| Section: |
23990 - MATH 399 - 01
|
| Meeting Times: |
Normally 1:00-1:50pm MWF in instructor's office.
|
| Instructor: |
Tom Kunkle, 327 RS Small,
kunklet@cofc.edu,
(843) 953-5921 (office), (843) 766-0943 (home),
kunklet.people.cofc.edu
|
| | Instructor's Office Hours: |
M 9-9:50am, T 12:30-1:30pm, W 2-3pm, F 12:00-12:55pm,
or by appointment.
|
| Text: |
Ideals, Varieties, and Algorithms, by Cox, Little, and O'Shea. Second edition or later.
We'll cover chapters 1-4 of the text and possibly parts of chapters 5,8, and 9.
Supplementary readings in abstract algebra will be assigned if necessary.
|
| Course Objectives: |
This tutorial is an introduction to elementary algebraic geometry, the study of polynomials (especially polynomials of more than one variable), systems of polynomial equations, and their solutions.
Topics covered will include
- ideals
- ideal membership and long division, Groebner bases
- elimination and extension theorems, resultants
- radical ideals and Nullstellensatz
- varieties in affine space
- unique factorization, irreducible varieties, and primary ideals
- topics in abstract algebra as needed.
Additional topics, time permitting
- quotient rings
- finite varieties
- varieties in projective space, Bezout's Theorem
|
| Student Learning Outcomes: |
The student is expected to display a through understanding of elementary algebraic geometry. In particular, she will
- complete assigned readings in Algebraic Geometry at an advanced undergraduate level and present these to the instructor in one-on-one meetngs;
- demonstrate a working understanding of the main techniques of proof in the subject;
- illustrate main ideas with examples computed both by hand and using Mathematica;
- demonstrate learning beyond the given definitions and theorems by completing of numerous theoretical exercises.
All proofs and examples completed by the student will be written in complete, gramatically and typographically correct sentences and paragraphs and typeset in TeX.
|
| Assessment: |
SLO's will be assessed through board discussions based on assigned readings
and the student's completed examples and theoretical exercises. The final outcome will be a written document in the form of a portfolio of completed exercises and proofs.
The course grade will be based on
Discussion (30%), Problem Solving (30%), and Final Portfolio (40%).
|
| Mathematics Program Student Learning Outcomes: |
This course can be used to satisfy some requrements of the undergraduate mathematics degree program,
for which there are also some standard goals; students will:
- use algebra, geometry, calculus and other track-appropriate sub-disciplines of mathematics to
model phenomena in mathematical terms;
- use algebra, geometry, calculus and other track-appropriate sub-disciplines of mathematics to derive correct answers to challenging questions by applying the models from the previous Learning Outcome; and
-
write complete, grammatically and logically correct arguments to prove their conclusions.
These outcomes will be assessed on final portfolio.
|
| Workload: |
The student is expected to work at least 150 hours, including
- discussions with the instructor in his office three times weekly, and
- solo reading, exercises, writing.
|
| Academic Integrety: |
Students in all classes will be held to
the College of Charleston's
Standards of Academic Integrity and Honor Code.
Solutions to most exercises in undergraduate mathematics can be found online.
The student may supplement the readings with other sources, but may
not substitute an internet search for actual problem solving,
nor present others' work as her own.
|
| Estimated Timetable: |
| week of 1/7 : |
1.1, 1.2, 1.3 |
| week of 1/14 : |
1.4, 1.5, 2.1 |
| week of 1/21 : |
2.2, 2.3, portfolio review |
| week of 1/28 : |
2.4, 2.5 |
| week of 2/4 : |
2.6, 2.7 |
| week of 2/11 : |
2.8, 3.1, portfolio review |
| week of 2/18 : |
3.2, 3.3 |
| week of 2/25 : |
(3.4) 3.5 |
| week of 3/4 : |
3.6, 4.1, portfolio review |
| week of 3/11 : |
4.2, 4.3 |
| week of 3/18 : |
Spring Break |
| Mar 25, 2019 is the last day to withdraw from the course with a grade of W |
| week of 3/25 : |
4.4, 4.5, portfolio review |
| week of 4/1 : |
4.6 (4.7) |
| week of 4/8 : |
to be detemined |
| week of 4/15 : |
to be detemined, portfolio review |
| week of 4/29 : |
submission of final portfolio |
|
| Assigned Problems: |
This list is subject to revision and is based on the second edition of the
text. The student will also receive credit for her solutions to
any additional problems she wishes to include in her portfolio.
* indicates an optional problem.
| 1.1: |
1, 2, 5, 6.
|
1.2: |
1-6, 8-10, 12, 15.
|
1.3: |
1-5, 6*, 8, 9, 11, 12, 14, 15, 16*.
|
| 1.4: |
1-11, 14, 15*.
|
1.5: |
1-6, 11, 12, 14-16, 17*.
|
2.1: |
1-3, 4*, 5.
|
| 2.2: |
1-5, 7, 8, 10, 11.
|
2.3: |
1-4, 6, 7, 9, 10.
|
2.4: |
1-5, 8-10.
|
| 2.5: |
1-4, 7-11, 13-17, 18*.
|
2.6: |
1-4, 5*, 6-10, 12.
|
2.7: |
1, 2, 3, 5*, 9, 10*, 11*.
|
| 2.8: |
1-5, 6*, 7, 8, 9*, 10*.
|
3.1: |
1-4, 7-9.
|
3.2: |
1-4, 5*.
|
| 3.3: |
1, 2, 4, 6, 7, 9, 10, 11*, 14*.
|
3.4: |
|
3.5: |
1-12, 15.
|
| 3.6: |
1-5, 7*, 8, 9, 10*, 11*, 13.
|
4.1: |
1, 2, 4, 5, 7, 9.
|
4.2: |
1-7, 8*, 14, 15*.
|
| 4.3: |
1-7, 8*, 9*, 10, 12, 13*, 14*, 15*.
|
4.4: |
1-5, 6*, 7*.
|
4.5: |
1,2,3,4*, 6, 7, 8, 11, 12.
|
| 4.6: |
1, 3, 4, 6*, 8*, 9.
|
4.7: |
|
|