Required texts:
Course Description: This semester we will study Gröbner bases, monoids and monomodules, and Hilbert functions.
Prerequisite: MAT 4/524 (Modern Algebra II). This may be waived in certain circumstances, though I can’t think of any offhand.Tentative schedule: While it is the instructor’s sincere desire to follow this schedule, a hallmark of good instruction is flexibility, as distinguished from rigidity. Therefore, the instructor may occasionally modify the schedule and content of the course, not as some sort of “right”, but as a duty to balance the tension between both presenting material sufficient to justify the course title (“Topics in Algebra”) and making the material as accessible as possible to the students in the course.
| Week | Sections (MN = “My Notes”, CCA = “Computational Commutative Algebra”) |
| 24-26 Aug | MN 10⋅1 Gaussian elimination, 10⋅2 Monomial orderings |
| 29 Aug - 2 Sep | MN 10⋅3 A triangular form for polynomial systems, 10⋅4 Nullstellensatz |
| 5-9 Sep | Labor Day, MN 10⋅5 Elementary applications, CCA 1 1⋅1 Polynomial Rings |
| 12-16 Sep | CCA 1 1⋅2 Unique Factorization, 1⋅3 Monomial Ideals and Monomial Modules |
| 19-23 Sep | CCA 1 1⋅4 Term Orderings, 1⋅5 Leading Terms |
| 26-30 Sep | CCA 1 1⋅6 The Division Algorithm, 1⋅7 Gradings |
| 3-7 Oct | CCA 1 catch-up, Test #1 |
| 10-14 Oct | CCA 1 2⋅1 Special Generation, 2⋅2 Rewrite Rules |
| 17-21 Oct | CCA 1 2⋅3 Syzygies, Fall Break |
| 24-28 Oct | CCA 1 2⋅4 Gröbner Bases of Ideals and Modules, 2⋅5 Buchberger’s Algorithm |
| 31 Oct - 4 Nov | CCA 1 2⋅6 Hilbert’s Nullstellensatz, CCA 2 4⋅1 Polynomial Rings Graded by Matrices |
| 7-11 Nov | CCA 2 4⋅2 Degree Forms and Macaulay Bases, 4⋅3 Homogenization |
| 14-18 Nov | CCA 2 4⋅4 Term Orderings of DegRev Type, 4⋅5 Homogeneous Gröbner Bases |
| 21-25 Nov | CCA 2 5⋅1 Basic Properties of Hilbert Functions, Thanksgiving break |
| 28 Nov - 2 Dec | CCA 2 5⋅2 Hibert Series, 5⋅3 Computation of Hilbert Series |
| 5 Dec - 9 Dec | CCA 2 5⋅4 Dimension, Multiplicity, and Hilbert Polynomials, catch-up |
| Week of 12 Dec | Final Exam |
Grading: Your grade will be averaged evenly between two groups of grades: in-class and homework. For the in-class work, you should expect two tests.
Late Assignments: Any assignment turned in past the specified due date and time will receive a grade lowered by ten percent for each school day late.
Makeup work: I do not give makeup tests/quizzes/etc. without an excused absence. If you must miss class, then you must also produce documentation of the reason for your absence. If you were sick, you can show me the receipt from the hospital or doctor; if you had a sports event, you can show me the schedule; if someone died, you can show me an obituary notice; if the tire on your car blew out, you can show me the receipt from the mechanic.
Type, or use a pencil. I do not accept work that is written in pen. It will receive a 10% deduction.
Mobile phones: Turn your phone off once class begins, and leave it off. If you absolutely must use the phone, please step out of class, take care of business, then return. If you use the phone in the class, I will ask you to leave. If you do not comply with this request, you will forfeit the next test. If you use the phone during a test, even as a calculator, you will forfeit the test.
Statement on academic integrity: All students at the University of Southern Mississippi are expected to demonstrate the highest levels of academic integrity in all that they do. Forms of academic dishonesty include (but are not limited to):Engaging in any of these behaviors or supporting others who do so will result in academic penalties and/or other sanctions. If a faculty member determines that a student has violated our academic integrity policy, sanctions ranging from resubmission of work to course failure may occur, including the possibility of receiving a grade of “XF” for the course, which will be on the student’s transcript with the notation “Failure due to academic misconduct.” For more details, please see the university’s Academic Integrity Policy: https://www.usm.edu/institutional-policies/policy-acaf-pro-12. Note that repeated acts of academic misconduct will lead to expulsion from the university.
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ADA Syllabus Statement If a student has a disability that qualifies under the American with Disabilities Act (ADA) and requires accommodations, he/she should contact the Office for Disability Accommodations (ODA) for information on appropriate policies and procedures. Disabilities covered by ADA may include learning, psychiatric, physical disabilities, or chronic health disorders. Students can contact ODA if they are not certain whether a medical condition/disability qualifies.
The University of
Southern Mississippi |