Basic Info

Instructor: Dave Feldman
Email: You can figure it out
Office: Second Floor, Turrets Annex
Office Hours: By appointment.
Help Sessions: Wednesdays, 6:30-8:00, Dining Hall

Course Goals


  • Gain a solid introduction to differential equations: learn what differential equations are, be able to formulate original mathematical models using differential equations, and be able to use a variety of techniques to solve and understand differential questions, including numerical and qualitative methods.
  • Gain experience conceptualizing, carrying out, and presenting research. This includes conceptualizing and framing research questions and problems, conducing a literature review, presenting results in a professional-style research talk, and writing up results in a style appropriate for publication.
  • Gain experience with scientific computing and writing programs to help you carry out mathematical analysis.

Textbook


There is no textbook for this course. We will use some online resources and I will distribute articles and perhaps some of my own notes.

Evaluation


Your evaluation will be based roughly as follows:

  • Homework assignments (30%)
  • Research project (65%)
  • Class participation (5%)

I recommend against grades; I believe they are more likely than not to interfere with genuine, reflective learning. However, I will assign grades (for those who so opt) by following the guidelines in the COA Course Catalog. I do not have any quota of A's, B's, etc.

Some Details

  1. Homework will usually be due Fridays at the end of the day. More than one unexcused late homework assignment will result in me mentioning this in your narrative evaluation and may result in a lowering of your grade.
  2. You are strongly encouraged to work together on homework. You can also consult me, other faculty, friends, and family. However, the homework you hand in should represent your own understanding. This means that if your friends get a homework problem and you don't understand how they did it, you shouldn't photocopy their solution and turn it in.
  3. You will need regular and reliable access to a computer that has python on it. If you need a loaner laptop from the school, please let me know.
  4. I will be sending out class info via email. Thus, it's important that you check your email.
  5. There will often be reading to do before class. It is important that you do the reading and come ready to discuss and ask questions.
  6. Academic misconduct—cheating, plagiarizing, etc.—is bad. Any cases of academic misconduct may result in a judicial hearing. Possible consequences range from failure of the assignment to expulsion.

Other Thoughts about the Course

  1. I've taught this class three times before, and each time it's gone quite well. This is a moderately revised version of the course, but it's sufficiently similar to the old version that I'm pretty confident of how most things will go.
  2. This class is listed as an intermediate/advanced (I/A) class. It's not I/A because we'll be doing super hard, super tricky math. Rather, it's I/A because of what you'll be asked to do with some mostly straightforward math. You'll be asked to think with math and to use math creatively. You will also be expected to think about the structure of research and improve research skills.
  3. This class makes extensive use of ideas from calculus. It's important that you've had both differential and integral calculus before and felt reasonably good about the experience, although it's ok if you've forgotten much of it.
  4. Falling behind in this course is not a good idea. If you're confused about something, it's very important that you seek help sooner rather than later.
  5. I do not expect all of the homework assignments to be easy; I don't expect you to be able to sit down and do them easily the first time. The style of homework assignments in this class might be very different from what you're used to. Don't let yourself get frustrated. I strongly suggest working with others and seeking help if you need it.
  6. There will be few, if any, quick plug-in assignments. This class is not so much about mathematical calisthenics. Rather, there will be only a few problems, but they will be longer, requiring multiple steps. On many occasions you will be asked to form your own mathematical model as part of a problem.
  7. You will learn some of the basic elements of the python programming language. This is not a full course in python by any means; my goal is to teach you just enough so you can use it for what we need in this course, and also give you enough fundamentals so that you can learn further on your own if you want. I'm confident that this will work well, although I'm not quite sure of the some of the details.
  8. This course is structured to allow you to do a significant project as part of the coursework. You will need to start early on this and keep moving throughout the term. There will be numerous check-points throughout the term, including an oral presentation during week five. Falling behind in your project can be disastrous.
  9. Throughout I will be taking a (hyper) "modern" approach to differential equations. I emphasize qualitative, global analysis of differential equations, rather than specialized analytic tricks. We will frequently rely on numerical solutions to differential equations. We will use computers without guilt or apology to help us do mathematics and science. All of this might give the course a different feel than other math classes you've taken.
  10. There are three parts to this course: differential equations, python, and research projects. These strands will flow more or less in parallel. I don't think this will be a problem, but it may be just a tiny bit schizophrenic at first.

Research Projects

This class is designed to help you transition from being a reader, to being read, from being someone who listens, to someone who is listened to.

Research is an essential component of almost any area of study and field of work. Learning what research is, how to think about it, and how to do it well, are extremely important skills---ones that you will use in many, many areas beyond differential equations and mathematics. I think research skills are important enough that it is worth devoting a considerable fraction of this class to them.

Overview: The research project should be an in-depth exploration of a particular application of differential equations. It may also be possible to do a more theoretical project. We will spend the first few weeks of the term talking about what research is, how to move from an area of interest to a topic, and how to relate a research question to a broader set of questions or concerns. There are three major "deliverables" associated with the project:

  1. Initial Presentation: During week five you will give an initial presentation in which you introduce your topic. You should give a sense of what you'll be doing the next several weeks and the questions you'll wish to address. There are three reasons for this initial presentation.
    1. It will force you to choose a topic early rather than later.
    2. It will let other people in the class know what you're up to, so you can get feedback and ideas from others.
    3. It will give you a chance to practice giving technical presentations. You'll get feedback on your presentation from both me and classmates.
  2. Final Presentations: Sometime during week 10 we will give presentations in a session open to the community. These talks will follow the format of contributed talks at a scientific meeting.
  3. Paper or Technical Report: Due on the last day of the term. This will be a 5-20 page paper on your research question. You should write in a style appropriate for publication in peer-reviewed journal of your choosing. (If you want to explore another form, this is possible, but you should clear this with me early on.) Your paper must be well-formatted, include figures and graphs as appropriate, and correctly use the citation format standard for whichever journal you choose.

Throughout the term we will spend time discussing a number of issues and ideas in research, and also cover some specific research skills. This will likely include:

  • The difference between interests, topics, questions, and problems
  • The mechanics of peer-reviewed journals and academic publishing
  • The rhetorical style of research papers
  • How to search for sources, including using the citation lists available in Google scholar
  • How to give an effective technical presentation
  • Ethics in research, plagiarism and academic fraud, issues surrounding research on human subjects
  • Software for organizing and automating bibliographies

I probably will also divide you up into small groups based on similarity of interests and ask your groups to meet and/or share resources.

A few of these discussions may sound not that interesting, but give them a shot. In the past students have found these topics to be useful and quite helpful.

It is inevitable that some of you will "get farther" in your projects than others, since some of you are farther along in your studies. Also, some research topics are harder to make progress on than others. This diversity is fine. The point of the projects is not to come up with publishable original research or even have an amazing and polished final result. Rather, the idea is to strive toward as good a result as possible, and learn a lot (and challenge yourself and have fun) while doing so.

Finally, as you may have guessed, I suspect that these research projects will be a lot of work. But with a good topic and the right attitude, I think this work can be very fun and rewarding. And I think doing a term-long project like this will be very helpful before undertaking your final project or a research internship somewhere.