**January 11
Matt Papanikolas (Brown University)**

*Periods of Drinfeld
modules with complex multiplication*

We investigate
transcendence properties of periods of Drinfeld modules with
complex multiplication. In particular we show that if such
Drinfeld modules have different CM fields then their fundamental
periods are algebraically independent over the algebraic numbers.
Joint work with Dale Brownawell.

**January 25
Robert Vaughan (PSU)**

*Waring's problem: A
Survey*

Recent work on* **G(k)
*in Waring's problem, jointly with T. D. Wooley, will be
described and placed in a historical context. Some
speculations will be made about future directions in the Hardy-Littlewood
method.

**February 1
No seminar: Peter Sarnak (Princeton University and the
Institute for Advanced study) is giving the Russell Marker
Lectures in Mathematics**

*L-Functions,
Arithmetic and Semiclassics*

Monday, January 29, 8:00 p.m.,
112 Osmond Laboratory, *Hilbert's Eleventh Problem*

Tuesday, January 30, 4:30 p.m.,
112 Osmond Laboratory, *Lp Norms of Eigenfunctions*

Wednesday, January 31, 4:30
p.m., 112 Osmond Laboratory, *Quantum Unique Ergodicity*

Thursday, February 1, 4:30
p.m., 112 Osmond Laboratory, *Families of L-Functions and
Symmetry*

**February 8
Sanju Velani (Queen Mary Westfield College, London), 103
McAllister**

*On simultaneously
badly approximable pairs*

For any pair $i,j \geq 0$
with $i+j =1$ let $\Bad(i,j)$ denote the set of pairs $(\a,\b)
\in \R^2$ for which $ \max \{ ||q\a||^{1/i}\, ||q\b||^{1/j} \}
> c/q $ for all $ q \in \N $. Here $c = c(\a,\b)$ is a
positive constant. If $i=0$ we identify the set $\Bad(0,1)$ with
$\R \times \Bad $ where $\Bad$ is the set of badly approximable
numbers. That is, $\Bad(0,1)$ consists of pairs $(\a,\b)$ with $\a
\in \R$ and $\b \in \Bad$. If $j=0$ the roles of $\a$ and $\b$
are reversed. We prove that the set $\Bad(1,0)\cap \Bad(0,1) \cap
\Bad(i,j)$ has Hausdorff dimension 2, i.e. full dimension. The
method easily generalizes to give analogous statements in higher
dimensions.

**February 15
Mark Watkins (PSU), 10:10, 103 Mcallister: NOTE earlier
time.**

*Special values of L-functions
and modular parametrisations of elliptic curves*

Formulae which relate L-values to arithmetic information can
be viewed in two directions: you can first compute the arithmetic
information to determine the size of the L-value, or conversely
you can compute the special L-value by a separate method, thus
gaining arithmetic information. The generic method for
computing special L-values (actually any L-value) goes back to
Cohen and Zagier in the 1970s, but only recently has it appeared
in full generality in print (appendix of Cohen's latest book).
We describe how their method works, and then use it in a specific
example, namely the computation of the degree of modular
parametrisation of an elliptic curve. In fact, we give data from
a large-scale project to compute modular degrees, with over 40000
curves considered.

**February 22
Edward Formanek (PSU)**

*A relation between
the Bezoutian and the Jacobian*

**March 1
Trevor Wooley (University of Michigan) : Colloquium**

*Slim exceptional sets
in Waring's problem*

A result of Hua from 1938
shows that the expected asymptotic formula holds in Waring's
problem for sums of four squares of primes for almost all
integers in the expected residue classes, in the sense that the
number of exceptions up to N is O(N(log N)^{-A}). This estimate
was recently improved by Liu and Liu to O(N^{13/15+epsilon}).
From a naive viewpoint, both conclusions are surprisingly weak,
in the sense that a similar conclusion holds already for sums of
three squares of primes, and the excess square of a prime brings
a negligible improvement in the estimate for the
exceptional set. This phenomenon permeates the subject,
especially when the variables under consideration are from such
exotic sets as the prime numbers or integers possessing only
small prime factors. We present a method for better exploiting
excessive variables, especially exotic variables, and thereby
slim down the available estimates for associated exceptional sets
in various problems of Waring type. By way of illustration, we
establish that the above exponent 13/15 may now be replaced by 13/30.
As with the best miracle diet plans, this slimming process

involves almost no effort.

**March 15
Tonghai Yang (University of Wisconsin)**

*Taylor expansion of
an Eisenstein series*

In this talk, we will give
an analogue of the well-known Kronecker limit formula for a
classical Eisenstein series (with character). In this case, the
Eisenstein series is holomorphic at its center and its central
value is given by theta functions via the Siegel-Weil formula. We
will give an explicit formula for its central derivative. We will
also use the formula to compute the central derivative of certain
Hecke L-functions, which are related to CM elliptic curves.

**March 22
Joel Anderson (PSU)**

**March 29
Scott Parsell (Texas A&M)**

*Pairs of additive
equations and inequalities*

We will discuss recent
progress on obtaining upper bounds for the number of variables
required to ensure that a pair of diagonal forms of differing
degree, satisfying appropriate local solubility conditions, has a
non-trivial integral zero. The arguments are based on the
Hardy-Littlewood method, and in particular on the iterative
methods of Vaughan and Wooley for estimating mean values of
exponential sums over smooth numbers. Our estimates can
also be applied to the corresponding problem for inequalities, in
which one tries to show that two forms with real coefficients
assume arbitrarily small values simultaneously at integral points.

**April 5
David Farmer (Bucknell)**

*Deformation of Maass forms*

Phillips and Sarnak conjecture that Maass forms on cofinite
subgroups of SL(2,R) are destroyed by almost all deformations of
the group. Some calculations will be described which
indicate that "almost all" cannot be replaced by "all."
Time permitting, the dynamics of the motion of the Maass forms
under deformation will also be discussed.

**April 12
William Stein (Harvard)**

*Visibility of Mordell-Weil groups*

I will introduce the notions of visibility and modularity of
Mordell-Weil groups of abelian varieties. My notion of
visibility is analogous and dual to Barry Mazur's notion of
visibility of Shafarevich-Tate groups. In my talk, I will
make conjectures about visibility of Mordell-Weil groups, prove
that Mordell-Weil groups of certain elliptic curves are visible
in an appropriate restriction of scalars, and give some explicit
examples. If time permits, I will discuss connections with
the Birch and Swinnerton-Dyer conjecture for elliptic curves of
analytic rank greater than one.

**April 19
Wenzhi Luo (Ohio State University)**

*Equidistribution of
Hecke eigenforms on modular surface*

For the holomorphic Hecke eigenforms of weight 2k, one can associate with it naturally a probability measure $mu _{k}$ on the modular surface X. We show that

\mu _{k}(A) = \mu (A) + O(k^{-1/2})

holds uniformly for any set
A on X as k tends to infinity, where $\mu$ is the invariant
measure associated to the Poincare metric. Moreover the above
decay rate is sharp. This equidistribution property of Hecke
eigenforms can be regarded as an analogue of ergodicity of
Laplacian eigenfunctions.

**April 26
David Boyd (University of British Columbia): Colloquium**

*Mahler's measure and the Bloch group*

** **