Stepping out of the lecture hall complex at IISER Pune after a class is a veritable treat to the eyes and the attentive mind. Beds of multicoloured flowers around the complex, which keep changing as per the season, help to relax and think about the lecture, what went well, what needs more work and how the course structure is developing. Sometimes, they may also remind one of larger teaching principles. For example, unlike previous years, some of the Gulmohar trees this year had low hanging branches with an abundance of flowers. While looking at one such flower from close quarters, I learnt for the first time that one of the five petals of this flower is larger than the other four, and has a unique layout with yellow and red streaks on a white background. These low hanging branches make one think about how to present a course so that students can get a better glimpse of the underlying beauty. Similarly, while walking past adjacent Tellicherry and the Plumeria trees with their own white flowers blending into each other, an observant passerby cannot help but reflect that any course that we teach is a seamless blend of many areas of mathematics, while also retaining its own unique character.
This semester, I taught a "topics" course in analytic number theory. This course, not a part of the regular curriculum, introduced students to a wide variety of analytic tools to understand and address some of the most mathematically exciting problems of our times, such as those related to the distribution of primes numbers. The advantage of teaching a topics course is that the students who take such a course genuinely want to explore the content with an open mind. Typically, a topics course is designed and offered by a faculty member who wants to introduce students to areas in which the faculty member works (or in some cases, new areas that they themselves want to learn). Thus, there is a synergy between students who want to learn and an instructor who has a deep desire to teach the subject. Long story short, my teaching duties this semester brought in huge returns on the "job satisfaction" front.
Number theory is a stream that offers the interested learner a large set of unanswered questions. These questions are easy to state and can often be explained at the level of high school mathematics. Can we write any even number larger than 2 as a sum of two primes? On a related note, can we write any odd number greater than 5 as a sum of three primes? Can we find infinitely many pairs of consecutive primes with gap 2 such as (3,5), (5,7), (11,13), (17,19) and so on? For that matter, can we find infinitely many pairs of consecutive primes with gap 4 such as (7,11), (13,17), (1, 23) and so on? This leads to a broader set of questions about the behaviour of gaps between consecutive primes. The "patron saint" of a first course in analytic number theory is a question that was literally asked by a high school student: a 15 year old German boy by the name of Carl Friedrich Gauss who made a guess about the asymptotic growth of prime numbers in 1792. His conjecture was proved more than a 100 years later, and is known as the prime number theorem. The proof came through an article by Bernhard Riemann in 1859 which provided a template and "vision document" to prove the prime number theorem (and more). Riemann described how such questions can be answered by the study of what is now called the Riemann zeta function as a function of complex variables. These ideas were developed into a complete proof of the prime number theorem in 1896 by Hadamard and de la Vallee Poussin independently. Riemann's ideas also provide a framework to address many arithmetic problems of interest in the current times. Today, "grand unification" themes in mathematics such as the Langlands programme seek to uncover the algebraic structures behind a large class of functions to which Riemann's ideas can be applied.
A typical first course in analytic number theory, aimed at a senior undergraduate student or a beginning postgraduate/PhD student, introduces Riemann's ideas and a complete proof of the prime number theorem along with more refined versions (error terms in the asymptotic distribution of primes) as well as interesting variants such as the distribution of primes in arithmetic progressions (sequences of terms with a constant difference, for example, a sequence with terms 5, 9, 13, 17, 21, 25, 29,....) . Depending on the level of preparation of the students and the goals of the instructor, the course can also aim at developing tools for other problems. For example, to answer questions about writing an even number larger than 2 as a sum of two primes, or any odd number greater than 5 as a sum of three primes, requires the introduction of a method called the circle method. To answer questions about gaps between consecutive primes requires a set of tools called sieve methods.
The students taking this course here were in the 3rd or 4th year of the BS-MS programme (the latter equivalent to the first year of a Master's programme), and many had not taken courses that are considered prerequisites for this course, such as complex analysis. At the same time, having taught many of them before, I could count on their enthusiasm and ability to pick up new ideas. So, it felt reasonable to design it like a standard course on this topic for a senior undergrad, keeping the prime number theorem and the mathematics around it as the primary goal. This provided clear learning goals for the course, while also giving us the space to spend time with topics that students had not seen before or were seeing simultaneously with a course in complex analysis. Indeed, extending the Tellicherry and Plumeria analogy, during many classes of our course, a passerby would not have been able to tell whether this was a course on complex analysis or analytic number theory.
As I mentioned above, a topics course provides a broad canvas for an instructor to introduce students to potential research areas. There were some attendees in the class who had an express intention to work with me. The course gave me an opportunity to introduce them to the kind of problems that I am interested in, without diluting or straying away from the course goals. One of the students, for example, will work on his MS thesis under my guidance. In the last few lectures of the course, we went over the theme of zero-free regions of the Riemann zeta function. When we sat down to go over a preparatory article that he has to read for his project, we noted how the paper felt like a redoing of the course for a different type of zeta function. With the basic analytic framework safely ensconced in his mind, he is well placed to appreciate the new ingredients and challenges that one encounters to study these new functions. Similarly, another student who wants to start his PhD with me, attended the lectures of this course. With foundational preparation in place, once he clears his preliminary requirements for PhD candidacy, we will be able to start working on a new project.
I love teaching. With every course I teach, I learn a new lesson to improve my teaching practice. Over the years, I have learnt the importance of being organized, properly prepared and of staying true to the course goals, while also leaving space for going beyond them. I have learnt that one earns the trust of genuine students through careful preparation and honest presentation, not hand waving or posturing. I have learnt that we can make mistakes during lectures, and these (mostly) do not cause the world to end. I have learnt that we must accept our mistakes, correct them and move on. I have learnt that practice leads to progress (and fewer mistakes).
Through years of study and research, we develop a "niche". This niche starts reflecting in our teaching. We owe it to the niche to make it reflect clearer and better. And that happens only with time, effort and practice.
