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.
Note special venue: This is Tuesday, 1:25pm, 107 Wartik.
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.