Scott Ahlgren
Congruences for modular forms of half-integral weight
Abstract: The Fourier coefficients of modular forms of half-integral
weight contain a tremendous amount of information. For example,
they carry information about special values of L-functions, about
elliptic curves, and about generating functions of combinatorial
interest. I will describe some recent results on such modular forms
(notably in the case of forms of level four) and their applications.
***************************************************************************
George Andrews
Combinatorial aspects of integer partitions
Abstract: Within the last decade, the theory of partitions has
flourished. This has primarily been an symbiotic interaction between
modular forms and classical q-series. I hope in this lecture to
focus on the q-series aspect of this exciting topic beginning with
a recent Inventiones paper (Inv. Math., 169(2007), 37-73). We will
then move on to parity questions related to B. Gordon's generalization
of the Rogers-Ramanujan identities. We conclude with open questions.
***************************************************************************
Bruce C. Berndt
Analysis in Ramanujan's Lost Notebook
Abstract: Published with Ramanujan's Lost Notebook are six partial
manuscripts of Ramanujan in the handwriting of G.N. Watson. (The
original manuscripts no longer exist.) We discuss some recently
examined entries in three of these partial manuscripts falling under
the purview of either classical analysis or classical analytic number
theory. In particular, we discuss a beautiful transformation formula
involving the Riemann zeta function and a certain class of integrals,
which can be regarded as analogues of either theta functions, Gauss
sums, or elliptic functions.
***************************************************************************
John Friedlander
Brinkmanship in the Semi-linear Sieve
Abstract: An interesting aspect, essentially unique to the semi-linear
(also known as half-dimensional) sieve, is that, when applied to
a sequence possessing optimal level of distribution, it comes within
a whisker of success. This opens
the possibility of gaining the goal by providing just a slight additional
input from other sources that exploit special properties of the
sequence. We shall exhibit some recently discovered instances of
this feature found in joint work with Henryk Iwaniec.
***************************************************************************
Steve Gonek
The First 150 Years of the Riemann Zeta Function
Abstract: This is the 150th anniversary of Riemann's pivotal 1859
paper "Ueber die Anzahl der Primzahlen unter einer gegebenen
Grosse" ("On the
number of primes less than a given magnitude"). In this survey
talk for a general mathematical audience, we describe the contents
of Riemann's paper, the early work it spurred in the theory of the
zeta function, and how the theory developed subsequently, down to
the present day. Along the way we give an overview of key features
of the theory and an idea of how some of the most important results
were achieved.
***************************************************************************
Habiba Kadiri
A bound for the least prime ideal in the Chebotarev density
problem
Abstract: A classical theorem due to Linnik gives a bound for the
least prime number in an arithmetic progression. Lagarias, Montgomery
and Odlyzko gave a generalization of this result to any number field.
Their proof relies on some results about the distribution of the
zeros of the Dedekind zeta function (zero free regions, Deuring-Heilbronn
phenomenon). In this talk, I will present some new results about
these zeros. As a consequence, we are able to prove an effective
version of the theorem of Lagarias, Montgomery and Odlyzko.
************************************************************************
Ram Murty
Special values of L-series
Abstract: We will discuss the transcendental nature of special
values of abelian and non-abelian L-series. The first half of the
lecture will survey results in this context since the time of Dirichlet.
In the second half, we will present new results with a special focus
on special values of Hecke L-series of imaginary quadratic fields.
************************************************************************
Nathan Ng
Moments of the Riemann zeta function
Abstract: In this talk I will discuss the problem of evaluating
integral and discrete moments of the Riemann zeta function. In 1918
Hardy and Littlewood introduced the integral moments in order to
study the Lindelof hypothesis. In the 1980's Gonek and Hejhal independently
introduced discrete moments in which the zeta function is averaged
over its zeros. These discrete variants are important for various
number theoretic applications such as finding simple zeros of the
zeta function or finding small gaps between the zeros of the zeta
function. I will discuss some of the different techniques for evaluating
these moments and the challenges we face in evaluating high moments
in both cases.
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Ken Ono
Generalized Borcherds products and two number theoretic applications
In his 1994 ICM lecture, Borcherds introduced a very strange construction
of certain modular forms. He obtained a vast generalization of the
classical fact that Delta(z), the normalized cusp form of weight
12 on SL_2(Z), is a simple infinite product. He constructed forms
with Heegner divisor as infinite products whose exponents are Fourier
coefficients of weight 1/2 modular forms. In joint work with Bruinier,
we have constructed "generalized Borcherds products" where
the infinite product exponents are coefficients of harmonic Maass
forms (a generalization of classical modular forms). Here we discuss
the construction and present two number theoretic
applications:
1) Parity of the partition function
2) Central values and derivatives of modular L-functions
***************************************************************************
Kannan Soundararajan
Quantum Unique Ergodicity and Number Theory
Abstract: A fundamental problem in the area of quantum chaos is
to understand the distribution of high eigenvalue eigenfunctions
of the Laplacian on certain Riemannian manifolds. A particular case
which is of interest to number theorists concerns hyperbolic manifolds
arising as a quotient of the upper half-plane by a discrete ``arithmetic"
subgroup of SL_2(R) (for example, SL_2(Z), and in this case the
corresponding eigenfunctions are called Maass cusp forms). In this
case, Rudnick and Sarnak have conjectured that the high energy eigenfunctions
become equi-distributed. I will discuss some recent progress which
has led to a resolution of this conjecture, and also on a holomorphic
analog for classical modular forms.
***************************************************************************
R. K. Guy and H. C. Williams
Some Interesting Divisibility Sequences
Abstract: A sequence of rational integers {A_n} is said to be a
divisibility sequence if A_m | A_n whenever m | n. If the divisibility
sequence {A_n} also satisfies a linear recurrence relation, it is
said to be a linear divisibility sequence. Linear divisibility sequences,
where the characteristic polynomial for the recurrence is irreducible
and of degree larger than 2, tend to occur rather infrequently.
However, consider the problem of enumerating the ways a(n) of tiling
a 4xn rectangle by dominoes. It turns out that {a(n)}satisfies a
linear recurrence with characteristic polynomial x^4-x^3-5x^2-x+1
with initial values 1, 5, 11, 36 for n=1,2,3,4, respectively. What
is remarkable about {a(n)} is that it is also a divisibility sequence.
In this talk we will describe the number theoretic properties of
a much more general quartic recurrence than {a(n)}. This sequence
is also a divisibility sequence and includes {a(n)} for certain
values of some parameters. Also, this general sequence shares many
properties with the well–known Lucas function U_n. We go on
to discuss a particularly remarkable octic linear divisibility sequence
which arises from counting the number of spanning trees of the graph
P_4 x P_n.
**************************************************************************
Contributed
Talks
***********************************************************************
Yumiko Ichihara
The first moment of the value of automorphic L-functions over
primitive forms on the critical line
Abstract: We are interested in the first moment of the value
of automorphic L-functions L_f (1/2+it), which is a sum over primitive
forms. In this talk, I will show an asymptotic formula for it, in
the case that weight k is an even integer satisfying 0 < k <
12 and level is p^a, where p is a prime number. This formula yields
a lower bound of the number of primitive forms which L_f(1/2+it)
are not vanish.
Marie Jameson and Robert Lemke Oliver
Proof of Alder's Conjecture
Abstract: Motivated by classical identities of Euler, Schur,
and Rogers and Ramanujan, H. L. Alder investigated $q_d(n)$ and
$Q_d(n),$ the number of partitions of $n$ into $d$-distinct parts
and into parts which are $\pm 1 \pmod{d+3},$ respectively. He conjectured
that $$q_d(n) \geq Q_d(n).$$ G. E. Andrews and A. J. Yee proved
the conjecture in the cases where $d = 2^s-1$ and $d \geq 32.$ We
complete the proof of Alder's conjecture by determining asymptotic
estimates for these partition functions with explicit error terms
(correcting earlier work of G. Meinardus).
Shanta Laishram
Irreducibility of generalized Hermite-Laguerre Polynomials
Abstract: Let $a\ge 0, a_0, a_1, \cdots , a_m$ be integers. Let
\begin{align*} f_a(x)=\sum^m_{j=0}\frac{a_jx^j}{(j+a)!}.
\end{align*}Schur(in 1929) proved that $f_0(x)$ with $|a_0|=|a_n|=1$
is irreducible $\forall m$. Schur's result has been generalized
by many authors by using $p-$adic methods of Coleman and Filaseta.
In this talk, I will give a survey of the some of these results
and prove some results on the irreducibility of generalized Hermite-Laguerre
Polynomials by combining $p-$adic methods with the greatest prime
factor of the product of terms of an arithmetic progression.
Youness Lamzouri
Distribution of values of L-functions at the edge of the critical
strip
Abstract: In 2003, Granville and Soundararajan computed large moments
of the family of Dirichlet $L$-functions of quadratic characters
at $s=1$ and deduced an asymptotic formula for the distribution
function of the $L$-values. They also proved analogous results for
the Riemann zeta function on the line Re$(s)=1$. Following their
ideas, Liu, Royer and Wu studied the distribution of values of $L$-functions
attached to holomorphic cusp forms in the weight aspect. In this
talk we will generalize these results, namely by constructing and
studying a large class of random Euler products. We then deduce
information of the distribution of values of families of $L$-functions
at the edge of the critical strip. Among new applications, we study
families of symmetric power $L$-functions of holomorphic cusp forms
in the level aspect (assuming the automorphy of these $L$-functions)
at $s=1$, functions in the Selberg class in the height aspect, and
quadratic twists of a fixed $GL(m)/{\Bbb Q}$ automorphic cusp form
at $s=1$.
Mathieu Lemire
Extensions of the Ramanujan-Mordell formula and representations
by quadratic forms
Abstract: The Mordell-Ramanujan formula concerns the number of representations
of a positive integer $n$ by the form \[ x_{1}^{2} + x_{2}^{2} +
\cdots + x_{k}^{2}. \] In a recent article, Alaca, Alaca and Williams
gave an elementary proof of the Mordell-Ramanujan formula when $k$
is a multiple of $4$ by making use of some basic properties of polynomials
associated with Eisenstein series. In this talk, we use similar
ideas to determine extensions of the Mordell-Ramanujan formula.
That is, we extend this formula to the form
\[ x_{1}^{2} + \cdots + x_{r}^{2} + 2 x_{r+1}^{2} + \cdots + 2 x_{r+s}^{2}
+ 4 x_{r+s+1}^{2} + \cdots + 4 x_{k}^{2}\] for an arbitrary positive
integer $k \equiv 0 (\text{\rm{mod}} \ 4)$ and integers $r$, $s$
and $t$ satisfying $r \geq 1$, $s \mbox{(even)} \geq 0$, $t \geq
0$, $r+s+t=k$. As a result, we obtain several new explicit formulae
giving the number of representations of an integer $n$ by forms
satisfying these conditions when $ k \in \{4,8,12,16\}$.
Matija Kazalicki
2-adic and 3-adic part of class numbers and central values of
$L$-functions
Abstact: In the early 80s, Williams showed that if $\epsilon=T+U\sqrt{p}$
is a fundamental unit of the real quadratic field $\Q(\sqrt{p})$,
and if $h(-p)$ is the class number of $\Q(\sqrt{-p})$ then $h(-p)
\equiv T + (p-1) \pmod{16}$, where $8|h(-p)$.
In this talk we study the connection between 2-part and 3-part of
class number $h(-d)$ and $h(-3d)$ and ray class groups of $\pd$
unramified outside $2$ (and $3$), when $d$ is prime or the product
of two primes. We obtain certain "reflection'' theorems, and
as an immediate consequence we reproduce the result of Williams
(and we get a similar result in the case when $d$ is the product
of two primes).
The main ingredients of the proof are certain congruences between
$L_2(1,\chi_d)$ (and $L_3(1,\chi_d)$) and $h(-d)$(and $h(-3d$))
modulo powers of $2$ (and $3$), which we prove using modular forms.
We also obtain similar congruences for the central values of $L$-functions
associated to Ramanujan's $\Delta$-function, and relate them to
the structure of $2$-adic and $3$-adic Galois representation attached
to the $\Delta$-function.
Sun Kim
Göllnitz-Gordon identities and parity questionsin partitions
(with Ae Ja Yee).
Abstract: Parity has played a role in partition identities from
the beginning. In his recent paper, George Andrews investigated
a variety of parity questions in partition identities. At the end
of the paper, he then listed 15 open problems. The purpose of this
paper is to to provide answers to the first three problems from
his list, which are related to the Göllnitz-Gordon identities
and their generalizations.
Abdellah Sebbar
Equivariant forms, construction and structure
Abstract: The notion of equivariant forms for a modular group
will be introduced. After constructing few examples, we will see
how to construct algebraic and geometric structures on these forms.
Thomas Stoll
Bounds for the discrete correlation of infinite sequences on
$k$ symbols and generalized Rudin-Shapiro sequences
Abstract: Pseudorandom sequences, i.e., deterministic sequences
with properties reminiscent of random sequences, are a well-studied
subject. In this talk, we study the discrete correlaton of infinite
sequences over a finite alphabet, where we just take into account
whether two symbols are identical. We show that the correlation
cannot be "too small" in some specific sense;
moreover, we construct a large class of sequences (generalizd Rudin-Shapiro
sequences) which achieve the bound, provided $k$ is prime or squarefree.
The proofs involve combinatorial sieving, the Lovasz local lemma
and exponential sums estimates. This is joint work with E. Grant
and J. Shallit.
Gary Walsh
Eclipsing Siegel's method on a family of Quartic Equations
Abstract: Shabnam Akhtari has recently refined Siegel's argument
in the case of certain quartic Thue equations, providing a means
to determine upper bounds for quartic equations of the form $X^2-dY^4=k$,
the subject of recent work by the speaker. We show that this method
can be improved substantially by way of an irrationality measure
due to Yuan.
Benjamin Weiss
Galois Groups of Random P-Adic Polynomials
Abstract: The space of fixed degree polynomials with p-adic coefficients
has a natural probability distribution. Each polynomial also has
an associated group which is the Galois group of its splitting field.
We will discuss the induced distribution on groups, and derive results
for the limiting distribution as p grows. Time permitting, we will
discuss a relationship to Serre's mass formulae for extensions of
local fields, and prove a complementary theorem to the Chebotarev
Density theorem. This work is joint with Chris Hall.
Thomas Wright
Adelic Singular Series and the Goldbach Conjecture
Abstract: The purpose of this paper is to show how adelic ideas
might be used to make progress on the Goldbach Conjecture. In particular,
we present a new Schwartz function which is able to keep track of
the number of prime factors of an integer. We then use this, along
with the Ono/Igusa adelic methods for Diophantine equations, to
present an infinite sum whose evaluation would prove or disprove
the veracity of the Goldbach conjecture. We also use this to compute
the singular series, a quantity which, if we make the highly non-trivial
assumption that the circle method of Hardy and Littlewood applies,
would indicate whether there are solutions to the Goldbach equation
for every even natural number. In particular, if the circle method
applies, this quantity would be sufficiently large to prove Goldbach
is true.
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