Course: MAT215
Instructor: Aretakis
S 2016

Description of Course Goals and Curriculum

MAT 215 functions as the introductory course for all potential math majors that do not have prior proof-based math experience. More generally, it serves as an introduction to proof-based mathematics. As a result, while in the fall semester the class is full of potential math majors, the spring semester has a lot of PHY, COS, and ECO majors who are considering graduate school or simply want to understand proof-based mathematics. The class textbook is Principles of Mathematical Analysis by Rudin. The class consists of 2 80-minute lectures each week, and attendance is crucial since the book itself is very dense. There are also 2 evening problem sessions a week—leave room in your schedule to attend at least one of them.

Learning From Classroom Instruction

It is important to attend each class—try not to miss a single one if possible. Read the relevant chapter in Rudin before going to class. It will make absolutely no sense, but then the professor will explain it by drawing/going through examples during lecture and you will be able to keep up with lecture. Afterwards, read through the theorems in Rudin once more and then start the problem set. Real Analysis essentially consists of you learning a few theorems a week and then connecting them to previous theorems to solve problems and prove new concepts. It is highly conceptual, and the results sometimes seem trivial (for instance, you will prove that x^2 > 0 for all x other than 0), but the point of the class is for you to learn to use rigorous logic to solve problems. This means that, when paying attention to class, you should focus less on the actual answers and more on the methods/way of thinking used to get to those answers. The same tricks/techniques will come up again on the problem set and on the exams.

Learning For and From Assignments

The assignments will take 15-20 hours a week, including time to re-read Rudin, go to office hours, and go to the problem sessions. The problems (there are only 5-6 of them on any given problem set) are designed to challenge you so that you can properly wrap your head around a concept. As a result, they will often be frustrating, but do not lose hope! There are resources (professor, pset groups, and TAs) at your disposal. It is super-important that you form a good problem set group during the first few weeks of the semester. Ideally, your partners will be at your level or slightly better, so that you can both learn from them and explain things to them (which helps you improve as well!). The exams will be difficult but in reality are easier than the problem sets. For instance, the Spring 2014 Final had 3-4 problems on it. One was a proof that we did in class, one was a proof from a homework problem, and two were challenging proofs that used techniques that we learned in class but appeared to be completely different. To prepare for them, memorize every single proof of every single theorem that you learned in the book, on the homework, and in class. Then do as many practice problems as possible, focusing on the strategies and not the actual answers. If you can solve these practice problems in a reasonable amount of time, you have the knowledge to take the exam.

External Resources

Your best resources are the professor (at office hours) and the TAs. Leave time in your schedule for as many office hours and TA problem sessions as possible. They will help you get past roadblocks in the problem sets and in midterm preparation. In order to take advantage of them, start the problem sets 1-2 days before going for help. In this way, you can ask for help with specific challenges instead of asking, “how do I do this problem?”

What Students Should Know About This Course For Purposes Of Course Selection

You do not need prior proof-based math experience, as MAT 215 assumes none. However, you do need to be quantitatively minded and be able to think in a logical manner. More importantly, only take MAT 215 if you are committed to doing well in the class. Block 15-20 hours of time a week for class, problem sets, office hours, etc., and you will do well. At the end of the day, MAT 215 is an incredibly rewarding class—it will teach you how to think rigorously and mathematically. You will leave the semester with an understanding of the world of math that is conceptual (as opposed to computational) and far superior to that of your classmates in other Physics, COS, Econ, and ORFE classes. If you want to challenge your brain and come away a better learner and a smarter person, take MAT 215.
Honors Analysis (Single Variable)

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