Help @ARC

The ARC provides the following free services:
- Peer tutoring for a wide range of courses
- Exam reviews
- Supplement Instruction
- Workshops and Seminars
- Group study
- Computing and Printing
Location: Hermann Hall Building-First Floor (Northwest Corner) Room HH-112
Telephone: (312) 567-5216
Email: arc@iit.edu

Software help

If you decide to use Macaulay2, you might want to consult a chapter by Bernd Sturmfels from a book on Macaulay2.
Information on how to use Mathematica/Maple for computations with Gröbner bases may be found in Appendix C of the textbook.
(Note: Maple packages tend to be rather slow in comparison with a dedicated system such as Macaulay2.)

Help with typing math: TeX, etc.

You are encouraged to type your assignments. You can access LaTeX in the computer labs; more information and help can be found on this departmental page. Note: for Macs, I recommend TeXShop.
You might also consider using the what-you-see-is-what-you-get text editor TeXmacs; it makes it unnecessary for you to learn the LaTeX typesetting language while producing output of comparable quality. The program is freely downloadable, available for various platforms, able to import and export LaTeX files, and offers a plugin for Macaulay 2.

Math 431/530: Computational Algebraic Geometry // Applied/Computational Algebra

Homework schedule

Homework assignments will be posted at least one week before the due date. It is your responsibility to check the course page on Piazza, which you can easily access by logging into Blackboard, to obtain the assigned problems.

You are expected to start working on the homework sets early (not the day they are due or right before). It is extremely difficult to answer last-minute homework questions; particularly if you have not been participating in the Piazza discussion beforehand.

Help with writing up assignments

To improve your mathematical writing quickly, start by writing draft solutions to homework early. A day or two later after you have had time to forget what you wrote, read it. If it doesn’t make sense or convince you, rewrite it. Writing a solution requires saying what you mean and meaning what you say. Be intellectually honest. Intellectual dishonesty includes: 1) stating a “reason” without understanding its relevance. 2) Claiming a conclusion when you know you haven’t proved it. 3) Giving an example and claiming you have proved the statement for all instances.
(This text borrowed from Prof. Kaul)

Lecture schedule

You are expected to cover (at least at a high level) the assigned readings before coming to the lecture. This will help you follow the course and organize your notes. In the reading schedule below, "CLO" refers to Cox, Little and O'Shea, Ideals, Varieties and Algorithms.

Homework problem sets are naturally related to the material covered in the course; hence, homework numbers are listed next to the corresponding topic.

Dates   Tentative topics covered Assigned reading Related homework
January 10&12 Topic 1: What is applied algebra? Preliminaries: basic introduction to algebraic structures (fields, rings). Polynomials and affine spaces. CLO: 1.1. Homework 1, due 1/17.
January 17&19 Topic 2: Affine varieties and their parametrizations. CLO: Sections 1.2 (finish) and 1.3 (start and mostly cover). Homework 2, due 1/24.
January 24&26 Topic 3: Ideals. CLO 1.4. Homework 3, due 1/31.
Jan31&Feb2 Topic 4: Polynomials in one variable. [Or: ways in which polynomial rings are like integers.]
Every ideal in k[x] is principal. Introduction to algorithms/pseudocode.
Brief introduction to Chapter 2.
QUIZ 1 in class this week. 		CLO 1.5. Homework 4, due 2/7.
February 7&9 Topic 5: Chapter 2: Background needed for Groebner basics:
Monomial orderings and division algorithm in many variables. 		CLO 2.1, 2.2, 2.3. Homework 5, due 2/14.
February 14&16 Topic 6: Chapter 2: Monomial ideals and Dicson's Lemma; The Hilbert basis theorem and consequences (including the ascending chain condition).
2/14: Brief project discussion.
2/17: DEADLINE to form project groups. 		CLO 2.4, 2.5 Homework 6, due 2/21.
February 21&23 Topic 7: Groebner bases and their properties. S-pairs.
2/23: Project Outline due. *EXTENDED TO 2/24, 2PM* 		CLO 2.5, 2.6. Homework 7, due 2/28.
February 28 & March 2 Topic 8: Buchberger's algorithm and first applicatin of Groebner bases. CLO 2.7 and 2.8. Homework 8, due 3/21.
March 7&9 Topic 8: continued from previous week; closing of chapter 2. Intro to Elimination Theory (start of 3.1).
Planned date for the MIDTERM EXAM: Tuesday March 7.
In class, details tbd. 		Topics Chapters 1 and 2, except material covered week immediately before the exam. Relevant homework sets: 1-7.
March 14&16 Spring break!
March 21&23 Topic 9: Elimination and extension theorems. Geometry of elimination. Implicitization problem. CLO Sections 3.1, 3.2, 3.3. Homework 9, due 3/28.
March 28&30 Topic 10: Implicitization problem and algorithms for polynomial and rational implicitization.
3/30: Project Rough Draft due. 		CLO section 3.3. Course notes essential; focus on examples! Homework 10, due 4/11.
April 4&6 Topic 10: Continued: Resultants and proof of Extension Theorem. Topic 11: Hilbert's NullStellenSatz
4/6: Feedback on Project Rough Draft due.
No other HW due this week!
NOTE: Graded project drafts will be available for pickup Friday 4/7 from 10am until 3pm. 		CLO 3.5, 3.6, and 4.1. Homework 10, due 4/11.
April 11&13 Topic 11, continued: Hilbert's NullStellenSatz Topic 12: Radical ideals, ideal-variety correspondence, radical membership. CLO 4.1 and 4.2. Homework 11 (short HW + extra credit) due 4/25.
April 18&20 Topic 13: Ideal operations.
4/18: Final Project Paper due. 		CLO 4.3. Homework 11 (short HW + extra credit) due 4/25.
April 25&27 Topic 13: Irreducible varieties, decompositions, and Zariski closure (time permitting).
4/27: Feedback on Final Project Paper due. 		part of CLO 4.4 and 4.5.
Comprehensive Final Exam 2:00 pm - 4:00 pm F RE 258 May 05, 2017