MATH 411 – Selected Topics in Advanced Mathematics
Spring 2009: Topics in Modern Algebra and its Applications
Text: Abstract Algebra, D. S. Dummit & R. M. Foote, 2nd or 3rd ed.
Prerequisites:C or better in MAT 336 or MAT 337 or MAT 347; or consent of instructor required.
Content: The course is meant as an opportunity for students to explore several topics drawn from modern algebra, with applications in physics and other sciences.
Objectives: There are three general goals for the course: 1. To immerse students in the exploration of mathematical phenomena in order to develop habits of mathematical thinking and writing. 2. To expose students to potential research problems in mathematics. 3. To provide mathematical content from abstract algebra, with applications to geometry and other sciences.
Course Format & Learning Cycle: You are responsible for reading assignments, and this is not a trivial task. Browsing assignments before the next scheduled meeting of the class is meant to prepare you to assimilate the explanations during the lecture (you need a map before going on the field trip). During the class, we will focus on understanding the concepts and methods. After the class read the assignment. After you are done and before attempting the homework, write one page outline of the corresponding section (include major concepts, methods and relations, with brief explanations). Check you have acquired the corresponding knowledge by reading the outline. Attempt all the homework problems before the next class period, and build your list of questions. Come to class, when you should turn in the outline for one point per submission. Before discussing the new assigned material, we will discuss your difficulties or questions regarding the theory and homework problems. You may choose to present solutions to the homework problems on transparencies, for “class participation” points. Attendance and active participation are expected. .
Homework: There will be daily homework assignments. Homework will be collected on Wednesdays. Please show all the work.
Exams: Attendance at exams is mandatory. It is highly recommended that you contact me before the exam, if an emergency has arisen.
Evaluation: Grades will be based on the following points:
Homework 100
Class activities (Attendance 50, outlines 30, and participation 20): 100
Midterm Exam 100
Final exam100
Total 400
The grading scale is based on: A [90-100%], B [80-90%), C [70-80%), D [60,70%), F [0,60%).
N.B.: Students who believe they may need accommodations in this class/program are encouraged to contact the DisabilityAccessCenter as soon as possible to better ensure that such accommodations are implemented in a timely fashion.
“Warning: Plagiarism and cheating are serious offenses. Penalties can range from a minimum of a zero grade on the invalid instrument to expulsion from the University”
Topics Outline
Topics in Group Theory- Group actions, Permutation representations
- Sylow Theorems
- Direct and semi-direct products and abelian groups
- p-Groups, nilpotent groups, solvable groups
- Free groups
- Representation theory and character theory
- Modules and Vector Spaces; Graphs and the associated complex
- Cycles, boundaries and homology
- Theorems of Green, Stokes, Gauss.
Topics in Ring Theory
- Ideals and rings of fractions
- Factorization and PIDs
- The Chinese reminder Theorem
- Basic theory of field extensions, algebraic extensions
- Cyclotomic polynomials and extensions,
- Galois theory
HW / Date / Section covered / Problems assigned / HW / Date / Section covered / Problems assigned
1 / M / 1/12 / Introd. / Outline keywords (Wiki) / M / 3/9 / No
W / 4.1 / W / Classes / Spring break/ HW due next W
F / 4.2 / … / F /
2 / M / 1/19 / 4.2 / 9 / M / 3/16 / [1] i
W / 4.3 / W / [1] ii
F / 4.3 / F / [1] iii
3 / M / 1/26 / 4.3 / 10 / M / 3/23 / [2] 12.1
W / 4.4 / W / 12.2 i
F / 4.4 / F / 12.2 ii
4 / M / 2/2 / 4.4 / 11 / M / 3/30 / 12.3
W / 4.5 / W / 12.4
F / 4.5 / F / r&p
5 / M / 2/9 / 4.6 / 12 / M / 4/06 / 12.5
W / 5.1 / W / 12.6
F / 5.2 / F / r&p
6 / M / 2/16 / 5.3 / 13 / M / 4/13 / 12.7
W / 5.4 / W / 12.8
F / 5.5 (I) / F / r&p
7 / M / 2/23 / 5.5 (II) / 14 / M / 4/20 / 14.1
W / W / 14.1
F / F / r&p
8 / M / 3/2 / R&p / 15 / M / 4/27 / 14.2
W / Review / W / Review
F / Midterm Exam / F / Final Exam