David
A. Cox,
Amherst College
Abstract: This lecture will explore the algebra that arises whendealing
with parametric curves in the plane and parametric
surfaces in 3-dimensional space. These are
standard objects in
computer-aided geometric design and have been used for the Boeing 777
and the Guggenheim Museum in Bilbao. The
surprise is that theseparametrizations lead to substantial questions in
commutative
algebra, including the Hilbert Syzygy Theorem,
the Serre Conjecture, local complete intersections, and resultants.
Abstract: When a system of polynomial
equations in several variableshas only finitely solutions over the complex numbers, one can
find
the solutions using eigenvalue methods
from linear algebra. From
the symbolic point of view, the key player is the characteristic polynomial. The surprise is that
these characteristic
polynomials have purely algebraic applications, including primary
decomposition, factoring of number fields, and Galois
theory.
Abstract: An important topic in
commutative algebra is the Reesalgebra of an ideal in a commutative
ring containing a
field. This graded algebra encodes a lot of
information about the ideal andcorresponds geometrically to a
blow-up. One can represent
the Rees algebra as the quotient of a
polynomial ring by an ideal, in whichcase the ideal is called the
"relation ideal" of the Rees algebra. The surprise is that the relation
ideal was discovered
independentlyby computer scientists working in computer-aided geometric design,where they used the term "moving curve ideal" when dealing with parametrized curves. This
lecture will explore the structure
of thisideal. There are magical determinantal formulas for
implicitization that arise from the interpretation of the moving curve ideal in
terms of saturation. We will also discuss how the singularities of the parametrized curve influence the moving curve ideal. A second surprise will be the appearance of
adjoint curves from the
classical theory of plane curves.