**January 20
Alexandre Borovik, UMIST**
*Note special venue: This is Monday, 4:40-5:30, 115 McAlliste*

*Groups of finite Morley rank and a strange question from number
theory*

Groups of finite Morley rank (FMR) naturally appear in model theory. For example, the simple groups of FMR can be characterised as those groups which admit a satisfactory description in the language of first order logic. In more formal terms, this means that, for the group G, there is a unique, up to isomorphism, group G^* of first uncountable cardinality with the same set of valid (first order) logic formulae.

Being defined by their "uniqueness", it is natural to believe that groups of FMR should turn out to be some familiar and central objects of Mathematics. Not surprisingly, the famous Cherlin-Zilber conjecture suggests that simple groups of FMR are simple algebraic groups over algebraically closed fields.

The talk will discuss some recent results by the speaker, Altinel and
Cherlin on special cases of this conjecture. We use methods (but
not the result itself) of the Classification of Finite Simple Groups. Our
work and a remarkable result by Frank Wagner lead to some strange questions
in number theory.

**January 23
Igor Pak, MIT**

*The nature of partition bijections*

Partition bijections arise in the study of various partition identities and often give the shortest and the most elegant proofs of these identities. These bijections are then often used to generalize the identities, find "hidden symmetries", etc. But to what extend can we use these bijections? Do they always, or at least often exist, and how do you find them? Why is it that some bijections seem more important than others, and what is the underlying structure behind the "important bijections"?

I will try to cover a whole range of partition bijections and touch
upon these questions. The basis of my observations is my recent survey
on the subject. Hopefully, the talk will be somewhatentertaining.

**January 30
Michael Hirschhorn, University of New South Wales**

*Partitions of a number into four squares of equal parity*

Inspired by a conjecture of William Gosper, we investigate the number
of partitions of a number into four squares of equal parity. We find various
relations, including one that proves, and indeed sharpens, Gosper's conjecture.
We also show that the number of partitions of $72n+60$ into four odd parts
is even.

**February 6** **
Alexander Borisov, Penn State**

*Special periodic orbits of algebraic maps over finite fields*

The talk will be focused on the following conjecture. Conjecture.
Suppose X is an algebraic variety over a finite field and f: X--> X is
a dominant map. Then the set of all algebraic points x in X, such that
f(x) is conjugate to x, is Zariski dense in X.

Together with Mark Sapir, we showed that this conjecture
has interesting applications to group theory. I will discuss an approach
to it based on Deligne's results on etale cohomology and intersection theory
of Fulton.

**February 13** **No
seminar this week.**

**February 18** **Bruce
Reznick, University of Illinois at Champaign-Urbana**
*Note special venue: This is Tuesday, 1:25pm, room 115 Osmond.
Bruce will also be giving the teaching seminar at 4:00. *CANCELLED
DUE TO WEATHER

*Patterns of Dependence among Powers of Polynomials*

The ticket $T(F)$ of a finite set $F = \{f_k\}$ of polynomials is defined
to be the set of all integers $m$ so that $\{f_k^m\}$ is linearly dependent.
We discuss some families with interesting or surprising tickets. Unsurprisingly,
$|T(F)|$ is bounded by $|F|$; however, every finite set of integers can
be a ticket. The motivating example goes back to Desboves (1880).
For $k = 0,1,2,3,$ let

$$

f_k(x,y) = i^k x^2 + i^{2k}\sqrt 2 xy - i^{3k} y^2,

$$

where $i^2 = -1$. Then $\sum_{k=0}^3 f_k^m = 0$ for $m = 1,2,5$. By
the end of the seminar, it is hoped that these identities will become obvious.

**February 20 Andreas Strombergsson,
IAS**

*Equidistribution of horocycles*

In my talk, I will briefly recall the celebrated theorem by Marina Ratner
on equidistribution of unipotent flows, and some of its applications in
number theory. I will then look at the special case of the horocycle flow
on the unit tangent bundle of a hyperbolic surface, and discuss some questions
which go beyond Ratner's result. One of these questions is related to the
pair correlation statistics for the sequence n^2x modulo 1.

**February 25 Sinnou David,
L'Institut de Mathématiques de Jussieu, Université Paris 7**
*Note This is a Tuesday: Room 116 McAllister*

*On the Mordell-Lang conjecture*

We shall discuss effectivity questions around the former Mordell-Lang
conjecture on counting algebraic points of a subvariety of an abelian variety.
Beside describing what is known on the subject, we shall suggest some stronger
conjectures dealing with uniformity properties. We shall also explain links
with questions about the existence of "small" points on such varieties.

**February 27 Scott Ahlgren,
University of Illinois at Champaign-Urbana**

*Arithmetic of singular moduli and class equations*

The values of the usual j-invariant at imaginary quadratic arguments
are known as singular moduli; these are algebraic integers which play many
important roles in number theory (e.g. in class field theory and in the
theory of elliptic curves). Here we investigate divisibility properties
of traces of singular moduli. We also investigate the arithmetic properties
of class equations (i.e. the minimal polynomials of singular moduli).
(This is joint work with K. Ono.)

**March 6** **Bruce
Berndt, University of Illinois at Champaign-Urbana**

*Theorems on Partitions from a Page in Ramanujan's Lost Notebook*

On page 189 in his lost notebook, Ramanujan recorded five assertions
about partitions. Two are famous identities of Ramanujan immediately
yielding the congruences $ p(5n+4) \equiv 0 \pmod5 $ and $ p(7n+5) \equiv
0 \pmod7 $ for the partition function $ p(n)$. Two of the identities,
also originally due to Ramanujan, were rediscovered by M.~Newman, who used
the theory of modular forms to prove them. The fifth claim is false,
but Ramanujan (almost) corrected it in his unpublished manuscript on the
partition and $\tau$-functions. A complete proof of a correct version
of Ramanujan's assertion was recently given by Scott Ahlgren and Matthew
Boylan. In this talk, we indicate elementary proofs of all four correct
claims. In particular, although Ramanujan's elementary proof for
his identity implying the congruence $ p(7n+5) \equiv 0 \pmod7$ is sketched
in his unpublished manuscript on the partition and $\tau$-functions, it
has never been given in detail. This proof depends on some elementary
identities mostly found in his notebooks; new proofs of these identities
are given. This is joint work with Ae Ja Yee and Jinhee Yi.

**March 20 Robert Griess,
University of Michigan**

*Pieces of Eight*

We present a new theoretical foundation of the Leech lattice, Golay
code, Conway groups and Mathieu groups. The traditional way to see
and prove uniqueness of the Leech lattice, L, was to find a sublattice,
say M, which is orthogonally decomposable as a direct sum of rank 1 lattices,
then move from M to L by including more generators by formulas given by
the famous binary Golay code. One of the earliest uniqueness proofs
for L depended on uniqueness of the binary Golay code. We give a
uniqueness proof of the Leech lattice based on sublattices which are orthogonal
direct sums of scaled copies of the E_8-lattice. This approach implies,rather
than depends on, the uniqueness of the Golay code. Furthermore, we get
new proofs of many nice properties of Aut(L), the famous Conway group C_{O_0}
of order (2^22)(3^9)(5^4)(7^2)(11)(13)(23) which largely avoid special
counting arguments. Surprisingly, we can prove transitivity results
on configurations in L without use of "extra automorphisms" or even knowing
the order of Aut(L)! We get the existence, uniqueness and many properties
of the Golay code and Mathieu group as a corollary of our theory.
This reverses the customary logical development of these two generations
of the Happy Family.

**March 20** **Peter
Sarnak, Courant Institute**
*Note special venue: This is Thursday, 2:30, room 116 McAllister.*

*Classical versus quantum fluctuations for the modular surface*

In spite of the title this talk is all about L-functions.

**March 25
Gautam Chinta, Brown University**
*Note special venue: This is Tuesday, 2:30, room 115 McAllister*

*Non-vanishing twists of GL2 L-functions*

We discuss the problem of finding twists of a GL2 L-function by a character
of fixed order n (n>2) which are non-vanishing at the central point.
This has conjectural applications to ranks of elliptic curves via the Birch/Swinnerton-Dyer
conjecture. A result is given when n=3.

**March 27
Andrei Suslin, Northwestern University**
*On Grayson's Spectral Sequence*

The problem of constructing a spectral sequence relating algebraic K-theory
to motivic cohomology is part of Beilinson's original program of defining
"motivic cohomology" with resonable properties. This problem was resolved
(for fields) by S. Bloch and S. Lichtenbaum around 1993. Unfortunately
the preprint of Bloch and Lichtenbaum contained several minor errors and
what's worse is very hard to understand. A much clearer approach to the
construction of the motivic spectral sequence was suggested by D. Grayson.
Grayson's construction had however problems of its own: its second term
was given by certain cohomology groups which looked like motivic cohomology
groups but for a long time nobody was able to show that they
really coincide with motivic cohomology groups. In this talk we'll outline
the proof of the theorem asserting that Grayson's motivic cohomology coincides
with the usual motivic cohomology and hence Grayson's spectral sequence
gives a desired spectral sequence relating motivic cohomology to algebraic
K-theory.

**April 3
Robert Vaaughan, Penn State**

*Report on the "Elementaren und Analytische Zahlentheorie Tagung"
at Oberwolfach, 9th - 15th March 2003*

**April 10
David Terhune, Penn State**

*Double L-functions*

We generalize a result of Zagier concerning double zeta evaluations
to the double L-values. Time permitting, a method of numerical computation
of these numbers will also be discussed. This allows verification
of examples of the theorem.

**April 15
Hyman Bass, University of Michigan. Cancelled owing to indisposition.**

*The zeta function of a graph*

This talk is concerned about a generating function for the closed paths
in a finite graph. (It is a combinatorial analog of the Selberg zeta
function counting closed prime geodesics on a compact Riemann surface.)
The main theorem, which is more or less proved from scratch, says that
this function is a polynomial, and gives some information about the geometric
significance of its roots. The talk is slightly technical, but self-contained
and elementary. It is even accessible to advanced undergraduates.

**April 17
Jonathan Pila, Institute for Advanced Study, Princteon**

*Some diophantine geometry of subanalytic sets*

Let X be a compact subanalytic subset of \RR^n, and denote by tX its
homothetic dilation by t\ge 1. I will present various upper estimates for
the number of integer points on tX as t\rightarrow\infty, and for the number
of rational points on X of height \le H as H\rightarrow\infty. In particular,
when dim(X)=2, I will show that #tX(\ZZ) \le c(X,\epsilon)t^\epsilon for
all \epsilon>0 except for points that reside on a semialgebraic subset
of X of pure positive dimension. The union of such subsets I denote
X^{alg}. This result generalizes a result for dim(X)=1 obtained jointly
with E. Bombieri some time ago. I will present further conjectural estimates
in which X^{alg} plays a role as above analogous to the "special set" in
diophantine geometry.

**April 17
Jeff Lagarias, Information Sciences Research, AT&T Labs-Research**
*This is an additional lecture: 2:30pm, 116 McAllister.*

*Wavelets, Tilings, and Number Theory*

This talk considers orthonormal wavelet bases of the Hilbert space of
square-summable functions on n-dimensional Euclidean space. These are orthonormal
bases formed by translates and dilations of a single function; the Haar
basis is the prototypical example. Such wavelets are specified by a scaling
function, which is a solution of a functional difference equation, called
a dilation equation. This equation involves a dilation map which takes
x to Mx, where M is an integer n by n matrix which is expanding, meaning
all its eigenvalues are of length exceeding one. Ingrid Daubechies showed
there exist orthonormal bases of compactly supported wavelets of arbitrary
smoothness for dilations taking x to 2x on the line. Do such wavelets exist
for all dilation matrices M? We consider the case of Haar-type wavelets.
Their existence is related to radix expansions to base M having nice tiling
properties. These lead to problems in number theory, some solved and some
unsolved.

**April 18
Jeff Lagarias, Information Sciences Research, AT&T Labs-Research**
*This is an additional lecture: 9:05, 202 Osmond Laboratory.*

*De Branges Hilbert Spaces Of Entire Functions And L-functions*

This talk reviews the de Branges theory of Hilbert spaces of entire
functions, and explains its possible relevance to the study of the zeros
of Dirichlet $L$-functions. de Branges' theory involves a mixture of complex
function theory and operator theory. On the operator theory side it concerns
a class of symmetric operators of deficiency index $(1,1)$, and gives a
canonical invariant subspace decomposition for such operators. Although
this may appear a quite narrow subject, it is not. It includes a notion
of integral transform generalizing the Fourier transform. It includes as
special cases several well known theories, e.g. orthogonal polynomials
on the line.

**April 24 ** **Damien
Roy, University of Ottawa**

*Diophantine approximation in small degree*

One objective of this talk is to show that

(3+sqrt(5))/2 = 2.618...

is the optimal exponent of approximation of a transcendental real number
by algebraic integers of degree at most 3. Although it was shown
by Davenport and Schmidt in 1969 that this exponent is at least 2.618...,
the natural conjecture was that the best exponent should be 3. Surprisingly,
the same number is also the optimal exponent for a Gel'fond type criterion
in degree 2 (the natural conjecture was 2) while (-1+sqrt(5))/2 = 0.618...
is an optimal exponent for simultaneous rational approximation of a transcendental
real number and its square (the natural conjecture was 1/2). We will
explain the connections between these problems and describe some properties
of the corresponding extremal numbers

(see arXiv:math.NT/0303150).

**April 29
Ling Long, Institute for Advanced Study, Princeton**
*Note special venue: This is Tuesday, 11:15am, 116 McAllister.*

*Elliptic pencils and Torelli theorem*

An elliptic pencil is a fiber space over a Riemann sphere whose generic
fibers are elliptic curves. Elliptic K3 surfaces are examples of elliptic
pencils. The weak Torelli theorems for K3 surfaces states that two K3 surfaces
are isomorphic if there exists a Hodge isometry between the second cohomology
groups of these surfaces. We will talk about some applications of Torelli
theorems of K3 surfaces and discuss some potential generalizations of these
applications to elliptic pencils.

**April 29
Bruce Reznick, University of Illinois at Champaign-Urbana**
*Note special venue: This is Tuesday, 1:25pm, 115 Osmond.
Bruce will also be giving the teaching seminar at 4:00.*

*Patterns of Dependence among Powers of Polynomials*

The ticket $T(F)$ of a finite set $F = \{f_k\}$ of polynomials is defined
to be the set of all integers $m$ so that $\{f_k^m\}$ is linearly dependent.
We discuss some families with interesting or surprising tickets. Unsurprisingly,
$|T(F)|$ is bounded by $|F|$; however, every finite set of integers can
be a ticket. The motivating example goes back to Desboves (1880).
For $k = 0,1,2,3,$ let

$$

f_k(x,y) = i^k x^2 + i^{2k}\sqrt 2 xy - i^{3k} y^2,

$$

where $i^2 = -1$. Then $\sum_{k=0}^3 f_k^m = 0$ for $m = 1,2,5$. By
the end of the seminar, it is hoped that these identities will become obvious.

**May 1
Dorian Goldfeld, Columbia University**
*Note: Dorian is also giving the Mathematics Departmental
Colloquium today.*

*On the average number of occurrences of a generator in words in
a group*

We consider an abstract group defined by generators and relations. Every
word or element in the group can be expressed as a product of the generators,
but the representation is not unique. In certain cases the number of occurrences
of a particular generator in an arbitrary word may be a well defined function,
and it is then an interesting question to explore the average value. In
joint work with C. O'Sullivan, we introduce a new method in analytic number
theory to study this question. The main tool is the theory of Eisenstein
series twisted by modular symbols.