Algebra and Number Theory Seminar, Spring 2003

Thursdays, 11:15-12:05, Room 116 McAllister.

         


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.