Meetings: MWF 2:10pm-3:00pm, PDSB 1025.
Instructor: Jesús A. De Loera.
email: deloera@math.ucdavis.edu
http://www.math.ucdavis.edu/~deloera/TEACHING/AI4MATHLectures/
Office hours: Monday and Wednesday after class or by appointment. My office is 1013 PDSB. I will be glad to help you with any questions, concerns, or problems.
Text and References: I will list useful references as I go along in my lectures, but here are a few useful resources.
Software: I will use MAPLE/SAGE as the software for class discussion, tests, homeworks, projects, etc. software will be allowed. A very useful resource, an e-book about MAPLE, is accessible to all UC Davis students for free in the electronic book (you do not need to buy this book!):
Maple and Mathematica :a problem solving approach for mathematics by Inna Shingareva and Carlos Lizarraga-Celaya, Springer, 2009, online resource (xviii, 483 p.).
To access the book there is a SpringerLink free to all UC campuses
e-book about MAPLE
If you wish to access the book from outside campus internet, then you can do
this using the VPN link of the library (go to the UCD library link).
Finally (NOT required) but a great text for all about MAPLE is
``An introduction to MAPLE'', by A. Heck, Springer, 2006.
Description of this Course: :
This course has two goals:
1) To introduce students to Algebraic/Symbolic
Computation. This is the part of mathematics dedicated to
algorithms where the answer is to be computed exactly.
This is complementary to the area of numerical analysis
where answers are computed with limited precision and error.
2) It is now undeniable that computers are useful tools for finding
counterexamples, discover patterns, and even proof theorems! For example, the
proof of the four color theorem, investigation of fractals, etc.
Thus, the second goal of the course is to learn how computers are useful tools
for mathematical research, experimentation and can even help to generate
formal proofs automatically. In fact, knowing how to use computers
can go a long way toward solving a math problem (e.g. see the wonderful
site of Project Euler ).
Lectures outline:
(Lecture 1) Propositional calculus, SAT solvers.
(Lectures 2,3) Algebraic Algorithms for Automatic Theorem proving.
Hilbert's Nullstellensatz, positivestellensatz
(Lectures 4,5,6) Ideals and Varieties, Multivariate systems of polynomial equations,
monomial ideals, term orders and Multivariate Division Algorithm. Elimination theory
(Lecture 7) Groebner bases and Buchberger's algorithm.
(Lecture 8-9) Automatic Conjecturing and Automatic Search for Examples.
FINAL PROJECT PRESENTATIONS
Here is the maple worksheet I showed in my introduction on the first day of the class.
Please read Computers and
the meaning of Proof this article from Scientific American
(a bit old now)
Here is the second MAPLE worksheet that I showed to introduce you to MAPLE
on our first LAB session. maple_crashcourse.
Here is a third MAPLE worksheet about univariate polynomials..
Here you can also download more code to manipulate
univariate polynomials, but not as a worksheet.
Here is pseudocode of the
extended euclidean algorithm Also you can find
a step by step example computation
with some suggestion of MAPLE commands that
will be useful when you implement it.
homework 2 Contains more problems for
the first midterm
Here is a piece of MAPLE code that should help you with using Groebner
bases
Here is a list of some review problems for the first midterm
----------------MIDTERM 1 up to here-----------------------------
homework 3 contains problems
for the second midterm.
homework 4 last set of problems
to be covered in second midterm
Here are two pieces of MAPLE code that were demonstrated in class.
The first is on how to use Groebner bases to solve system of equations
Groebner commands and applications. You
can also download code for the cyclohexane equations.
Finally here is a second worksheet with applications to
Graph theory and Logic
We are having our second Midterm on Friday May 20, at the
computer laboratory. Here are some review
problems for second midterm. Solutions for the computational
problems using MAPLE are posted here
Here is the MAPLE code that solves the numerical problems, first in
mw format,
----------------MIDTERM 2 up to here-----------------------------
homework 5 Final homework!
We learned about Hilbert's Nullstellensatz and its applications to
logic, graph theory, automatic theorem proving. Here are the
lecture transparencies used in class
to discuss the Linear Algebra approach to decide feasibility of
polynomial systems
Here are the MAPLE solutions for
problems 3, 2B,2A of midterm 2. Here is the text version.
-----------------------------------------------------------------
Our final exam will be a final project. You can choose from
one of the 2 projects proposed in class (logic computation or
integer programming) OR you can make your own project! For details
and rules please consult the MAPLE
worksheet about the final projects . NOTE: You must receive
my approval if you are doing a project you designed.
You can find my transparencies about the
logic satisfiability problem Davis Putnam and NP-completeness .
Even ore information about the Davis-Putnam algorithm implementation to solve logic
satisfiability problem is also available. You can also download the
brilliant survey lecture by Bernd Sturmfels
about Groebner bases . It includes our short introduction to
integer programming.
All final Projects are due at 12:00noon June 9. They MUST be submitted
in a single maple worksheet that will run at least 3 examples when
executed.
Homeworks and Handouts
and solving systems of polynomials Groebner MAPLE code .
then in a text version
(not for xmaple) text .