David A. Cox
Professor of Mathematics Emeritus
Fellow of the American Mathematical Society
Lecture Notes
Here are some postscript or pdf files containing lecture notes for
various lectures given between 2001 and 2012..

Gröbner Bases: Quick Updates and Extended Snapshots. This
lecture highlights recent developments in the theory of Gröbner
bases and gives fun applications. The lecture was given in Alcalá de
Henares, Spain, at the 2012 meeting of EACA (Encuentro de Álgebra
Computacional y Aplicaciones). Part of this lecture is based on the
Sampler of Recent Developments from my 2007 ISSAC Gröbner Bases
Tutorial (see below).

Évariste Galois and Solvable Permutation Group. This lecture
discusses the amazing work of Galois on solvable permutation groups. The
lecture was given in May 2012 at the University of the Basque Country in
Bilbao, Spain, as part of their celebration of the 200th anniversary of
the birth of Galois.

Geometric Modeling and Commutative Algebra and
The Surface Case.
These are the slides for the first lecture and part of the third lecture given at the PanAmerican Advanced Study Institute,
held in Olinda, Brazil, in August 2009.

Gröbner Bases Tutorial: The Geometry of Elimination and
Gröbner Bases Tutorial: A Sampler of Recent Developments.
These are the slides for the Gröbner bases tutorial given at ISSAC
in July 2007 in Waterloo, Canada.

Lectures on Toric Varieties. These lectures discuss
toric varieties in terms cones, fans, homogeneous coordinates and polytopes.
They also touch on some of the commutative algebra involved in toric geometry.
The lectures were written for the CIMPA School on Commutative Algebra given
in Hanoi in December 2005.

What is a Toric Variety? These notes introduce the idea of a toric
variety and discuss cones, fans, polytopes, and homogeneous coordinates.
Click
here for slides based on these notes. The slides are for a
lecture given at the Workshop on Algebraic Geometry and Geometric
Modeling to be held in Vilnius, Lithuania in the Summer of 2003.

Introduction to Algebraic Geometry. These notes cover abstract
varieties and topics such as normality and smoothness. They also discuss
Weil and Cartier divisors, invertible sheaves and line bundles. The notes
are based on lectures given in Grenoble at the Toric Summer School in the
Summer of 2000.

Minicourse on Toric Varieties. These are
notes for three lectures covering various ways to define of toric
varieties (fans, homogeneous coordinates, and toric ideals), results
related to polytopes (the DehnSommerville equations, the Ehrhart polynomial,
and the BKK bound), and an introduction to how toric varieties are used in
Mirror Symmetry (the Batyrev mirror construction). The minicourse was
given at the University of Buenos Aires in the Summer of 2001.
 What is
the Multiplicity of a Base Point? These are
the slides for an expository talk given on the definition of multiplicity
at the XIV Coloquio Latinoamericano de Algebra in
the Summer of 2001 in Cordoba, Argentina.

Newton's Method, Galois Theory, and Something You Probably Didn't Know About
A_{5}. These are
the slides for an expository talk given on the DoyleMcMullen Theorem, which
relates Newton's Method to Galois Theory. (Note that the postscript file
given here is missing the illustrations.) This lecture as given at the XIV
Coloquio Latinoamericano de Algebra in the Summer of 2001 in Cordoba,
Argentina.
Recent Papers
Click here for some of my papers, posted on the Amherst College Octagon, which is a collection of open access articles written by Amherst College faculty.
Applications of Polynomial Systemss
Click here
for the web page for my book Applications of Polynomial Systems, with contributions by Carlos D'Andrea, Alicia Dickenstein, Jonathan Hauenstein, Hal Schenck and Jessica Sidman. This book is based on 10 CBMS lectures given in June 2018 and explores some wonderful applications of algebraic geometry. It is published by the American Mathematical Society.
Toric Varieties
Click here
for the web page for my book Toric Varieties, written with John Little and Hal Schenck. This book is about a wonderful part of algebraic geometry that has deep connections with polyhedral geometry. It is published by the American Mathematical Society.
Galois Theory
Click here
for the web page for my ≈çbook Galois Theory. This book is about the wonderful interaction between group theory
and the roots of polynomials. It is now in its second edition and is published by John Wiley
& Sons. The book has been translated into Japanese.
Primes of the Form x^{2} + ny^{2}
Click here
for the web page for my book Primes of the Form x^{2}
+ ny^{2}, with contributions by Roger Lipsett. This book is about Fermat, class
field theory, and complex multiplication, and was written for
anyone who loves number theory. The third edition is published by the American Mathematical
Society and includes complete solutions, written by Roger Lipsett and David Cox.
Click here
for the web page for my book Ideals, Varieties and Algorithms,
written with John Little and Don O'Shea. This book is an introduction
to algebraic geometry and commutative algebra, and was written
for undergraduate math majors. It is now in its fourth edition
and is published by SpringerVerlag. The book has been translated into Japanese, Russian and Persian. In January 2016, Ideals, Varieties and Algorithms was awarded the Leroy P. Steele Prize for Mathematical Exposition by the American Mathematical Society.
Using Algebraic Geometry
Click here
for the web page for my book Using Algebraic Geometry,
also written with John Little and Don O'Shea. This book is an
introduction to Gröbner bases and resultants, which are two
of the main tools used in computational algebraic geometry and
commutative algebra. It also discusses local methods and syzygies,
and gives applications to integer programming, polynomial splines
and algebraic coding theory. It is published by SpringerVerlag
and is available in hardcover and paperback. The second edition appeared in the Spring of 2005.
The book has also been translated into Japanese.
Mirror Symmetry and Algebraic Geometry
Click here
for the web page for my book Mirror Symmetry and Algebraic
Geometry, written with Sheldon Katz. This monograph is an
introduction to the mathematics of mirror symmetry, with a special
emphasis on its algebrogeometric aspects. Topics covered include
the quintic threefold, toric geometry, Hodge theory, complex and
Kähler moduli, GromovWitten invariants, quantum cohomology,
localization in equivariant cohomology, and the work of
LianLiuYau and Givental on the Mirror Theorem. The book is written
for algebraic geometers and graduate students who want to learn
about mirror symmetry. It is also a reference for specialists
in the field and background reading for physicists who want to
see the mathematical underpinnings of the subject. It is published
by the American Mathematical Society.
You can contact me at
dacox@amherst.edu