Teaching philosophy

Since 2013, I've been teaching mathematics courses as an instructor at IIT Hyderabad.

Calculus, multivariate analysis, real analysis, functional analysis, complex variables, numerical analysis, topology, measure theory and integration, and other mathematical courses are among the subjects I've taught. Without a doubt, lecturing on these topics improved my comprehension of the subjects and allowed me to go further into the significance of the topics.

I enjoy teaching because it allows me to share my knowledge of mathematics with others while also allowing me to revisit previous materials and improve my own comprehension. Some topics bring back memories for me, and I feel as if I'm taken back to the times when I was studying them.

I prefer to teach a single subject from numerous perspectives; beginning in one direction and progressing to a higher level may not be the same the next time. By means of what I've demonstrated may be an assumption, and I'll need to verify the facts based on my previous assumptions. 

I feel that the conceptual and problem-solving elements of mathematics are complementary in general. Indeed, having a good conceptual knowledge makes problem-solving strategies become more natural, making them easier to assimilate. Students may mistakenly regard these procedures as mysterious algorithms that they must memorize in absence of its understanding. The more capable students may still be able to solve these tasks, but the process has lost its purpose and is unlikely to be remembered after the course gets over. Although, like with any skill, a great deal of problem-solving experience helps to develop a conceptual knowledge.

I understand that viewing and going through a number of instances helps to improve one's mental picture of an abstract idea. Thus, my goal is to emphasize both sides of mathematics as clearly and concisely as possible, so that I maximize the student’s potential for success. My first course in IIT Hyderabad was a undergraduate Calculus course. I started my teaching when a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is said to be differentiable at some vector in its domain. I defined the differentiability of such function is similar to have a linear map between $\mathbb{R}\rightarrow\mathbb{R}$, with certain assumptions, which of course depends on the vector and the map f. This took a long time for me to address with the students. I spent time (even outside class, over emails etc) and finally enabled to give a glimpse that $f^\prime (x_0)$ is none other than a linear map between $\mathbb{R}^n$ to $\mathbb{R}$. I don't feel bad if I don't have enough time to cover more problem-solving techniques for other topics, while spending more time on a single topic. I found in some Calculus books the derivative of such a map is defined as a vector in $\mathbb{R}^n$ which in some sense I do not like.

I understand teaching/ proving a theorem in an undergraduate class is really a challenging job. Students are more oriented to learn problem solving methods. In some possible cases (in \cite{cal18, cal17} etc) I turn a theorem into an exercise and put it as an assignment. For instance in the chapter of Riemann Integration it is a well-known fact that Any continuous (or monotonic) function is Riemann integrable. When I put them as exercise I found it is more effective than teaching them as theorems. In my Masters' classes (like \cite{ra18, mi15}) whenever these discussions come I prefer to include that  Riemann Integrable functions are precisely those which are continuous almost everywhere (in some sense) in its domain. I had a plan to introduce the notion of a measure from this point, ie. a measure zero set is that where there exists a Riemann integrable function which is discontinuous in it. 

On the other hand in Masters level classes it is bit difficult for an instructor to spend more time on solving problems. I felt in this case a set of recorded lectures can be provided in advance before commencement of the class. Students can come to the class to get a better understanding and solutions for their assignments. I prefer to give some instances I discuss in various Masters level classes viz. \cite{ra18, mi15} etc.

In 1983 Hermite made a comment: I turn away in fright and horror from this lamentable plague of functions that do not have derivatives. I enjoy to surprise my class by saying that a function what one can draw in a paper is differentiable almost everywhere (in dictionary meaning). I can draw a function and in order to make it non differentiable it is enough to add some sharp corners into it. But such a function is very much differentiable except a few points. The example that comes to our mind as a non differentiable function is $f(x)=|x|$. But this function has difficulty in terms of differentiability only at x=0 and at all other points the function behaves like a affine function, which is certainly differentiable (infinite times).  Surprisingly, human eyes can not observe a nowhere  differentiable function. In 1872 Weierstrass demonstrated that such a function exists and can be constructed. Before Weierstrass people used to believe any continuous curve can be plotted graphically. One can see that a similar idea is being implemented (that to make some sharp corners) in this construction. The idea is one can create infinite ups and down in a neighbourhood to destroy the differentiability. I discussed these phenomenon in my class (in [5]) to give a glimpse to the beauty of abstract Mathematics. I feel enthusiastic to surprise students that the family of such functions is a fat set (in the set of continuous functions over a compact interval) and contains any separable Banach space isometrically!

In my Measure theory classes I prefer to start my classes with a revision of {\it Riemann Integration}. It still surprise me what a marvelous idea that Riemann had achieved in his time when there were no concepts of Topology,  Measure theory etc. The core idea hidden in the entire process of Riemann integration is the 'upper sum' (and the 'lower sum') decreases (increases) for a bounded and  nice function for a finer partition. I relate this fact with the theorem that comes in their initial classes like A bounded function can be approximated pointwise by a sequence of simple functions. The students get thrilled when they realize that they have already experienced this fact in their Riemann integration class viz. the upper sum (and the lower sum) converges pointwise to the original function and these sums can be viewed as a sequence of simple functions. So the mechanism of Measure theory allows one to replace intervals with more abstract sets (like Borel sets etc). This allows one to capture more complicated functions (like Dirichlet function etc) to make them integrable, which were non integrable in the sense of Riemann. All of the topics that are supposed to be covered in this course are very basic and can be applied to almost any area of mathematical analysis. 

In my Banach space theory course, I aim to emphasize the importance of developing a basic understanding of Schauder bases, reflexivity, smoothness, and convexity in infinite dimensional normed vertor spaces.

My goal is to instill knowledge in my students' mind that will enable them to extract geometry from whatever analysis they learn in the future.