Welcome to Topology! This syllabus is meant to introduce you to both the structure and motivation for this course. Please take a close look and be sure to have it easily accessible during the semester. I’m looking forward to working with you!
Picture a “torus”, which you can imagine as the shape of the outside of a bagel. We often encounter surfaces as the graphs of functions with two variables, but it would be quite difficult to express a torus in this way. Even if we could find a way to do this (by piecing together the graphs of several functions), it would be difficult to see the essential features of the torus (e.g. two-dimensional, no boundary, one hole) just by looking at the expressions. These are the “topological” features of this surface, and in order to articulate them, we need to develop some new tools.
In this course, we will introduce the idea of a topological space and study several properties which we can view as generalizations from real analysis: continuity, connectedness, compactness, and others. Our explorations will include several examples of topological spaces and some side-excursions into topics such as surfaces, polytopes, configuration spaces, and simplicial complexes. We will also discuss some applications of topology.
- Understand the topological (rather than geometric) notion of “shape”
- Improve mathematical writing and communication skills, especially with respect to proof writing and presentation.
- Identify the role of topology as a subfield of the broader mathematical landscape
- Develop strong topological intuition via in-depth exploration of examples
- Develop topological intuition in arbitrarily many dimensions
After completing this course, students will be able to:
- Articulate the notion of continuity in the setting of topological spaces
- Describe various properties of topological spaces (e.g. connectedness, compactness) and exhibit examples of spaces which do or do not hold these properties
- Apply topological concepts to particular classes of examples, including surfaces and simplicial complexes
- Describe several applications of topology both in and outside of mathematics
- Classroom: Pardee 112
- My office: Pardee 229
- Office Hours: M 11am-12pm, T 2-4pm, F 10-11am
- Email: email@example.com
- Course website: https://moodle.lafayette.edu/course/view.php?id=23426
Basic Topology by M.A. Armstrong
Here, I’ve described some of the structure for our course. Take a look to see what you can expect throughout the semester.
For a few hours each week, I will be available to chat about the course in my office (Pardee 229). While it is often helpful to come with a specific concept or question you’d like to discuss, I’m also happy to just chat about how things are going. You do not need to request time for my scheduled office hours - you can just drop by! If you cannot make it to my scheduled hours, let me know and I will do my best to find a time when we can meet.
We will have weekly homework assignments, posted on Moodle and submitted via Gradescope. These assignments will be the primary way for you to test your understanding of the course material and develop your knowledge. I highly recommend that you work on the assignments with your classmates, although your submitted work must be your own. Remember - the goal of these assignments is to help you learn!
Throughout the semester, we will have several opportunities for presentations. You will each be expected to give one or two brief presentations during the course, and you’ll be given the topic at least a week in advance.
This course includes both a midterm exam and a cumulative final exam. At the moment, I am planning on both exams being held in-person. The midterm is tentatively scheduled for Monday, October 3 during our usual class time, although this is subject to change.
The graded work in this course includes homework, presentations, a midterm exam, and a final exam. These categories combine to form your final grade according to the following breakdown:
- Homework: 50%
- Presentations: 10%
- Midterm: 20%
- Final: 20%
Policies and Resources
No syllabus would be complete without the fine print! These sections are important, though - please be sure to familiarize yourself with it and keep an eye out for resources which may be helpful to you.
In an effort to ensure everyone’s safety, mask-wearing is required in class and during office hours. Please be sure to bring a well-fitting mask which covers your nose and mouth. We may revise this policy later in the semester, depending on how the covid situation evolves.
Diversity and Inclusion
We value diversity and inclusion, and are committed to a climate of mutual respect and full participation both in and out of the classroom. This class strives to be a learning environment that is equitable, inclusive, and welcoming, regardless of race, ethnicity, religion, gender, sexual orientation, disability, socioeconomic background, and nationality. If you anticipate or experience any barriers to learning, please don’t hesitate to discuss your concerns with me.
Discussing mathematics can often be difficult - it takes practice! Please work hard to be considerate and respectful when talking to your classmates. Remember that we are not just machines for solving math problems. We are humans as well!
Lafayette is committed to providing support and reasonable accommodations for students with disabilities who self-identify with Accessibility Services. Students requesting accommodations to alleviate the impact of their disability should register their needs as soon as possible with the Accessibility Services Office, which is housed in the Academic Resource Hub (firstname.lastname@example.org). Once registered, students should request their accommodation letters to provide notification of their needs to their professors, on a semester by semester basis. If you have questions or concerns pertaining specifically to your accommodations within this course, please contact me to discuss them.
If you anticipate that you will have an unavoidable course conflict due to a religious observation, please meet with me as soon as possible so that we can make appropriate arrangements.
While working collaboratively on homework is (strongly) encouraged, the submission of work which is not your own is strictly prohibited. This includes (but is not limited to) copying answers from peers or the internet. Explicitly, you may discuss assignments together, but your written work must be your own. When you do work with someone else or get help from an outside source, please cite it accordingly.
Federal Credit Hour Compliance
The student work in this course is in full compliance with the federal definition of a four credit hour course. Please see the Lafayette College Compliance webpage (http://registrar.lafayette.edu/additional-resources/cep-course-proposal/) for the full policy and practice statement.