[Alta-Logic] FW: PIMS Distinguished Lecturer - Dr. Noam Elkies - Jan 14 and 17

[Kristine Bauer]

Richard's recent advertisement made me remember that some of the Peripatetic seminar participants might be interested in these two talks as well.  See abstracts below.

Cheers,
Kristine


PIMS Number Theory CRG Distinguished Lecture<http://math.ucalgary.ca/news-events/events/pims-events/pims/crg/areas-rational-triangles-pims-crg-number-theory-distinguished-lecture>

January 17, 2011    15:00
ICT 114

On the areas of rational triangles
Dr. Dr. Noam Elkies, Harvard University
Abstract:
 By a "rational triangle" we mean a plane triangle whose sides are  rational numbers.  By Heron's formula, there exists such a triangle  of area sqrt(a) if and only if  a > 0  and  x y z (x + y + z) = a  for some rationals x, y, z.  In a 1749 letter to Goldbach,  Euler constructed infinitely many such  (x, y, z)  for any rational  $a$ (positive or not), remarking that it cost him much effort, but not  explaining his method.  We suggest one approach, using only tools  available to Euler, that he might have taken, and use this approach  to construct several other infinite families of solutions.
 We then reconsider the problem as a question in arithmetic geometry:  xyz(x+y+z) = a  gives a K3 surface, and each family of solutions is  a singular rational curve on that surface defined over Q.  The structure of the Neron-Severi group of that K3 surface  explains why the problem is unusually hard.  Along the way  we also encounter the Niemeier lattices (the even unimodular  lattices in R^24) and the non-Hamiltonian Petersen graph.

Previously advertised…..

January 14, 2011  15:00
ICT 114

How many points can a curve have?
Dr. Noam Elkies, Harvard University

 Abstract:
 Diophantine equations, one of the oldest topics of  mathematical research, remain the object of intense and fruitful study.  A rational solution to a system of algebraic equations is tantamount to  a point with rational coordinates (briefly, a "rational point") on  the corresponding algebraic variety V.  Already for V of dimension 1  (an "algebraic curve"), many natural theoretical and computational  questions remain open, especially when the genus g of V exceeds 1.  (The genus is a natural measure of the complexity of V; for example,  if P is a nonconstant polynomial without repeated roots then the equation  y^2 = P(x) gives a curve of genus g iff P has degree 2g+1 or 2g+2.)  Faltings famously proved that if g>1 then the set of rational points  is finite (Mordell's conjecture), but left open the question of how  its size can vary with V, even for fixed g.  Even for g=2 there are  curves with literally hundreds of points; is the number unbounded?

 We briefly review the structure of rational points on curves of  genus 0 and 1, and then report on relevant work since Faltings on  points on curves of given genus g>1.














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