
\documentclass{article}
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\title{Syllabus for Mathematics 517\\
Foundations of Computational Mathematics\\
CALL Number 5681}
\author{Instructor: Andrew Sommese}
\date{MWF at 10:40-11:30}
\maketitle
The course is a solid theoretical introduction to numerical
analysis. Topics covered will include:
\begin{enumerate}
\item Polynomial interpolation (including generalized
Hermite interpolation); two dimensional interpolation; splines;
trigonometric interpolation
\item least squares and the basic theory of orthogonal functions
\item numerical integration in one variable, including
adaptive methods; Romberg integration; Gauss quadrature and the
relations to orthogonal functions
\item numerical linear algebra
\begin{enumerate}
\item direct methods and the analysis of error based on
the condition number;
\item basic numerical factorizations of matrices and the singular value
decomposition;
\item iterative methods, e.g.,the Jacobi method and the method of
successive over-relaxation; and
\item methods to find eigenvalues and eigenvectors such as the
QR Method, power method, and inverse power method.
\end{enumerate}
\item methods to solve systems of nonlinear equations, e.g., Newton
like methods and constrained Newton's methods such as homotopy
continuation;
\item numerical solution of ordinary differential equations by
marching methods, multistep methods, finite differences, and the
finite element method (including such variants as the Galerkin and
Rayleigh-Ritz method);
\item solution of some simple partial differential equations
by difference methods;
and by the finite element method.
\end{enumerate}
We will devote at least a week to an extended discussion of what
is known about the numerical solution of systems of polynomials in
several variables.
\begin{thebibliography}{11111}

\bibitem{davis} P.J. Davis, {\em Interpolation and approximation}, Dover, 1975.
\bibitem{deBoor} C. De Boor, {\em A practical guide to splines}, Springer, 1978. .
\bibitem{IK} E. Isaacson and H.B. Keller, {\em Analysis of
numerical methods}, Dover Publications, 1994.

\bibitem{iserles}A. Iserles, {\em A first course in the numerical analysis of
differential equations}, Cambridge University Press, 1996.
\bibitem{johnson} C. Johnson, {\em Numerical solution of partial differential
equations by the finite element method}, Cambridge University Press, 1987.
\bibitem{morgan} A.P. Morgan, {\em Solving polynomial systems using continuation
for
engineering and scientific problems}, (1987) Prentice-Hall,
Englewood Cliffs, New Jersey.


\bibitem{StB}J. Stoer and R. Bulirsch, {\em Introduction to numerical analysis}, 2nd Edition,
Texts in Applied
Mathematics 12, Springer-Verlag, 1993, New York.


\bibitem{Tyr} E.E. Tyrtyshnikov, {\em A brief introduction to numerical
analysis}, Birkh\"auser, 1997.
\end{thebibliography}
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%%%%%%%%%%%%%%%%%%%%Math 622 Syllabus%%%%%%%%%%%%%%%%%%%




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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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