18.704 fall 2008

Seminar in Algebra and Number Theory

Computational Commutative Algebra and Algebraic Geometry

Web pages: http://web.mit.edu/18.704/www or http://math.mit.edu/18.704.

Meetings: TR 1-2:30; Room 2-255.

Instructor: Steven L. KLEIMAN, 2-278, x3-4996, kleiman@math.mit.edu.

Instructor's office hours: TR 8:30-9:30 and 12:30-2:30 and by appointment.

TA: Martina Balagovic martinab@math.mit.edu.

TA's office hours: 8:00-9:00pm on Wednesdays in 2-251, and by appointment.

Text: "Ideals, Varieties, and Algorithms" by Cox, Little, and O'Shea, UTM Springer, third edition, 2007. Google books has most of the book online at the URL

http://books.google.com/books?id=CTzRhJjLTJ0C&dq=using+algebraic +geometry&pg=PP1&ots=QsAPebPWsF&sig=CgUReyDJrX6IyAV3lrFlAwdypiU&prev=htt p://www.google.com/search%3Fhl%3Den%26safe%3Doff%26q%3Dusing% 2Balgebraic%2Bgeometry%26btnG%3DSearch&sa=X&oi=print&ct=title&cad=one- book-with-thumbnail

Grades: Based equally on classwork, homework, and the term paper; no exams or final. Written work work will be graded in part on the quality of the writing. The term paper is to be ten-pages long in TeX: a full outline is due by Thursday, 30 October; a complete first draft, by Thursday, 20 November; and a revised final draft, by Tuesday, 9 December.

Homework: Problems with numbers between braces are to be written up formally in TeX and passed in by Thursday of the week after they are assigned; they may be emailed either in TeX form or dvi form directly to the TA.

Assignments
Date First lecture Second lecture
1. R. 9/4 Steve Kleiman Sec. 1-1p.5: 2, {6b} Steve Kleiman Sec. 1-2 p.12: {6}, 8, 10
2. T. 9/9Steve Kleiman Sec. 1-3p.22: 1,3,4, {11} ShinnYih Huang Sec. 1-4p.34: 3b, {9}, 10
3. R. 9/11Yongjin Kim Sec. 1-5p.46: {9}, 10 Charles Tam Sec. 2-1,2p.52: 5; p.60: {2}, 10
4. T. 9/16Oleg Kozhushnyan Sec. 2-3p.68: {1} Jiawei Chiu Sec. 2-4p.73: 3, {9}
5. R. 9/18Ye Luo Sec. 2-5p.80: {1}, 10 Qingchun Ren Sec. 2-6p.87: 1, {12}
6. T. 9/23Amrik Kochhar Sec. 2-7p.94: 2a, {3a} Pablo Spivakovsky-Gonzalez slides Sec. 2-8p.100: {1}, 7
7. R. 9/25Steve Kleiman Sec. 3-1p.121: 1, {6} Yongjin Kim Sec. 3-2p.127: {3}, 5
8. T. 9/30Charles Tam Sec. 3-3p.135: 9, {11} Oleg Kozhushnyan Sec. 3-4p.148: 4, {10}
9. R. 10/2Jiawei Chiu pp.150--153botp.159: 1, {4} Ye Luo pp.153bot--159p.159: 11, {17}
10. T. 10/7Qingchun Ren Sec. 3-6p.166: {2}, 3, 8 Amrik Kochhar Sec. 4-1p.174: 1, {2}
11. R. 10/9Pablo Spivakovsky-Gonzalez slides Sec. 4-2p.182: 2, {7a} ShinnYih Huang Sec. 4-3p.191: 1, {11a-d}
T. 10/14 Columbus Day -- No class
12. R. 10/16Steve Kleiman Sec. 4-4p.197: {1}, 4 Charles Tam Sec. 4-5p.203: 5, {12}
13. T. 10/21Oleg Kozhushnyan Sec. 4-6p.209: 1, {9} Jiawei Chiu Sec. 5-1p.220: 8, 10
14. R. 10/23 Ye Luo Sec. 5-2p.228: 1, 6 Qingchun Ren Sec. 5-3p.237: 6, {8}
15. T. 10/28Pablo Spivakovsky-Gonzalez slides Sec. 5-4p.246: 8, 9 Amrik Kochhar Sec. 5-5p.256: {9}, 10
16. R. 10/30Steve Kleiman Sec. 5-6. Outline due.p.263: {5}, 8 ShinnYih Huang Sec. 6-1,2p.269: 4; p.277: {4}
17. T. 11/4Charles Tam Sec. 6-3p.287: {3}, 6 Oleg Kozhushnyan Sec. 6-4p.303: 9, 13d
18. R. 11/6ShinnYih Huang Sec. 6-5p.315: 3, 6a Jiawei Chiu Sec. 7-1p.324: {8}, 14
19. T. 11/11 Ye Luo Sec. 7-2p.333: 11, 14 Qingchun Ren Sec. 7-3p.342: {5}, 7
20. R. 11/13 Amrik Kochhar Sec. 7-4p.354: 6, 13 Pablo Spivakovsky-Gonzalez pp.357--371botp.375: 4, {7}
21. T. 11/18 ShinnYih Huang pp.371bot--385. First draft due. p.385: 15; p.385:{10} Charles Tam Sec. 8-4p.391: {2}, 9
22. R. 11/20 Oleg Kozhushnyan pp.393-400topp.406: 7, {9} Jiawei Chiu / Charles Tam pp.400top-406p.406: 17, 18
23. T. 11/25 Ye Luo Sec. 8-6p.419: 7, 10, {15} Qingchun Ren Sec. 9-1p.442: {2}, 5
R. 11/27 Thanksgiving -- No class
24. T. 12/2 Pablo Spivakovsky-Gonzalez Sec. 9-2p.453: 3, 12 Amrik Kochhar Sec. 9-3p.465: 10, {12}
25. R. 12/4 ShinnYih Huang / Charles Tam Sec. 9-4. Sec. 9-4; split at Prop.5, p.471. Deadline for late homework. p.474: 6, 10 Oleg Kozhushnyan / Jiawei Chiu Sec. 9-5; split p.479midp.482: 7, 13
26. T. 12/9 Ye Luo / Qingchun Ren Sec. 9-6; split p.477mid. Final draft due. p.493: 13, 17 Pablo Spivakovsky-Gonzalez / Amrik Kochhar Sec. 9-7; split at Thm.6, p.500 p.504: 6, 13