Max Alekseyev:
On the number of two-dimensional threshold functions
A two-dimensional threshold function of $k$-valued logic can be viewed as
coloring of the points of a $k\times k$ square lattice into two colors such that there exists a
straight line separating points of different colors. For the number of such functions only asymptotic
bounds are known. We give an exact formula for the number of two-dimensional threshold
functions and derive more accurate asymptotics.
Jim Brown: Level lowering for half-integral weight modular forms
Let g be a half-integral weight eigenform of level Mp^{n}, gcd(p,M) = 1. Under a mild hypothesis we show
there exists a half-integral weight eigenform of level M that has eigenvalues congruent to those of g
modulo p. The main tools involved are the Shimura correspondence and Khare-Wintenberger's recent proof of Serre's conjecture. This is joint work with Yingkun Li and arose out of his summer undergraduate research project during the summer of 2008.
Joshua Cooper: The Discrepancy of the Lexicographically Least de Bruijn Cycle
A (binary) de Bruijn cycle of order $n$ is a cyclic binary
word of
length $2^n$ so that every binary word of length $n$ appears as a subword
exactly once. There are many constructions for de Bruijn cycles, but one
stands out for its extraordinary properties. The lexicographically-least de
Bruijn cycle, sometimes called the Ford sequence, is also the de Bruijn
cycle
generated by the "greedy" algorithm, the cycle generated by a linear-shift
feedback register of minimum weight, and the string of all Lyndon words of
length dividing n arranged in lexicographic order. It is not hard to see
that the Ford sequence is very "unbalanced" in the sense that there is an
excess of 0's near the beginning and an excess of 1's near the end. It is
possible to quantify this intuition by defining the discrepancy of a
sequence
to be the maximum, over all initial segments, of the difference between the
number of 0's and 1's. We show that the discrepancy has order $2^n \log
n/n$, answering a question of Ron Graham.
James Carter: On the restricted Hilbert-Speiser and Leopoldt properties
Let $G$ be a finite abelian group. A number field $K$ is called a
Hilbert-Speiser field of type $G$ if for every tame $G$-Galois extension $L/K$,
the ring of integers $\mathcal{O}_L$ is free as an $\mathcal{O}_K[G]$-module.
If $\mathcal{O}_L$ is free over the associated order $\mathcal{A}_{L/K}$ for every
$G$-Galois extension $L/K$, then $K$ is called a Leopoldt field of type $G$.
It is well-known (and easy to see) that if $K$ is Leopoldt of type $G$, then
$K$ is Hilbert-Speiser of type $G$. The converse does not hold in general. However, we show that a modified version does hold for
many number fields $K$ (in particular, for $K/\mathbb{Q}$ Galois) when $G=C_{p}$ has prime order. Finally, we show that even the
modified converse is false in general, and we give examples which show that the modified converse can hold while the original does
not.
Igor Erovenko: Bounded generation and second bounded cohomology of
wreath products
We show that the (standard restricted) wreath product of groups is boundedly
generated if and only if the bottom group is boundedly generated and the top
group is finite. We also establish a criterion for triviality of the singular
part of second bounded cohomology of wreath products. (Joint work with Nikolay Nikolov and B. Sury.)
David Ford: Complexity of Ideal Factorization in an Algebraic Number
Field
Given
- $F(x)$, an irreducible monic polynomial in $\Zx$,
- $p$, a rational prime,
- $K$, the extension of $\Q$ generated by a root of $F$,
- $\cO$, the ring of integers of $K$,
the Montes algorithm (1999)
computes $\ldss{e_1}{e_r}$
and $\ldss{f_1}{f_r}$
such that
\[
p\msp\cO = \mfp_{1}^{e_1} \cdots \mfp_{r}^{e_r}
\]
is the complete factorization of $p\msp\cO$
as a product of prime ideals in $\cO$,
with $\ldss{f_1}{f_r}$ the respective degrees of $\ldss{\mfp_1}{\mfp_r}$.
\newpar
Examples of very high degree are known for which the Montes algorithm
terminates quickly, but a complexity estimate for the performance
of the algorithm in the general case has not been made.
\newpar
Variants of the Zassenhaus ``Round Four'' algorithm can provide this
factorization, via the construction of an integral basis for $K$.
Pauli (2001)
has given a complexity estimate
for the ``two-element'' variation.
\newpar
The Montes algorithm avoids the construction of an integral basis,
so there is reason to suspect that its complexity is
lower than that of Round Four.
\newpar
We analyze a simplified variant of the Montes algorithm, which
determines whether $F(x)$ is reducible in $\Zpx$.
We assume (without proof) that this is the worst case for the Montes
algorithm, since in the irreducible case
the algorithm would construct the most ``levels''.
\newpar
We show that
the bit-complexity of the modified Montes algorithm is
\[
O\bigp{\,(\deg F)^{\ep{3}}\,v_p(\disc F)^{\ep{2}}\,}.
\]
Paul Gunnells: Multiple Dirichlet Series
Multiple Dirichlet series are generalizations of L-functions
that involve several complex variables. While the functional equation
of a usual L-series is an involution s -> 1-s, a multiple Dirichlet
series satisfies a group of functional equations that intermixes all
the variables. Siegel constructed the first example of such a series
in 1956 by taking the Mellin transform of a half-integral weight
Eisenstein series. In recent years multiple Dirichlet series have
been intensively studied in a variety of contexts and with many
applications in mind.
In this talk we will present an overview of this subject. We show how
these series provide natural tools to address questions in analytic
number theory. We then describe a construction of a class of multiple
Dirichlet series attached to Dynkin diagrams, where the resulting
group of functional equations is the associated Weyl group. These
series are expected to be Fourier-Whittaker coefficients of
metaplectic Eisenstein series.
Michael Mossinghoff: The distance to an irreducible polynomial
Given an integer polynomial f(x), how many coefficients
do you need to adjust before you are assured of finding an irreducible
polynomial of the same degree or smaller? More precisely, does there
exist an absolute constant C so that for every
f(x) in Z[x] there exists an irreducible
g(x) in Z[x] with deg(g) ≤ deg(f) and L(f-g) ≤ C, where
L(h) denotes the sum of the absolute values of the
coefficients of h(x)? This question was first posed by
Turán in 1962, and it remains unsolved. We discuss some
algorithms designed to investigate this question, and report on the
results of some recent computations.
Carlos Nicolas: k-Triangulations and k-Splitters
So far k-triangulations have been defined only for points in convex position in the plane (as maximal sets of diagonals of the
n-gon without any k+1 of them mutually crossing). I will introduce the concept of a k-triangulation for sets of points in
general position in the d-dimensional Euclidean space. Roughly speaking, a k-triangulation is a way to break down the
(k-1)-splitters of the entire set into a sum of splitters for blocks of a certain size. The definition agrees with the usual
triangulations of points when k=1 and also with the k-triangulations of the n-gon. I will discuss the problem of constructing
k-triangulations in two dimensions using a continuous motion argument.
Andrew Sills: On the Rogers-Selberg Identities and Gordon's Theorem
The Rogers-Ramanujan identities are among the most famous in the theory of
integer partitions. For many years, it was thought that they could not be
generalized, so it came as a big surprise when Basil Gordon found an
infinite family of partition identities that generalized Rogers-Ramanujan in 1961.
Since the publication of Gordon's result, it has been suspected that a certain special
case of his identity should provide a combinatorial interpretation for a set of three analytic
identities known as the Rogers-Selberg identities. I will discuss a bijection between two relevant
classes of integer partitions that explains the connection between Gordon and Rogers-Selberg.
This work appeared in JCTA 115 (2008) 67-83.
Ethan Smith: The Lang-Trotter Conjecture ``on average"
Abstract: The Lang-Trotter Conjecture concerns a prime distribution
problem related to elliptic curves. In this talk, I will describe the
original conjecture of Lang and Trotter as well as a version for
elliptic curves defined over number fields. I will also give a survey
of several results (by various authors) which may be interpreted as
saying that the conjecture holds ``on average."
Cliff Smyth: Paths in Line Arrangements
Consider an arrangement $A = \{l_1, \ldots , l_n\} of $n$ lines in the
plane. A path in $A$ of length $N$ is an alternating sequence $P = (L_0,
P_1, L_1, P_2, \ldots , P_N, L_N)$ of lines $L_i$ from $A$ and points
$P_i$, where each $P_i$ is the intersection the lines $L_{i-1}$ and $L_i$.
A path is monotone if the $x$ coordinates of the $P_i$ are increasing with
$i$. It is conjectured that the maximum possible length $N$ of a monotone
path in any line arrangment of $n$ lines is $N = o(n^2)$. For any
$\epsilon > 0$, we show the maximum length is $N =
\Omega(n^{2-\epsilon}).$
This is joint work with Joszef Balogh, Oded Regev, William Steiger and
Mario Szegedy.
Gary Walsh: A new approach to an old Diophantine problem
Our goal is to obtain an upper bound for the number of
integer solutions to the equation $x^2-dy^4=k$,
in which the upper bound depends as lightly as possible on $d$ and $k$.
A result in this direction was proved recently by the speaker,
where the dependence on $k$ was given in terms of the
number of its prime factors. We refine these results for a particular
case in which $k$ is a power of $2$. This is joint work with Pingzhi Yuan.
John Webb: Partition Values and Modular L-Functions
Using Waldspurger's theorem, interesting congruences can be
built relating values of the partition function to central critical
values of particular modular L-functions. This expands upon the work of
Guo and Ono from the late 1990's.
Qingquan Wu: The different exponent of Artin-Schreier extension towers
Let $K$ be a function field over a perfect constant field of
positive characteristic $p$, and $L$ the compositum field of some
(degree $p$) Artin-Schreier extensions of $K$. We describe how places
of $K$ split in this elementary abelian $p$-extension $L/K$. One
consequence is that the converse of Abhyankar's Lemma is also true
here. It turns out that a place of $K$ totally ramifies in $L/K$ (is
inert, splits completely, resp.) if and only if it totally ramifies (is
inert, splits completely, resp.) in all of the intermediate degree $p$
Artin-Schreier extensions over $K$. Similar scenarios are rare in number
field extensions. But in the function field setting, examples are given
to show that all possible decompositions are indeed possible. These
examples are independent of the function field structure of $K$ and its
characteristic $p$. Our main result is to compute the different exponent
of such extension towers.
This is joint work with Renate Scheidler.
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