MAT 431 with a grade of C or better.
Topological spaces, continuous maps, homeomorphism, metric spaces, connectedness, compactness, product spaces, quotient spaces, elementary geometric topology.
The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
Loosely speaking, topology is the branch of mathematics concerned with the properties of space that are preserved under continuous deformations, such as stretching, bending, and twisting, but not ripping or gluing. Topology is sometimes referred to as “rubber sheet geometry”, but truth be told, this only vaguely captures one aspect of the subject. The driving force behind topology is a desire to distill familiar mathematical concepts such as convergence, connectedness, continuity, and dimension down to their essence. Formally, topology is the study of properties of topological spaces that are invariant under continuous maps. Of particular interest are the properties that are preserved under homeomorphisms, which are invertible continuous maps with continuous inverses. Homeomorphisms are isomorphisms between topological spaces. Chapter 1 of our textbook “Topology Through Inquiry” contains the following beautiful quote:
Topology is a subject whose power arises from the impulse to abstract essential features from complex situations and then to let our curiosity roam while striving to truly understand what is essential about fundamental ideas.
This is not a lecture-oriented class or one in which mimicking prefabricated examples will lead you to success. You will be expected to work actively to construct your own understanding of the topics at hand with the readily available help of me and your classmates. Many of the concepts you learn and problems you work on will be new to you and ask you to stretch your thinking. You will experience frustration and failure before you experience understanding. This is part of the normal learning process. If you are doing things well, you should be confused at different points in the semester. The material is too rich for a human being to completely understand it immediately. Your viability as a professional in the modern workforce depends on your ability to embrace this learning process and make it work for you.
In order to promote a more active participation in your learning, we will incorporate ideas from an educational philosophy called inquiry-based learning (IBL). Loosely speaking, IBL is a student-centered method of teaching mathematics that engages students in sense-making activities. Students are given tasks requiring them to solve problems, conjecture, experiment, explore, create, and communicate. Rather than showing facts or a clear, smooth path to a solution, the instructor guides and mentors students via well-crafted problems through an adventure in mathematical discovery. Effective IBL courses encourage deep engagement in rich mathematical activities and provide opportunities to collaborate with peers (either through class presentations or group-oriented work). If you want to learn more about IBL, read my blog post titled What the Heck is IBL?.
Much of the course will be devoted to students presenting their proposed solutions/proofs on the board and a significant portion of your grade will be determined by how much mathematics you produce. I use the word “produce” because I believe that the best way to learn mathematics is by doing mathematics. Someone cannot master a musical instrument or a martial art by simply watching, and in a similar fashion, you cannot master mathematics by simply watching; you must do mathematics!
In any act of creation, there must be room for experimentation, and thus allowance for mistakes, even failure. A key goal of our community is that we support each other—sharpening each other’s thinking but also bolstering each other’s confidence—so that we can make failure a productive experience. Mistakes are inevitable, and they should not be an obstacle to further progress. It’s normal to struggle and be confused as you work through new material. Accepting that means you can keep working even while feeling stuck, until you overcome and reach even greater accomplishments.
Don’t fear failure. Not failure, but low aim, is the crime. In great attempts it is glorious even to fail.
Furthermore, it is important to understand that solving genuine problems is difficult and takes time. You shouldn’t expect to complete each problem in 10 minutes or less. Sometimes, you might have to stare at the problem for an hour before even understanding how to get started. In fact, solving difficult problems can be a lot like the clip from the Big Bang Theory located here.
In this course, everyone will be required to
As the semester progresses, it should become clear to you what the expectations are.
Class meetings will consist of discussion of problems, student-led presentations, and group work focused on problems selected by the instructor. A typical class session may include:
The impediment to action advances action. What stands in the way becomes the way.
As a student in this class, you have the right:
In our classroom, diversity and individual differences are respected, appreciated, and recognized as a source of strength. Students in this class are encouraged and expected to speak up and participate during class and to carefully and respectfully listen to each other. Every member of this class must show respect for every other member of this class. Any attitudes or actions that are destructive to the sense of community that we strive to create are not welcome and will not be tolerated. In summary: Be good to each other.
Students are also expected to minimize distracting behaviors. In particular, every attempt should be made to arrive to class on time. If you must arrive late or leave early, please do not disrupt class. Please turn off the ringer on your cell phone. I do not have a strict policy on the use of laptops, tablets, and cell phones. You are expected to be paying attention and engaging in class discussions. If your cell phone, etc. is interfering with your ability (or that of another student) to do this, then put it away, or I will ask you to put it away.
The textbook for the course is Topology Through Inquiry by Michael Starbird and Francis Su. The book is brand new and not yet in print. The good news is that the author and publisher are allowing us to use the book for free! The bad news is that the book will only be available in digital format and it cannot be posted publicly. A PDF of the book is available in BbLearn.
I will not be covering every detail of the textbook and the only way to achieve a sufficient understanding of the material is to be digesting the reading in a meaningful way. You should be seeking clarification about the content of the material whenever necessary by asking questions. Here’s one of my favorite quotes about reading mathematics.
Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?
Reviewing material from previous courses and looking up definitions and theorems you may have forgotten is fair game. However, when it comes to completing assignments for this course, you should not look to resources outside the context of this course for help. That is, you should not be consulting the web, other texts, other faculty, or students outside of our course in an attempt to find solutions to the problems you are assigned. On the other hand, you may use each other, the textbook, me, and your own intuition. If you feel you need additional resources, please come talk to me and we will come up with an appropriate plan of action.
There will be two midterm exams and a cumulative final exam. Each exam will consist of an in-class portion (weighted 70%) and a take-home portion (weighted 30%). The in-class portions of the midterm exams are tentatively scheduled for Friday, February 22 (week 6) and Friday, April 12 (week 12). The take-home portions of midterm exams will typically be due the following Wednesday. Each midterm exam will be worth 20% of your overall grade. The final exam is schedule for Wednesday, May 8 at 10:00AM-12:00PM and is worth 25% of your overall grade. Make-up exams will only be given under extreme circumstances, as judged by me. In general, it will be best to communicate conflicts ahead of time.
You are allowed and encouraged to work together on homework. However, each student is expected to turn in their own work. In general, late homework will not be accepted. However, you are allowed to turn in three late homework assignments with no questions asked. Unless you have made arrangements in advance with me, homework turned in after class will be considered late. When doing your homework, I encourage you to consult the Elements of Style for Proofs. Your overall homework grade will be worth 20% of your final grade.
On each homework assignment, please write (i) your name, (ii) name of course, and (iii) Daily/Weekly Homework number. You can find the list of assignments on the homework page. I reserve the right to modify the homework assignments as I see necessary.
Homework will be assigned each class meeting, and students are expected to complete (or try their best to complete) each assignment before walking into the next class period. All assignments should be carefully, clearly, and cleanly written. Among other things, this means your work should include proper grammar, punctuation, and spelling. You should write a draft of a given solution before you write down the final argument, so do yourself a favor and get in the habit of differentiating your scratch work from your submitted assignment.
The Daily Homework will generally consist of completing exercises and proving theorems from the textbook. In addition to completing the assigned problems, I also want you to assess your own work. Next to each problem, I want you to write down a score between 0 and 4 that represents your perception of the validity and quality of your proposed solution/proof. Consider using the rubric given below in the description of the Weekly Homework assignments. Not completing the self-assessment step may impact the score on your homework.
On the day that a homework assignment is due, the majority of the class period will be devoted to students presenting some subset (maybe all) of the proofs/solutions that are due that day. Daily Homework will be graded using a $\checkmark$-system. You are allowed (in fact, encouraged!) to modify your written work in light of presentations made in class; however, you are required to use the colored marker pens provided in class. This will allow me to differentiate the work done in class versus the work you completed before class. The grade you receive on an assignment will be determined by the work you completed prior to class. I will provide more guidance with respect to this during the first couple weeks of the semester.
In addition to the Daily Homework, we will also have Weekly Homework assignments. For most of these assignments, you will be required to submit 2-3 formally written proofs. Some or all of these problems will come directly from the Daily Homework assigned the previous week. You are required to type your submission using $\LaTeX$. You can either submit a hardcopy of your assignment or email me the PDF of your completed work. If you email me the PDF, please name your file as
X is the number of the assignment and
LastName is your last name. Notice there are no spaces in the filename. Each problem on the Weekly Homework assignments is subject to the following rubric:
|4||This is correct and well-written mathematics!|
|3||This is a good piece of work, yet there are some mathematical errors or some writing errors that need addressing.|
|2||There is some good intuition here, but there is at least one serious flaw.|
|1||I don't understand this, but I see that you have worked on it; come see me!|
|0||I believe that you have not worked on this problem enough or you didn't submit any work.|
The problems chosen for presentations will come from the Daily Homework assignments. Though the atmosphere in this class should be informal and friendly, what we do in the class is serious business. In particular, the presentations made by students are to be taken seriously since they spearhead the work of the class. Here are some of my expectations for the presenter:
Presentations will be assessed using the following criteria.
|U||Unsatisfactory. Minimal progress was made that included relevant information or the student was unprepared.|
|I||In progress. The student made an honest attempt at the problem but recognized a flaw that prevented them from being able to complete the problem during the presentation. Alternatively, the student reported on their current progress on a problem and attempted to convey where or why they are currently "stuck".|
|M||Meets Expectations. The student demonstrated an understanding of the problem and presented the key ideas. Perhaps some details were omitted or interesting mistakes were made. The presentation led to fruitful class discussion.|
|E||Exceeds Expectations. The presentation was flawless and the student demonstrated keen insight into the problem. The presentation led to fruitful class discussion.|
You should aim to avoid unsatisfactory (U) presentations. An in progress (I) presentation should not be viewed as a bad thing as each of us will occasionally get stuck. However, you should strive for the majority of your presentations to meet (M) or exceed (E) expectations. Most presentations will meet expectations (M) while presentations that exceed expectations (E) will be rare. You should not let the rubric deter you from presenting if you have an idea about a solution/proof that you’d like to present but are worried that your solution/proof is incomplete or you are not confident your solution/proof is correct. You will be rewarded for being courageous and sharing your creative ideas! In my view, an interestingly wrong solution or proof makes for the best presentation since it generates the best discussion. This is really what we are after. On the other hand, you should not come to the board to present unless you have spent time thinking about the problem and have something meaningful to contribute. I will provide a progress report concerning each student’s presentation history after each of the midterm exams.
In general, I will curate the list of student presenters each class meeting. A presenter is a student that either volunteered (V) or was chosen (C) by me. Volunteering is encouraged, but being chosen without volunteering isn’t bad. If more than one student volunteers for a specific problem, the student with the fewest number of presentations has priority. I reserve the right to decline your offer to present. This may happen if you are volunteering too often (and hence removing another student’s opportunity to present) or if I know in advance that another student’s presentation will lead to a fruitful discussion.
If you are chosen to present but would prefer not to present that particular problem, you can either negotiate presenting a different problem or take a pass (P). You may elect to pass at most two times during the semester, after which a presentation will be deemed unsatisfactory (U). By default, if you have an unexcused absence on a day when you have been chosen to present, then your presentation will be recorded as a pass (P) unless you have already exhausted your two passes, in which case the presentation will be recorded as unsatisfactory (U).
|V||Volunteered. Student volunteered during class or in advance to present.|
|C||Chosen. Student was selected by the instructor and agreed to present.|
|P||Pass. Student was selected to present, but asked to take a pass. Allowed at most two.|
In summary, for each student presentation, I will record one of V, C, or P. In the case of V or C, I will also record one of U, I, M, or E based on the rubric given above. The most common pair will likely be CM (i.e., student was chosen to present and presentation met expectations).
You are expected to respectfully participate and contribute to class discussions. This includes asking relevant and meaningful questions to both the instructor and your peers. Moreover, you are expected to be engaged and respectful during another student’s presentation. Your class participation will be assessed as follows.
|U||Unsatisfactory. Student was often disengaged or disrespectful. Alternatively, the student regularly missed class.|
|M||Meets Expectations. Student was consistently respectful, engaged, and contributed to meaningful class discussions. In addition, the student regularly attends class.|
|E||Exceeds Expectations. Student's presence in the classroom truly enhances the learning environment.|
Your Presentation and Participation grade is determined by your frequency and ability to foster productive class discussions through presentations and audience participation. The greatest determining factor in your Presentation and Participation grade is your willingness to present often. You should aim to present at least twice prior to each midterm exam. The table below provides a summary of how your Presentation and Participation grade will be determined.
|90-100%||Student receives M or E for participation. Student averages at least 3 presentations prior to each of the exams. Student often volunteers to present and some of these problems are challenging. Most presentations receive M or E.|
|80-89%||Student receives M or E for participation. Student averages at least 2 presentations prior to each of the exams. Student occasionally volunteers to present. Most presentations receive M.|
|70-79%||Student receives M for participation. Student averages less than 2 presentations prior to each of the exams. Student rarely volunteers to present and actively avoids presenting challenging problems. Some presentations receive U.|
|60-69%||Student receives U for participation. Student rarely presents and actively avoids presenting challenging problems. Some presentations receive U.|
|Below 60%||Student receives U for participation. Student rarely or never presents and has completely disengaged from the class community.|
I anticipate that most students will fall in the 80-89% range. Your Presentation and Participation grade is worth 15% of your overall grade.
I must not fear.
Fear is the mind-killer.
Fear is the little-death that brings total obliteration.
I will face my fear.
I will permit it to pass over me and through me.
And when it has gone past I will turn the inner eye to see its path.
Where the fear has gone there will be nothing.
Only I will remain.
Regular attendance is expected and is vital to success in this course, but you will not explicitly be graded on attendance. Yet, repeated absences may impact your participation grade (see above). Students can find more information about NAU’s attendance policy on the Academic Policies page.
The only thing I will award extra credit for is finding typos on course materials (e.g., textbook, exams, syllabus, webpage). This includes broken links on the webpage. However, it does not include the placement of commas and such. If you find a typo, I will add one percentage point to your next exam. You can earn at most two percentage points per exam and at most five percentage points over the course of the semester.
In summary, your final grade will be determined by your scores in the following categories.
|Homework||20%||A combination of Daily & Weekly Homework|
|Presentations & Participation||15%||See above for requirements|
|Exam 1||20%||February 22|
|Exam 2||20%||April 12|
|Final Exam||25%||Wednesday, May 8 at 10:00AM-12:00PM|
You are responsible for knowing and following the Department of Mathematics and Statistics Policies (PDF) and other University policies listed here (PDF). More policies can be found in other university documents, especially the NAU Student Handbook (see appendices) and the website of the Office of Student Life.
As per Department Policy, cell phones, MP3 players and portable electronic communication devices, including but not limited to smart phones, cameras and recording devices, must be turned off and inaccessible during in-class tests. Any violation of this policy will be treated as academic dishonesty.
Here are some important dates:
There are many resources available to get help. First, I recommend that you work on homework in small groups as much as possible and to come see me whenever necessary. I am always happy to help you. If my office hours don’t work for you, then we can probably find another time to meet. It is your responsibility to be aware of how well you understand the material. Don’t wait until it is too late if you need help. Ask questions! Lastly, you can always email me.
Any changes to this syllabus made during the term will be properly communicated to the class.
Mathematics & Teaching
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Flagstaff and NAU sit at the base of the San Francisco Peaks, on homelands sacred to Native Americans throughout the region. The Peaks, which includes Humphreys Peak (12,633 feet), the highest point in Arizona, have religious significance to several Native American tribes. In particular, the Peaks form the Diné (Navajo) sacred mountain of the west, called Dook'o'oosłííd, which means "the summit that never melts". The Hopi name for the Peaks is Nuva'tukya'ovi, which translates to "place-of-snow-on-the-very-top". The land in the the area surrounding Flagstaff is the ancestral homeland of the Hopi, Ndee/Nnēē (Western Apache), Yavapai, A:shiwi (Zuni Pueblo), and Diné (Navajo). We honor their past, present, and future generations, who have lived here for millennia and will forever call this place home.