**September 9 Luis Gallardo, Brest
**

**Note: Special Time and Place, 4:00-5:00, 201 Thomas.**

**September 16** **Joe Hundley, PSU
**

Let $\pi$ be an automorphic representation of a split reductive algebraic
group $G$ over a number field $F$. Langlands has associated to $G$ a
complex Lie group $^LG$, and defined, for every finite dimensional representation
$r$ of $^LG$, an $L$-function $L(\pi,r,s),$ which is conjectured to have
analytic properties similar to those of the Riemann zeta function and/or
Dirichlet $L$-functions, and contain interesting information about the
representation $\pi.$ In the Rankin-Selberg method, analytic properties
of $L$-functions are proved by expressing the $L$-functions in terms
of integrals containing Eisenstein series, whose analytic properties
are known. We will consider a new integral on the similitude orthogonal
group GSO(8), in which the representations $r$ that arise are the standard
and half-spin representations of SO(8,C), and use it to relate poles of
$L$-functions with the existence of a non-trivial functorial lift to the
exceptional group $G_2.$

**September 23** **Alexander Borisov, PSU**

**Two elementary results concerning rational polynomials **

The two elementary results in the title are part of my recent work. Both
have to do with the wonderful fact that in polynomials with integer coefficients
both the coefficients and the powers are taken from essentially the same
set. Most of the talk will be accessible to most graduate students.

The first result is the following theorem.

Theorem. A polynomial with rational coefficients divides a derivative of
the polynomial which is split over rationals if and only if all of its irrational
roots are real and simple.

I will explain the motivation as well as the proof of this result, which
resembles the famous three point theorem of Belyj. The second result is
related to the classification of the set of solutions of the system of functional
equations satisfied by the so-called quantum integers with respect to multiplication.

**September 30 Robert Vaughan, PSU**

**Report of the Oberwolfach meeting on the Riemann zeta and allied
functions
**

** October 7 David Terhune, PSU
**

*Evaluations of a class of double L-values*

An anlaytic proof for the `convolution' -type double L-values is given.
Along the way, Dirichlet character analogues of generalized single and
double polylogarithms are defined, and the monodromies of these new functions
play a pivotal role.

October 14 Scott Parsell, Butler U.

**Paucity Problems**

In analogy with Fermat's Last Theorem, one typically expects that a diophantine
equation in few variables (relative to the degree) will have few, if any,
non-trivial solutions. When counting the number of solutions in a box,
one can frequently show that the number of non-trivial solutions grows more
slowly than the number of trivial ones as the size of the box tends to infinity,
and in this case we say that the equation (or system of equations) exhibits
a "paucity" of non-trivial solutions. We will present several examples
of this phenomenon in symmetric systems and discuss some recent results
in this direction.

**
October 21 Bill Hoffman, LSU
**

**Modular forms on noncongruence subgroups and Atkin-Swinnerton-Dyer
relations **

This is a joint work with Liqun Fang, Ben Linowitz, Andrew Rupinski, and
Helena Verrill. We give new examples of modular forms on noncongruence
subgroups whose $l$-adic representations are modular and whose expansion
coefficients satisfy Atkin-Swinnerton-Dyer congruences.

**October 28 Cam Stewart, Waterloo U.**

**On intervals with few prime numbers**

In this talk we shall survey the known results on large gaps between consecutive
prime numbers and we shall then discuss recent work of Maier and the speaker
on long intervals with fewer than the expected number of primes.

October 29 Vladimir Retakh, Rutgers

**Note: Special Day, Friday, 11:15-12:05, 320 Whitmore. **

**Universal algebra of pseudo-roots of non-commutative polynomialsa
and non-commutative symmetric functions
**

A generic monic polynomial $P(t)\in R[t]$ where $R$ is a (non-commutative)
division ring admits many different facotrizations into a product of linear
polynomials $t-a$. Elements $a$ are called pseudo-roots of $P(t)$ and
may be considered as generators of a quadratic algebra. This algebra has
a deep structure. I will talk about this structure anbd its relations to
other problems.

**November 4**** Dale Brownawell, PSU**

*Independence of derivatives of Carlitz periods*

Carlitz defined an exponential function in the setting of function fields
F_q(t) over finite fields F_q. In particular, its periods are all multiples
of a fundamental period \pi_q which is the analogue of the usual 2{\pi}i,
whose transcendence was shown by Carlitz's student Wade. Recently Laurent
Denis has investigated the algebraic independence of the derivatives of \pi_q.
Alf van der Poorten and I have taken up this investigation.

**November 11** **Katherine Hurley, PSU**

*Vertex Operator Algebras and Spherical Harmonics*

This talk introduces the axioms for a vertex operator algebra and gives
the examples of Heisenberg and Lattice VOAs. It then discusses the use of
harmonic polynomials (spherical harmonics) to construct highest-weight vectors
for the Virasoro Lie algebra and determine their graded traces on the vertex
operator algebra.

November 18 Robert Vaughan, PSU

*Diophantine Approximation on Planar Curves*

Let $\cal C$ be a non--degenerate planar curve and for a real, positive
decreasing function $\psi$ let $\cal C(\psi)$ denote the set of simultaneously
$\psi$--approximable points lying on $\cal C$. In joint work with Beresnevich,
Dickinson and Velani, we show that $\cal C$ is of Khintchine type for divergence;
i.e. if a certain sum diverges then the one-dimensional Lebesgue measure
on $\cal C$ of $\cal C(\psi)$ is full. We also obtain the Hausdorff measure
analogue of the divergent Khintchine type result. In the case that $\cal
C$ is the unit circle the convergence counterparts of the divergent results
are also obtained. Furthermore, for functions $\psi$ with lower order in
a critical range we determine a general, exact formula for the Hausdorff
dimension of $\cal C(\psi)$. These results constitute the first precise
and general results in the theory of simultaneous Diophantine approximation
on manifolds.

**December 2** **Ulrike Vorhauer, U. of Michigan**

**Comparative prime number theory
**

Chebyshev conjectured that $\pi(x;4,3) \ge \pi(x;4,1)$ for all $x\ge 3$.
Although $x$ are now known for which this fails, and it is known that the
difference $\pi(x;4,3)-\pi(x;4,1)$ changes sign infinitely many times, for
the general modulus it is still not known that $\pi(x;q,a) - \pi(x;q,b)$ changes
signs for arbitrarily large $x$ when $(a,q) = (b,q) = 1$. We survey the
existing knowledge, and add a new result in this area.

**December 9 George Andrews, Penn State**

**Partitions with Short Sequences and Ramanujan's Mock Theta Functions
**

In a recent paper "Integrals, Partitions and Cellular Automata" in the
Transactions of the American Mathematical Society, Holroyd, Liggett and Romik
evaluated an intriguing definite integral and applied it to a variety of probability
models. The application to integer partitions concerned partitions in which
no sequence of consecutive integers of length k appears (k=2,3,...). The
authors note that in one instance, a proof of their result can also be based
on a little known partition theorem of P.A. MacMahon. Our object in this
talk will be to introduce these ideas and to develop the study of such partitions
from a purely combinatorial, q-series point of view. Surprisingly one of
Ramanujan's mysterious mock theta functions arises.