AMAT 342 : Elementary Topology

Fall 2020, Class #1886

Tuesday, Thursday 10:30-11:50 on Zoom

Last revision: 8/24

Instructor: Michael Lesnick
Office Hours: Monday, Wednesday 1:00-2:00, and by appointment.
mlesnick [at] albany [dot] [the usual thing]

I typically respond to emails within a day (two days on the weekend).

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Prerequisites:
AMAT 214, AMAT 220, and (especially) AMAT 299.
IMPORTANT: You cannot take this course without my permission if you have not passed all of the prerequisites.

About this Course:
This course will introduce some of the key ideas of topology. Topology studies the properties of geometric objects that are preserved under "continuous deformations." Informally speaking, continous deformations include bending, twisting, and stretching of the object, but not tearing or puncturing.

Topology is one of the major areas of modern mathematics. It is a beautiful subject in its own right and it plays an important role in other parts of mathematics, the sciences, and engineering.

Topics may include: Course Logistics:
The course will be fully online, in a (partially) synchronous format. I will cover the core course content in a series of short, pre-recorded videos, which you will be required to watch before each class meeting. In-class sessions will happen live on Zoom, and be more interactive, devoted to questions, exercises, review, and additional examples. We will have a Blackboard forum where students can ask and answer questions, and in-class activities will be guided in part by what is posted on the board. The live sessions will be recorded and available to the class (not to anyone outside of the class). Most or all course content will be available through Blackboard.

I understand that logistical issues may sometimes make attandance of the live sessions difficult for some students. Thus, attendance of the live classes and active participation is strongly encouraged, but not required. Students who miss the in-class sessions are strongly encouraged to watch the recording.

To participate in the course, you will need to have a way to create legible scans of homework and exams, and to submit these electronically via Blackboard. It will also be helpful to have a working webcam and a decent internet connection.

Course Materials:
There is no required textbook. I will provide handwritten lecture notes which will closely follow the content of the pre-recorded videos.

Supplemental Materials (Not Required):
For those looking to explore course topics in more depth, there are many good textbooks on topology, though some are pitched at a somewhat higher level than this course. Two standard choices are "Topology" by Munkres, and "Basic Topology" by Armstrong.

For those interested in learning about applications of topology to the sciences, I recommend "Topological Data Analysis for Genomics and Evolution: Topology in Biology" by Blumberg and Rabadan. The first half of the book is an accessible introduction to applied topology.

Exams:
My tentative plan is to exams, on the folowing days:
Midterm I: Thursday October 1,
Midterm II: Tue, November 5.
Final: TBD
The exams will happen during the scheduled class time, and like everything in this course, be conducted remotely/online. Special arrangements can be made for students who cannot take an exam at these times, provided this is discussed with me well in advance of the exam.

Exam rules: For each exam, you will be allowed to prepare one page of handwritten notes, front and back, to use during the exam. You will electronically submit a copy of these notes to me before the start of the exam. During the exam, no use of any other materials (inlcuding the internet) will be allowed, and naturally, any communicaton with your classmates (or any mathematical communication with anyone) is strictly prohibited.

At my discretion, some or all exams may also have an oral component, to occur after the written portion of the exam.

Homework:
Each set of online lectures will typically come with a relatively straightforward "small homework", due before class. This is mostly to incentivize you to keep up with the lectures, and think about what you are learning. In addition, we will have more substantial problem sets ("big homework") due once every 1-2 weeks).
Homework rules: Late homeworks will not be accepted for credit. For small homeworks, no collaboration/discussion with anyone except the instructor is allowed before submission. You may discuss the big homework with others, but you must complete all of the assignments yourself. Copying any part of a homework solution (whether from a classmate, a website, or anywhere else) is strictly prohibited, and sharing the solution is also prohibited.

Grading:
The class will use the university's A-E grading scheme.

20%: Big homework
10%: Small homework
20%: Midterm I
20%: Midterm II
25%: Final
5%: Class engagement/effort.

Each big homework will be weighted equally, and each small homework will be weighted equally. The lowest big homework score and the lowest two small homework scores will be dropped. The midterm and final may be curved, but not downward.

The engagement/effort score will take into account in-class attendance, participation in class sessions, activity on the class forum (both asking and answering questions), and participation in office hours. Students do not need to engage in all of these ways to have a high engagement/effort score, but should actively engage through some of these avenues.

Expectations:
Topology is a fun and fascinating subject, but it is also a challenging one. The material is inherently very abstract, and it takes time and careful thought to become comfortable with the ideas. As a first eposure to topology, this class tries to introduce the ideas in a gentle, friendly way. Still, most students will find the material to be a good challenge, and to have a good experience in this class, they will need to put in substantial work. As a rough estimate, I expect students to put in at least 8-10 hours of work per week on this course, aside from time spent in our synchronous classes. This time should be spent watching the pre-recorded lectures, reading the notes, doing homework, visiting office hours, using the message board, discussing the material with your classmates, and thinking about the material on your own.

Pandemic-Related Challenges:
The pandemic--and the move to online instruction in particular--creates a complex set of potential difficulties for students. I intend to hold this class to a high standard of effort, but at the same time, I am mindful of the unique challenges our situation presents, and I intend to conduct the class accordingly. If you are dealing with issues created or exacerbated by the pandemic that risk getting in the way of your being a focused, active participant in this class, please let me know.

Academic Regulations:
Naturally, SUNY Albany's Undergraduate Academic Regulations apply to this course, and students are expected to be familiar with these. The regulations concern academic integrity and missing/rescheduling exams, among other things. Students should also be aware of the University's Medical Excuse Policy.

There will be no leeway on missed exams or last-minute exam rescheduling, except as noted in the regulations. If you anticipate an issue with the timing of an exam, please let me know as soon as possible.

Cheating: Given the unsual circumstances surrounding this course, there may be some unique temptations this semester to cheat. Please know that cases of cheating are a major headache for instructor, typically creating hours of extra work to resolve. Please also know that it is often a lot easier to detect cheating than students realize. If you are caught cheating, your grade will pay a heavy price, and there is a good possibility that you will fail the course. Your cheating may also be reported to the university, as described in the Academic Regulations.