Spring 2017

Course number: 22156

Meeting Tu Th 11:00 - 12:15

GMCS 405

San Diego State University

I will usually be available Tu Th 1-2:30. On other days I am often available in the afternoon, but it is best to email me in advance to schedule a meeting.

SCHEDULE | |

ASSIGNMENTS | |

PROJECTS(pdf) |

We will study algebraic geometry, one of the oldest and richest areas of mathematics. During the 20th century, the theoretical and very abstract side of the subject was prominent, but with the availability of computers, the computational roots have been reinvigorated. This course will develop the theory behind the computational tools.

What is algebraic geometry? Think back to high-school algebra where
you graphed polynomial equations and perhaps found the intersection
of plane curves defined by a line and a parabola or more general
curves defined by polynomials.
Now think about higher dimensional space and consider intersections of
hyper-surfaces defined by polynomial equations.
Such objects are called *algebraic sets* or *algebraic varieties*.
What is the dimension? How many components are there? What is the
simplest way to describe the intersection? These are some
of the *geometric questions* arising in algebraic geometry.

The fundamental result in algebraic geometry is the algebra-geometry
"dictionary" which gives a precise relationship between geometrical
objects and algebraic ones: between *varieties* in n-dimensional
space and *radical
ideals* in the polynomial ring in n variables.
Algebra provides tools for formalizing and being precise about
geometric concepts, which can be rather intuitive.
Conversely, algebraic results have a geometric interpretation that
brings richness to abstract formulas.

The fundamental tools in computational algebraic geometry are Groebner bases for ideals and Buchberger's algorithm to compute them. Groebner bases are a generalization of the greatest common divisor of integers. Just as the Euclidean algorithm may be used to compute the gcd, Buchberger's algorithm is used to compute a Grobner basis for an ideal.

In the last few decades, numerous applications of algebraic geometry have been discovered: in coding theory, cryptography, robotics, object recognition, engineering, genomics etc. Some links that show the scope of recent work are: The Society for Industrial and Applied Mathematics Activity Group on Algebraic Geometry, The Special Semester on Grobner Bases and Related Methods; The Thematic Year on Applications of Algebraic Geometry at the Institute for Mathematics and Its Applications; and the work of Bernd Sturmfels. Powerful computational software has also been developed. See for example Sage , Macaulay 2, Singular, and Magma. These computational tools are of great importance in application.

Cox, Little, O'Shea * Ideals, Varieties, and Algorithms: An
Introduction to Computational Algebraic Geometry and Commutative
Algebra* 4th Ed. 2015.

William A. Stein et al. Sage Mathematics Software The Sage Development Team, 2011, http:www.sagemath.org

The text is a well written book that is one of the standard references in computational algebraic geometry. The authors just won the 2016 American Mathematical Society Steele Prize for Mathematical Exposition We will cover the core material on Grobner bases (chapters 1-3), the algebra-geometry dictionary (chapter 4) and, in less detail, functions on a variety (chapter 7), and projective space (chapter 8). I will include some material on general (commutative) ring theory. Student interest will also guide the course.

Sage is an open source mathematics software package that incorporates numerous other open-source packages into a unified package. The Sage tutorial will help you get started.

- Properties of the Integers: The division theorem and divisibility, the Euclidean algorithm, unique factorization, modular arithmetic.
- Polynomial Ring in One Variable: The division theorem, greatest common divisor, the Euclidean algorithm, unique factorization. The correspondence between factors and roots. Polynomial rings modulo a polynomial.
- Commutative Rings and ideals: The general language of rings and ideals. Integral domains, the quotient of an integral domain by an ideal, homomorphisms.
- Linear Algebra: nullspace, subspace, dimension, basis.

There will be two exams: Tues Feb 21 and Thurs Mar 23.

There will be a final project, with a great deal of latitude in choice of topic. You may focus on theoretical questions, implementation of an algorithm, an applied problem, or some combination. I have plenty of references, including recent research, that should be accessible to you by the end of the course. You may also develop an educational module for advanced high-school students. More information about the project will be provided later in the semester. The final grade will be apportioned as indicated in the table +/- 5 points for each item.

Problem Sets | 30% |

Midterms | 40% |

Final Project | 30% |