive integers $g$ and $N$, a set $\mathcal{A}\subset
[1,N]\cap {\Bbb{Z}}$ is called a $B_2[g]$ set if every integer $n$ has at
most $g$ representations as $n=a+b$, where $a\leq b$ and
$a,b\in\mathcal{A}$. According to this definition, a $B_2[1]$ set is thus
a Sidon set. In the talk, I will introduce some new ideas to improve the
current best upper bounds for $B_2[g]$ sets.
NO SEMINAR
USC Fall Break
Mark Kozek, USC.
Title: On Coverings of the Integers and 2 1/2 Conjectures About
Numbers of the Form: k^r*2^n+1 or k^r-2^n.
Abstract:
I will start with a glance at Selfridge's argument that 78557 is
a Sierpinski number. Using it as a classical example of a situation where
coverings of the integers can be applied to establish a number's
non-primality, I will pose the question, "When else can coverings of the
integers be useful tools to resolving non-primality questions?" In
particular, based on joint research with Filaseta and Finch, I will use
similar covering arguments to resolve the conjecture that, for any postive
integer r there exist infinitely many positive odd numbers k such that
k^r*2^n+1 has at least two distinct prime factors for all postive integers
n, and to address two "interesting" cases of r for the conjecture that, for
any postive integer r there exist infinitely many positive odd numbers k
such that k^r-2^n has at least two distinct prime factors for all postive
integers n. Only a basic elementary number theory background is needed and
as such this talk should be accessible to all graduate students and to
advanced undergraduates.
John Brillhart, University of Arizona.
Title: The power of 2
Abstract: The prime 2 is very special in many ways that translate
into appropriate handling of problems and proofs and the construction of
efficient algorithms. In the talk I'll present a diverse collection of
examples in which 2 plays a basic part in some way. The talk will be at an
elementary level and will hopefully be hearer-friendly.
NO SEMINAR
Integers Conference 2005 (In Celebration of Ron Graham's 70th Birthday), Oct. 27-30, Carrollton, Ga.
Dave Penniston, Furman University.
Title: The arithmetic of some Calabi-Yau varieties.
Abstract: Since Wiles' proof of Fermat's Last Theorem, in which he
essentially settled the question of the modularity of elliptic curves
defined over the rational numbers, there has been increased interest in the
modularity of other geometric objects. In this talk we will discuss the
arithmetic of certain Calabi-Yau varieties, and draw connections between
these and number theoretic objects such as partition functions.
Clayton Petsche, University of Georgia.
Title: The Equidistribution of Small Points on Elliptic Curves
Abstract: Let E be an elliptic curve defined over a number
field k. In this talk I will discuss recent joint work with Matt Baker, in
which we establish an adelic equidistribution principle for algebraic points
of E with small Neron-Tate height. We deduce from our main inequality a number
of consequences. For example, we give a new and simple proof of the
Szpiro-Ullmo-Zhang equidistribution theorem for elliptic curves, and we
also prove a non-archimedean analogue of the Szpiro-Ullmo-Zhang theorem
which takes place on the Berkovich analytic space associated to E. I will
include a brief introduction to the theory of Berkovich spaces associated
to curves. Finally, I will discuss some quantitative
`non-equidistribution' theorems--which are consequences of our main
inequality--concerning totally real or totally p-adic small points. The
results for totally real points imply similar bounds for points defined
over the maximal cyclotomic extension of a totally real field.
Gang Yu, USC.
Title: Divisibility of class numbers of real quadratic fields
Abstract: Suppose $g>2$ is an odd integer. For real number $X>2$,
define $S_g(X)$ the number of square-free integers $d\leq X$ with the class
number of $\Bbb{Q}(\sqrt{d})$ being divisible by $g$. By constructing the
discriminants based on the work of Yamamoto, we prove a lower bound
$S_g(X)\gg X^{1/g-\epsilon}$ for any fixed $\epsilon>0$, which improved a
result of Murty.
Sungkon Chang, Armstrong Atlantic State University.
Title: Twists of Elliptic/Superelliptic curves
Abstract: In connection with the Birch and Swinnerton-Dyer Conjecture,
Goldfeld conjectured that the average Mordell-Weil rank in quadratic twists
$E_D$ of an elliptic curve $E/\mathbb{Q}$ is $1/2$. There are few
unconditional results in the literature about this average, though, and in
this talk we introduce a result, as a motivation, on the average
Mordell-Weil rank in a family of quadratic twists of an elliptic curve $E :
y^2=x^3-A$. This result is based on Schaefer's method of computing Selmer
groups, and it seems that this method is very useful for studying Selmer
groups of twists of superelliptic curves, in general. Using this method, we
also obtain results about the distribution of number of rational points on
twists of superelliptic curves over a global field.
NO SEMINAR
Thanksgiving Break
Matt Boylan, USC.
Title: Introduction to modular Galois representations
Abstract: In the late 1960's and early 1970's, Serre conjectured and
Deligne proved a relationship between representations of the absolute Galois
group of Q (the rationals) in the finite field of l elements (l is prime) and
the Fourier coefficients of modular forms.
In this talk, I will introduce the relevant objects and give examples. In
particular, I will describe (following the works of Serre and
Swinnerton-Dyer) how Deligne's theory explains congruences for the
coefficients of modular forms. I will also mention my contributions.