2021 Spring Course Syllabus - Mathematics 9072.001
Course: Mathematics 9072.001.
Course Title: Differential Topology.
How this course will be taught: Virtual.
Time: MW 10:30 am - 11:50 am.
Place: Zoom link found in Canvas.
Instructor: Matthew Stover.
Instructor Office: Virtual.
Instructor Email: mstover@temple.edu
Instructor Phone: NA.
Office Hours: By appointment.
Prerequisites: Basic knowledge of group theory, smooth manifolds, and algebraic topology, e.g., from the relevant qualifying exam courses.
Textbook: NA.
Technology specifications for this course: A working computer with a reliable internet connection, a Webcam, and audio capability. Recommended Internet Speed: 8mbps download & 5mbps upload. You can test your connection at https://www.speedtest.net. Please note: Hard-wired connections are more consistent than Wi-Fi for Zoom sessions. A scanning app such as AdobeScan or CamScanner is required as is access to Zoom and Canvas (the Canvas app is also recommended).
Limited resources are available for students who do not have the technology they need for class. Students with educational technology needs, including no computer or camera or insufficient Wifi-access, should submit a request outlining their needs using the Student Emergency Aid Fund form. The University will endeavor to meet needs, such as with a long-term loan of a laptop or Mifi device, a refurbished computer, or subsidized internet access.
Course Goals: Learn about discrete groups and hyperbolic manifolds.
Topics Covered: This is a course about explicit constructions of hyperbolic 2- and 3-manifolds. We will cover: 1. Generalities on (X,G) structures and discrete groups; basic examples 2. Hyperbolic structures on surfaces by gluing polyhedra 3. Hyperbolic structures on 3-manifolds and Thurston's Dehn surgery theorem The primary goal of this course is to provide explicit, hands-on constructions of hyperbolic manifolds of finite volume. We will start with the foundations of (X,G) structures and actions of discrete groups of isometries. Then, as a warm-up for dimension 3, we will consider cut-and-paste constructions of hyperbolic surfaces, being very precise about details that are quite often thrown under the rug. We will then put a complete hyperbolic structure on the figure-eight knot complement, introduce Dehn filling, and prove that all but finitely many Dehn fillings on the figure-eight knot complement are closed rational homology 3-spheres admitting a complete hyperbolic metric. We may explore other topics, like higher-dimensional hyperbolic manifolds, if time permits. This course will contain a great deal of active/inquiry-based learning, in addition to ordinary lecture. During most classes students will work in small groups on proving various results or working important examples, and will present their findings to the class. Previews of lecture material will be made available on Canvas, typically within 12 hours of the next class, and all lecture notes will be posted on Canvas. Lastly, the course will close with a joint project: The Whitehead link complement. Students will work together (in class and outside class) on writing a joint paper, modeled as closely as possible on a mathematical research paper. The topic of this paper will be generalizing all our results for the figure-eight knot complement for the Whitehead link complement (namely, existence of a complete hyperbolic structure of finite volume, and proving that all but finitely many Dehn fillings produce hyperbolic 3-manifolds). The goals will be to cement the key ideas behind putting complete hyperbolic structures on knot and link complements and Thurston's Dehn surgery theorem, along with gaining experience in both collaborating on mathematics and writing a mathematical paper.
Course Grading: 34% Class participation 33% Joint paper 33% Homework.
Exam Dates: NA.
Remote proctoring statement: Zoom, Proctorio or a similar proctoring tool may be used to proctor exams and quizzes in this course. These tools verify your identity and record online actions and surroundings. It is your responsibility to have the necessary government or school issued ID, a laptop or desktop computer with a reliable internet connection, the Google Chrome and Proctorio extension, a webcam/built-in camera and microphone, and system requirements for using Proctorio, Zoom, or a similar proctoring tool. Before the exam begins, the proctor may require a scan of the room in which you are taking the exam.
Attendance Policy: NA.
The University's attendance policy (opens in new tab/window) has been standardized to accommodate students who are ill or are required to self-quarantine for a period of time due to the COVID-19 pandemic.
To achieve course learning goals, students must attend in-person classes, and/or participate in classes or portions of classes that are taught remotely, to the extent that they are able. Though increased flexibility will be granted, in all cases, course assessments such as assignments, tests and exams must be completed for learning goals to be reached.
In order to facilitate contact tracing, instructors are required to ensure that attendance is recorded for each in-person meeting using an online attendance system designated by the university. Students who are exhibiting symptoms such as cough, fever, shortness of breath, muscle or body aches, headache, chills, sore throat, congestion, or new loss of taste or smell, or who have been in close contact with others who have symptoms, or who are engaging in self-quarantine at the direction of the Philadelphia Health Department, Student Health Services, or any healthcare professional, should not attend in-person classes. Students will not be required to provide formal documentation from a healthcare provider for COVID-related absences. For more information, see the Student Health Services COVID-19 site (opens in new tab/window).
It is also important to foster a respectful and productive learning environment that includes all students in our diverse community of learners. Treat your classmates and instructor with respect in all communication, class activities, and meetings. All opinions and experiences, no matter how different or controversial they may be perceived, must be respected in the tolerant spirit of academic discourse. You are encouraged to comment, question, or critique an idea but you are not to attack an individual. Our differences, some of which are outlined in the University's nondiscrimination statement (opens in a new tab/window), will add richness to this learning experience.
Please consider that sarcasm, humor and slang can be misconstrued in online interactions and generate unintended disruptions. Profanity should be avoided as should the use of all capital letters when composing responses in discussion threads, which can be construed as "shouting" online. Remember to be careful with your own and others' privacy. In general, have your behavior mirror how you would like to be treated by others.
Online Classroom Etiquette: It is expected that each student attends every class on time for the full duration of each class and behaves, in the same professional manner, as if you are in a regular classroom. This refers in particular to your location and attire. It is not appropriate to eat a large meal, drink alcohol, smoke, or get up often during an online class.
Statement on recording and distribution of recordings of class sessions: Any recordings permitted in this class can only be used for the student's personal educational use. Students are not permitted to copy, publish, or redistribute audio or video recordings of any portion of the class session to individuals who are not students in the course or academic program without the express permission of the faculty member and of any students who are recorded. Distribution without permission may be a violation of educational privacy law known as FERPA as well as certain copyright laws. Any recordings made by the instructor or university of this course are the property of Temple University.
Homework: Occasionally, I will ask for a writeup of an exercise, example, or result from in-class group work. In addition, there will be ungraded (!) "responses" to lecture previews. Typically, at least 12 hours before each class a preview of lecture material will be posted to Canvas, and an assignment will be opened on Canvas giving students the opportunity to respond to the preview before class. Responses can range from "I have seen this material before, and am pretty confident with it" to "I don't understand the statement of Proposition Z" to "a picture would really clarify this section" - anything that will help make lecture portions of class more productive and easier to follow.
Additional resources: No text is required, but some excellent resources are: Foundations of Hyperbolic Manifolds (J. Ratcliffe) Fuchsian Groups (S. Katok) Geometry of Discrete Groups (A. Beardon) These books contain all the major results we will cover, particularly Ratcliffe, which is freely available for download through the Temple Library website.
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 100 Ritter Annex (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.
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).
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).
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).