2022 Fall Course Syllabus - Mathematics 9100.001

Course: Mathematics 9100.001.

Course Title: Topics in Algebra.

Credits: 3.

How this course will be taught: In person (hopefully).

Time: Mondays, Wednesdays 9-10:20am.

Place: Wachman 617.

Instructor: Vasily Dolgushev.

Instructor Office: Wachman 530.

Instructor Email: vasily.dolgushev@temple.edu

Instructor Phone: 215-204-7287.

Office Hours: Wednesdays 3-4:30pm, Fridays, 10:30am-12pm and by appointment.

Prerequisites: Math 8011 and 8012 or permission of instructor.

Course Materials: For the first part of the course, I will mostly use J.E. Humphrey's book "Introduction to Lie Algebras and Representation Theory". For the second part of the course, I will use the book by S. Chmutov, S. Duzhin, J. Mostovoy, "Introduction to Vassiliev Knot Invariants". The following books are also very useful: M.W. Lorenz, "A Tour of Representation Theory"; J.-P. Serre, "Complex Semisimple Lie Algebras"; J.-P. Serre, "Lie Algebras and Lie Groups".

Course Goals: To master the standard toolbox of Lie algebras and study its applications to finite type invariants of knots.

Topics Covered: This course is divided into two parts. The first part is devoted the standard toolbox of Lie algebras: basics about Lie algebras, nilpotent, solvable and semisimple Lie algebras, Engel's theorem, Lie's theorem, Levi decomposition, the universal enveloping algebra of a Lie algebra, the PBW theorem and free Lie algebras. In the first part of the course, I will also talk about the classification of semisimple Lie algebras (over algebraically closed fields of characteristic zero) and touch some aspects of representation theory of Lie algebras. The second part of the course is devoted to finite type invariants of knots. In this part, we will introduce chord diagrams, open and closed Jacobi diagrams and describe "links" between these objects. At the end of the course, I will show how Lie algebras can be used to study the objects related to finite type invariants of knots. If time permits, we will have sessions about Python packages for symbolic computing related to the topics of the course.

Course Grading: Your grade will be base on the homework assignments.

Exam Dates: Not applicable.

Attendance Policy: I am not going to monitor the attendance.

Attendance and Your Health: To achieve course learning goals, students must attend and participate in classes, according to your instructors' requirements. However, if you feel unwell or if you are under quarantine or in isolation because you have been exposed to the virus or tested positive for it, you should not come to campus or attend in-person classes or activities. Students have the responsibility to contact their instructors to create a plan for participation and engagement in the course as soon as they are able to do so and to make a plan to complete all assignments in a timely fashion when illness delays their completion.

Canvas page: I will use the Canvas page of this course to post homework assignments, announcements and additional materials.

Homework: Homework assignments (with due dates) will be posted on the canvas page of this course. Please feel free to submit the hard copy of your assignment. You may work together on homework assignments, but I expect everybody to write up their own solutions; obvious copies will not be accepted. In your homework assignments, please justify all your steps and please write legibly. Not justifying steps or not writing legibly may reduce your score. The lowest grade for the homework assignment will be dropped.

Late homework assignments: Please try your best to submit your homework assignments on or before the due date. Occasionally, I may be able to give extensions for your homework assignments.

Letter Grades: 93-100 A, 90-92 A-, 87-89 B+, 83-86 B, 80-82 B-, 70-79 C, 50-69 D, 0-49 F.

Disability Statement: Any student who has a need for accommodations based on the impact of a documented disability or medical condition should contact Disability Resources and Services (DRS) in Howard Gittis Student Center South, Rm 420 (drs@temple.edu; 215-204-1280) to request accommodations and learn more about the resources available to you. If you have a DRS accommodation letter to share with me, or you would like to discuss your accommodations, please contact me as soon as practical. I will work with you and with DRS to coordinate reasonable accommodations for all students with documented disabilities. All discussions related to your accommodations will be confidential.

Academic Freedom: Freedom to teach and freedom to learn are inseparable facets of academic freedom. The University has adopted a policy on Student and Faculty Academic Rights and Responsibilities (Policy # 03.70.02) which can be accessed here (opens in new tab/window).

Add/Drop Policy: Students will be charged for a course unless dropped by the Drop/Add deadline date. Check the University calendar (opens in new tab/window) for exact dates.

During the Drop/Add period, students may drop a course with no record of the class appearing on their transcript. Students are not financially responsible for any courses dropped during this period. In the following weeks prior to or on the withdrawal date students may withdraw from a course with the grade of "W" appearing on their transcript. After the withdrawal date students may not withdraw from courses. Check the University Calendar (opens in new tab/window) for exact dates. See the full policy by clicking here (opens in new tab/window).

Incomplete Policy: The grade "I" (an "incomplete") is only given if students cannot complete the course work due to circumstances beyond their control. It is necessary for the student to have completed the majority of the course work with a passing average and to sign an incomplete contract which clearly states what is left for the student to do and the deadline by which the work must be completed. The incomplete contract must also include a default grade that will be used in case the "I" grade is not resolved by the agreed deadline. See the full policy by clicking here (opens in new tab/window).

Student Support Services: The following academic support services are available to students (all links open in a new tab/window):
    The Math Consulting Center
    Student Success Center
    University Libraries
    Undergraduate Research Support
    Career Center
    Tuttleman Counseling Services
    Disability Resources and Services
If you are experiencing food insecurity or financial struggles, Temple provides resources and support. Notably, the Temple University Cherry Pantry and the Temple University Emergency Student Aid Program are in operation as well as a variety of resources from the Division of Student Affairs.