MW 2:30 p.m. - 3:45 p.m. in Vincent Hall 207
Math 2263 or 2374 or 2573, Math 2243 or 2373 or 2574
Christine Berkesch
Office: Vincent Hall 254
Email: cberkesc -at- umn.edu (For faster response time, please include "Math5385" in the subject line.)
Office hours:
M 11:00 a.m. - 12:00 p.m., W 12:00 p.m. - 1:00 p.m., or by appointment (in Vincent Hall 254).
Ideals, Varieties, and Algorithms by David Cox, John Little, and Donal O'Shea, 2007. (On reserve in the Mathematics Library, Vincent Hall 310.) 3rd ed. errata and 4th ed. errata are here.
This is an introductory course in algorithms for solving systems of nonlinear polynomial equations; applications in geometry, algebra, and other areas; Gröbner basis methods. A suitable software package (e.g. CoCoA, Macaulay2, Singular, Sage) will be used to explore applications.
Getting help:
  • You are strongly encouraged to work in study groups and learn from each other, although all final work submitted must be your own. Lind Hall 150 is available for small group meetings and individual study.
  • Free tutoring services are available. See Tutoring Resources for more information.
  • Hire a tutor. A list of tutors is available in Vincent Hall 115 or by email request at ugrad@math.umn.edu.
  • Attend my office hours (see above).
  • Email me (above) short questions or comments. Please allow at least one working day for a response.

Weekly problem sets are posted in the course schedule. Problem sets will typically be collected at the beginning of class on Wednesday. Late homework will not be accepted, but early hard copy submission is welcomed. Your best 11 problem sets will determine your homework grade.

The homework in this course is intended to be challenging; it is training to help you build stronger mathematical muscles. There will be problems whose solutions are not immediate; there will be times that you will even need to sleep on a problem before fully grasping its solution. With this in mind, I recommend that you give yourself a full week to do the homework, so that only a few challenges remain by the Monday before the assignment is due. This gives you time to discuss any difficulties with your classmates and attend office hours in order to complete the problem set.

While you are encouraged to consult with your classmates on the homework, your final work must be your own. Copying a classmate's work constitutes plagiarism and violates the University of Minnesota Student Conduct Code. When you collaborate to reach an answer, include the name(s) of your classmate(s) with your solution.
Project info is available here. The project includes a written paper and an oral presentation.
Advice on presenting mathematics is available here: Halmos on writing, Kleiman on writing, and Halmos on talking.
Your grade is based on homework and the project, which will be weighted as follows:
Homework   60%
Project   40%
Written work:
We write to communicate. Please keep this in mind as you complete written work for this course. Work must be neat and legible in order to receive consideration. You must explain your work in order to obtain full credit; an assertion is not an answer. The logic of a proof must be completely clear in order to receive full credit.
You will find the lectures easier to follow if you spend time with your textbook before class. The course schedule section will tell you the topics for the coming class meetings. Before class, skip proofs, but seek to understand the "big idea" of each section, the key definitions, and statements of the main theorems. After class, read all statements and proofs carefully, and stop to identify useful proof techniques along the way.
You should expect to spend about 9 hours a week outside of class on reading and homework for this course. This course builds upon itself, so in order to be successful, it is important to not fall behind.
Students with disabilities who will be taking this course and may need disability-related accommodations are encouraged to make an appointment with me as soon as possible. Also, please contact UMN's Disability Services to register for support.
Academic integrity:
It is the obligation of each student to uphold the University of Minnesota Student Conduct Code regarding academic integrity. You will be asked to indicate this on your homework assignments. Students are strongly encouraged to discuss the homework problems but should write up the solutions individually. Students should acknowledge the assistance of any books, software, students, or professors.

Date                     Topic                                                             Read                                     Links                                      

01-17-2018 Overview
Fields and rings [IVA] §A1.1
Affine space [IVA] §1.1

01-22-2018 Affine varieties [IVA] §1.2 Email
Parametrizations [IVA] §1.3
01-24-2018 Ideals [IVA] §1.4 Problem Set 1
Projects M2

01-29-2018 Polynomials in one variable [IVA] §1.5, §A1.2
Monomials [IVA] §2.2
01-31-2018 Division algorithm [IVA] §2.3 Problem Set 2

02-05-2018 Chain conditions [IVA] §2.5 Topic due by email (5 p.m.)
Gröbner basics [IVA] §2.6
02-07-2018 Buchberger's criterion [IVA] §2.7 Problem Set 3

02-12-2018 Projections and graphs [IVA] §3.2
Implicitization [IVA] §3.3
02-14-2018 Singular points [IVA] §3.4 Problem Set 4

02-19-2018 Applications [IVA] §2.8 Outline
02-21-2018 Extension Theorem, Proof I [IVA] §3.5 Problem Set 5

02-26-2018 Resultants [IVA] §3.6
02-28-2018 Extension theorem, Proof II [IVA] §3.6 Problem Set 6

03-05-2018 Nullstellensatz [IVA] §4.1
03-07-2018 Radicals [IVA] §4.2 Problem Set 7

Spring break

03-19-2018 Operations on ideals [IVA] §4.3
03-21-2018 Zariski closure [IVA] §4.4 Problem Set 8

03-26-2018 Irreducible varieties and prime ideals [IVA] §4.5 Rough draft (3 hard copies)
03-28-2018 Minimal decomposition [IVA] §4.6
Primary decomposition [IVA] §4.8 Problem Set 9

04-02-2018 Projective space and projective varieties [IVA] §8.2 Feedback
04-04-2018 Projective algebra-geometry correspondence [IVA] §8.3 Problem Set 10

04-09-2018 Projective closure [IVA] §8.4
Example presentation
04-11-2018 Projective elimination [IVA] §8.5 Problem Set 11

04-16-2018 Intersection multiplicities [IVA] §8.7
04-18-2018 Bezout's theorem [IVA] §8.7 Problem Set 12

04-23-2018 Presentations
04-25-2018 Presentations Final paper

04-30-2018 Presentations Problem Set 13
05-02-2018 Presentations Presentation feedback


Christine Berkesch  ***  School of Mathematics  ***  University of Minnesota