Description of Course Goals and CurriculumNote: Course number and name was previously MAT 215 Honors Analysis I An introduction to the mathematical discipline of analysis, to prepare for higher-level course work in the department. Topics include rigorous epsilon-delta treatment of limits, convergence, and uniform convergence of sequences and series. Continuity, uniform continuity, and differentiability of functions. The Heine-Borel Theorem. The Riemann integral, conditions for integrability of functions and term by term differentiation and integration of series of functions, Taylor's Theorem. This course is intended to be a first (or early) experience in rigorous mathematical thinking. In some sense, the curriculum assumes minimal background as it attempts to construct some of the most fundamental tools in mathematical analysis. The first couple of weeks cover basic set theory, abstract algebra, and the construction of the integers, rationals, and real numbers. As the course moves into introductory point-set topology, it seeks to make students comfortable with thinking about and working with abstract mathematical objects, specifically by constructing rigorous arguments. Comfort with producing mathematical proof and thinking about mathematical concepts in a foundational manner is essential for further work in the math department or in understanding and appreciating the field of mathematics in general. It is exactly this comfort that the department and instructor hope to build in MAT 215 and the later courses in the MAT 200 sequence. As mentioned before, the main vehicles for accomplishing this are linear algebra and topology. These topics will introduce a host of definitions for mathematical concepts students have likely already been exposed to but have never had rigorously defined (such as open and closed sets, limit convergence, continuity, etc.). Some experience in introductory linear algebra is helpful in order to concentrate on the more abstract points of the proofs rather than the mechanics of matrix algebra. Even so, Professor Gunning assumes little to no background even in calculus.
Learning From Classroom InstructionProfessor Gunning typically speaks quite quickly in lecture. He writes fast as well. Occasionally his words and his writing don’t match up perfectly. Although this style does call for significant concentration from the student, there are certain advantages. Professor Gunning has ample time to give a short review of material from the previous lecture in the first few minutes, while also providing numerous examples of new concept he defines. Starting lecture unsure of where the course left off can make the rest of the 80 minutes less valuable, as they might appear to have little connection to previous topics. Meanwhile, examples of a new definition or theorem are often more illustrative than the statements themselves. If a particular statement in class does not make sense at first, finding specific examples (opposed to general statements) can be helpful. Having a well-rounded understanding of different statements is essential to being comfortable using them in one’s own proofs. Professor Gunning’s pace in class allows him to include a large amount of content and examples, but as a result there will be pieces that will not be picked up entirely in lecture. While these items can sometimes be clarified by a question during class, Professor Gunning provides his students will an excellent additional resource in his course notes (which could more appropriately be called his book). Nicely written and formatted (and nearly 300 pages), these notes form the basis of his lectures, but they are also extremely helpful to look back on after lecture. Also, while many courses have certain expectations about learning some ideas in lecture, others in precept, and yet others from readings, Professor Gunning removes such ambiguity from his class. With the exception of a couple of examples omitted for the sake of time, the content of lecture and the book is nearly identical (even the notation). All along, one must remember though that the course is less about the content of Gunning’s lectures and books and more about learning to engage with abstract mathematical objects and use them in the context of different mathematical argument. Learning all of the finer points of topology is not the point, while gaining the experience and comfort of using a set of previously proven mathematical results to prove a new one is.
Learning For and From AssignmentsThe weekly assignments in MAT 215 are challenging but rewarding and overall are successful at gradually making students more comfortable with increasingly complex proofs. Professor Gunning typically splits each assignment into Group I problems and Group II problems. Group I mostly includes specific examples in analysis, algebra, or topology that one must verify satisfy certain definitions and theorems. Group II problems are more complicated, sometimes trickier examples, but often involve proving corollaries of theorems proven in lecture and/or in Gunning’s book. Since the style of these problems is much different than most students have seen in high school (or even other lower division math courses at Princeton), the assignments can take many hours to complete. However, these hours are undoubtedly the most valuable of the course. If possible proofs do not come to mind immediately, recall the definitions of the basic objects in question (bijective mapping, skew-symmetric form, compact space, etc.) and try to connect these to the proofs of theorems proven in class. Also, when proving general statements (such as something that is true for all compact spaces), it often helps to try to understand first why it is true in particular examples (for example, why it might be true in a closed interval of the real line). Exams are very brief relative to the problem sets. With a time limit of three hours, proofs must be kept rather short. Thus, problems typically require one or two clever applications of theorems from the semester but not much more. The problem sets generally provide sufficient practice, but finding some basic exercises in Rudin (see External Resources section) or online can provide good additional practice. Remember the course is more about learning to do math in the form of abstract reasoning and proof writing, while it is less about learning actual mathematical facts. While all the problems’ terminology will come from course content, the exams reflect this purpose.
External ResourcesWikipedia can serve as an important mathematical resource. While there might exist some questions of reliability or bias on many articles, the pages on math are generally quite well done. Reading definitions or theorems worded slightly differently or using different terminology can also make one’s understanding of a particular statement more flexible and make clearer how to apply it in a greater variety of proofs. There are also numerous mathematical texts on the subject of introductory analysis. MAT 215 in previous years was often taught from Rudin’s Principles of Mathematical Analysis. Some of the exercises in the book can be helpful for portions of the course, but Gunning takes a much more topological approach to the content than does Rudin as well as including much more linear algebra, so Rudin may not be as insightful for these portions of the semester. Additionally, the math department offers multiple problem sessions each week with older undergraduates who have been successful in the course. These sessions are generally well-attended, and one can glean a lot simply from listening to the conversation of other classmates and how they reason through particular problems. Learning alternative proofs or equivalent statements can lead to insights for other problems. Sometimes a slight reframing of a definition makes clear its role in a proof. This is why it is so important to read statements worded differently and understand a variety of examples.
What Students Should Know About This Course For Purposes Of Course SelectionGiven the emphasis on mathematical proof, MAT 215 is mostly intended for those looking to study math in detail in the future. These students are most frequently freshmen and sophomores likely to concentrate in mathematics, but the course can also be helpful to many students studying physics or economics. Regardless of a student’s background or academic bent, the course is among the most demanding prerequisite courses offered at the University, requiring many hours of careful thought and concentrated study. However, at the same time, many who take it find it to be a very rewarding and exciting experience, giving them an introduction into what abstract mathematics looks and feels like.
Accelerated Honors Analysis I