 Math 5385, Introduction to Computational Algebraic Geometry
 Lectures:
 MW 2:30 p.m.  3:45 p.m. in Vincent Hall 207
 Prerequisites:
 Math 2263 or 2374 or 2573, Math 2243 or 2373 or 2574
 Instructor:
 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).
 Textbook:
 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.
 Description:
 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 at the Undergraduate Math page 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.
 Homework:
 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 Wednesday 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.
 Projects:
 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.
 Grading:
 Your grade is based on homework and exams,
which will be weighted as follows:
Homework 60% Projects 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.
 Reading:
 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.
 Workload:
 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.
 Disabilities:
 Students with disabilities who will be taking this course and may need disabilityrelated 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 01172018 Overview Fields and rings [IVA] §A1.1 Affine space [IVA] §1.1 01222018 Affine varieties [IVA] §1.2 Email Parametrizations [IVA] §1.3 01242018 Ideals [IVA] §1.4 Problem Set 1 Projects M2 01292018 Polynomials in one variable [IVA §1.5, §A1.2 Monomials [IVA] §2.2 01312018 Division algorithm [IVA] §2.3 Problem Set 2 02052018 Chain conditions [IVA] §2.5 Topic due by email (5 p.m.) Gröbner basics [IVA] §2.6 02072018 Buchberger's criterion [IVA] §2.7 Problem Set 3 02122018 Projections and graphs [IVA] §3.2 Implicitization [IVA] §3.3 02142018 Rational maps [IVA] §3.4 Problem Set 4 02192018 Applications [IVA] §2.8 Outline Common roots [IVA] §3.5 02212018 Resultants [IVA] §3.5 Problem Set 5 02262018 Elimination theory [IVA] §3.6 Extension theorem [IVA] §3.6 02282018 Nullstellensatz [IVA] §4.1 Problem Set 6 03052018 Radicals [IVA] §4.2 03072018 Operations on ideals [IVA] §4.3 Problem Set 7 Spring break 03192018 Zariski closure [IVA] §4.4 03212018 Minimal decomposition [IVA] §4.6 Problem Set 8 03262018 Primary decomposition [IVA] §4.7 Rough draft (3 hard copies) 03282018 Coordinate rings [IVA] §5.4 Problem Set 9 04022018 Morphisms [IVA] §5.5 Feedback 04042018 Projective elimination [IVA] §8.5 Problem Set 10 04092018 Quadric hypersurfaces [IVA] §8.6 04112018 Hilbert functions [IVA] §9.3 Problem Set 11 04162018 Intersection multiplicities [IVA] §8.7 04182018 Bezout's theorem [IVA] §8.7 Problem Set 12 04232018 Presentations 04252018 Presentations Final paper 04302018 Presentations Problem Set 13 05022018 Presentations Presentation feedback
Links