If you call yourself an analyst, you have to play with mud

 

So I am taking this Advanced PDE course this semester, which, perhaps unexpectedly, has been the most fun and useful course. Well the fun part was not perhaps unexpected due to it being taken by R Sir. He is perhaps the best teacher I have ever had. He will go lengths to make things rigorous.  Evans' PDE may be such that it has a reputation that if " you want to follow on with a more rigorous one, you can't beat Evans" according to one fellow (or as R Sir would put it, Chap) on Stackexchange, but it is but the most unsatisfactory for him. But since, as per him, basically every PDE book is shamelessly copied from Evans, he does the heavy lifting himself and basically writes 500 pages of notes just to teach us. Fortunately he also taught us PDE and before it Real Analysis (or is it called Metric Spaces?) in the first semester. Add to this that he is basically like Sheldon in the clip below. 

I initially (the idea came to me during class) thought I would not write stuff about him here as it would anger him, but then I thought:

1. Nobody I know reads this.
2. Its very very funny.
3. Nobody at all reads this.

Edit: Initial thoughts win.

He has this British accent and looks somewhat British too. But is cent per cent Bengali and son of an IPS babu on top of that. But he does have a very classy way of speaking and drops many bangers in class that I duly note down.

No, nothing should convice you   - R Sir 9/9/25

Or one that highlights his attention to detail,

 This is very boring torture, but needs to be done. If I did not do this, it will be cheating.  - R Sir 4/9/25

 This was when talking about the translation of convolutions of distributions and Swaqrtz class functions (ignore the terms should they mean nothing, then mean nothing to a majority of people, you will need a much lower amount of maths knowledge to know stuff where I actually need you to know maths). 

In short, his classes are the best. Still sometimes advanced PDE can feel like a slog. Our primary textbook is  Kesavan's Topics in Functional Analysis and Applications (which, despite the name, is a PDE text, I mistakingly bought it earlier but now can use it) and many a lectures are just an endless seige of statements of theorems and lemmas and propositions interleaved with some small Red Cross supplies consisting of Sir's motivations behind those. But I like to get my hands dirty, bring out those $\epsilon$ and $ \delta$.  Then only can I feel my wounds from the statement heal. That only fortifies my mind. 

So we were discussing Trace Theorem (Theorem 2.8.1 in the 3rd edition) for some days and today we finally reduced it to a statement about the density of smooth functions with compact support in $L^p$. And even further we were down to provinf that if $u \in L^p (\mathbb{R}^N)$ then $$ \int_{ \vert x \vert > k} \vert u \vert^p \, dx \rightarrow 0 $$             

Now, being who I am, I just took the interval as an indicator function and it followed with a simple DCT argument which is standard. But how Sir saw it was illuminating. He saw the problem as tail of a series. Now I know this from the Good Kenrels, but this was even finally putting that motivation into words. And in class I thought "hmm, series convergence should have a DCT argumnet htne too". After all summation is but integration with the discrete measure on natural numbers. And voila it is indeed, it works, This is no great discovery, but Sir and I saw the problem thorugh diffrent lenses, and then I find of pullbacked my meathod into's his. Thats the beauty of maths.

To the every abest reader, I was initially going to write only about the theorem and it does sounded much more grand in my head at that time, but putting it into works makes it sound meh. But still you will find i intresting if you look deeper., Anyway life is what it is and I will just end with another of his quotes from today itself about this proof:

If you call yourshelf an analyst, you have to know this. If you go outsidem you need to play with mud.