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\title{Math 622: Numerical Algebraic Geometry}

\author{Instructor: Andrew Sommese (631-6498)\\
       sommese@nd.edu}%


\date{Time: MWF 10:40-11:30}%

\maketitle

\thispagestyle{empty} Algebraic geometry is an ancient subject,
with intimate connections to science, engineering, and the rest of
mathematics.  One original motivation for the subject is the
explicit solution of systems of polynomials - often motivated by
engineering.   In the last few years powerful numerical analysis
techniques, based on ideas from algebraic geometry, have been
developed to numerically solve polynomial systems and manipulate
the possibly positive dimensional solution sets of such systems.

This course  will cover those parts of algebraic geometry that
have been useful in the recent numerical work, plus the new
algorithms for numerically solving systems of polynomials.
Polynomial systems from engineering are often sparse, and possess
special structure, that efficient algorithms  take advantage of.

Though, previous knowledge of algebraic geometry will be useful,
this introductory course does not assume it. There will be an
emphasis on the concrete description of algebraic geometric
objects.  For the first part of the course, we will follow D.
Mumford's ``Algebraic geometry I : complex projective varieties,''
Grundlehren Math.  Wiss. 221, Springer-Verlag, New York, (1976).
Ideas from numerical analysis and several complex variables will
be developed as needed.



Specifically the will course cover the following topics and
material around the topics.
\begin{enumerate}
\item Basic correspondence, including the Nullstellenzatz, between affine
and projective algebraic sets and ideals;
\item the irreducible/primary decomposition;
\item local theory of algebraic sets;
\item sheaf cohomology and coherent algebraic sets; projection formula;
\item basic examples such as ruled surfaces and their very ample line bundles;
\item numerical solution of polynomial systems; and
\item numerical manipulation of irreducible algebraic sets.
\end{enumerate}


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