General information

Instructor: Doug Knowles
Email: Douglas.D.Knowles.GR@dartmouth.edu
Lectures: We will meet 3 times a week for 65 minutes in Zoom. Prior to each course period, I will post a recorded lecture about 15-20 minutes in length that will include several guided questions to think about. You may expect the recorded lectures to posted sometime in the evening of prior course period (i.e. about 36 hours before class). The course period will then consist of discussion of any questions that arose from the recorded lecture, followed by more involved proofs and examples done either as a class or in breakout rooms for smaller examples. The course periods will be recorded for any students who cannot make it for any reason.
Zoom Classroom: Click Here [Password: Math54]
Class Meeting Time: 2:10-3:15 MWF
X-period: 3:00-4:00 on Thursdays (used for oral presentation of HW)
Office Hours: Tuesdays 3-5 PM and Fridays 9-11 EST, or by appointment
Questions and Concerns: Due to the unique time constraints of this term, I will be available through email at most reasonable hours into the early evening (say 8 PM EST) for any questions or requests involving lecture timing, homework, absences, etc. I will also keep an eye on email throughout the weekend, though I'm frequently out hiking and cannot guarantee a timely response at those times.
Proofs Practice: For those who unfamiliar with, or simply want more practice with, proofs, a number of sources can be found here.
Homework Solutions: Will generally be posted by end of the weekend after an assignment is turned in, and can be found here.
Important Course Policies & Information: Are found at the bottom of the course homepage, or by clicking here.

Course Weeks

Click to See Each Week's Course Materials

Course Overview & Objectives

On a mathematical and professional level, by the end of this course students will:

gain an understanding of foundational topological concepts
improve proof-writing skills relative to their starting point
learn to independently research and grasp mathematical ideas outside their current knowledge
develop collaboration skills on complex group problems
practice presenting difficult problems and their solutions to those who are unfamiliar with them

Topology is the study of spaces and objects up to continuous maps that may change the size or shape of said spaces and objects. In layman's terms, you can stretch it, shrink it, and twist it, but you can't break or cut it, nor can you glue two pieces together.

Source

This course will cover the fundamental concepts of point-set topology: abstract topological spaces, continuity, compactness, connectedness, manifolds, metric spaces, and homotopy theory. Applications of topology include knot theory, modelling of molecular structures, protein folding, classification of low-dimensional spaces, and many high-level theoretical physics applications.

Important Course Information

Homework
Textbook
Grading
Honour Principle, Student Accessibility, Mental Health and Religious Observances
Statement of Consent to Record

There will be homework due every week prior to X-hour. It will typically cover material from the week prior to it being assigned, and you can expect a mix of proofs with a small amount of computation. The homework assignments are posted here. You are allowed and encouraged to discuss and work together on this homework, but all submitted solutions should be your own work. The goal of the homework is for you to practice proof-writing technique at a high-level, so there will be an emphasis on the quality of your exposition as well as the correctness of your proofs. Homework will be graded over the weekend and feedback submitted back to you by the start of the next week. There will also be a set of challenge problems every two weeks. You will be split into groups of 2-3 (TBD how these groups will be assigned) and given a single problem different from the other groups to work on for the week. Unlike the standard homework, you will come up with a single joint solution. The week they are due, each group will present their solutions in full during X-hour, with the oral presentation being graded. The objective is that you will develop the skills to capably present solutions in such a manner that someone who has never seen the problem will have at least a surface-level understanding of how to go about the problem. These are invaluable skills whether you're going into grad school or the workforce. For those who are familiar with it, LaTeX is preferred, but not required. Turn in your assignments online via canvas, either as a scanned file or as a Latex (or other typeset) document. Assignments are due by the beginning of X-hour each week. Here you can find some tips on writing proofs. I also plan to post a short video sometime in the first week illustrating some proof techniques firsthand.

James R. Munkres, Topology, Second Edition, Pearson (available from sellers such as Amazon). The book is strongly suggested for supplemental reading to the class, particularly given the online nature of the class. Homework problems may be drawn from the book, but will always be stated in full as opposed to simply quoted as a problem number. 41fcv42eH5L._SX365_BO1,204,203,200_.jpg

The course grade will be based on weekly homework (60%), oral presentation of challenge problems during x-hour (20%), and a final presentation (20%).

The final presentation will be a 15-minute presentation on a topic of your choice, either from the list provided below or one you choose yourself with my approval:

Knot Theory

-Alexander Polynomial

-Jones Polynomial

-Colourings

-Applications

Algebraic Topology

-Simplicial Homology

-Fundamental Group

-Classification of Surfaces

You are expected to research your chosen topic independently, though I will be available to help you find sources or help answer questions that may arise during your research. Each student will make an appointment with me between the last day of class and the last day of finals to give their presentation, which other students are welcome to attend as time permits. The presentations will be recorded as a reference for grading. Each presentation will be followed by a round of follow-up questions from myself, and you will be graded on the accuracy of your research and the quality of your presentation. A rubric will be made available as we get closer to finals.

Grading Policy

We are returning to the standard letter grade system this term as opposed to the Credit/No Credit system used in the spring term. However, the options for NRO have been expanded. If you elect to use the NR option for this course, it may not be used to count towards the Mathematics major.

Remember that Dartmouth's academic honour principle applies to this course.

Students requesting disability-related accommodations and services for this course are encouraged to schedule a phone/video meeting with me as early in the term as possible. This conversation will help to establish what supports are built into my online course. In order for accommodations to be authorized, students are required to consult with Student Accessibility Services (SAS; student.accessibility.services@dartmouth.edu; SAS website; 603-646-9900) and to email me their SAS accommodation form. We will then work together with SAS if accommodations need to be modified based on the online learning environment. If students have questions about whether they are eligible for accommodations, they should contact the SAS office. All inquiries and discussions will remain confidential.

The academic environment at Dartmouth is challenging, our terms are intensive, and classes are not the only demanding part of your life. There are a number of resources available to you to support your wellness, including your undergraduate dean, Counseling and Human Development, and the Student Wellness Center.

Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please meet with me before the end of the second week of the term to discuss appropriate accommodations.

Additionally, some students may wish to participate in protests or other such movements during the term. I fully support this use of your time and will work with you to accommodate missed lectures or alternative homework submissions. I cannot however fully excuse students from homework as it is the main form of assessment for the course in determining what will be a crucial grade for those considering graduate school.

By enrolling in this course, I hereby affirm that I will not under any circumstance make a recording in any medium of any one-on-one meeting with the instructor without obtaining the prior written consent of all those participating, and I understand that if I violate this prohibition, I will be subject to discipline by Dartmouth up to and including expulsion, as well as any other civil or criminal penalties under applicable law.