सब सेट है, पर ठीक नहीं — पंद्रहवीं रैंक का इवान्

 हाल में ही CSIR Net का Result निकला। देश भर में रैंक 15 है। ख्याल था कि लेक्चरशिप का कटऑफ होगा, और असल में 15 रैंक है।

न उम्मीद थी, न कुछ और। कैसे हुआ, ये भी पता नहीं। यह तीसरी दफ़ा थी जब इस इम्तिहान का फॉर्म भरा था। पहली बार 2024 में था। उस समय तो पोस्टपोन हो गया था, NTA में चीटिंग वगैरह का कुछ मामला था। उस समय थोड़ी-बहुत तैयारी की थी, लेकिन सेंटर नोएडा में आया था। जब एग्जाम हुआ तो मेरी छुट्टियाँ खत्म हो गई थीं और मैं इधर ओडिशा में था, इसलिए दे नहीं पाया। उसके बाद दिसंबर 2024 में बाय-लॉज़ में कुछ बदलाव किए गए, जिससे हम लोग एलिजिबल ही नहीं रहे, तो फिर नहीं दे पाया। फिर गर्मी 2025 में दिया, कुछ नंबर से लेक्चरशिप रह गई। फिर अब दिया। तैयारी तो नहीं की थी। सिलेबस का तो अभी तक मुझे ज्ञान ही नहीं हुआ है। हफ्ते भर पहले तक CAM में था, फिर आकर एक शादी में गया था। कब ही पढ़ता? और वैसे भी न पढ़ने की इच्छा थी, न इस बार देने की। DD ने करीब-करीब ब्लैकमेल करके ही फॉर्म भरवाया था।  लोग मानते नहीं, सोचते हैं अकड़ रहा हूँ। मुँह पर तो कोई बोल नहीं पाता, लेकिन सोचना लाज़िमी है। अब खैर, ये कि DJ Khalid का Suffering From Success है। मज़ाक अपनी जगह, सच में मैंने कुछ नहीं पढ़ा था। हाँ, 3 दिन, और सिर्फ 3 दिन, DD के साथ पेपर बनाने बैठा था, लेकिन फिर मनमुटाव से, समय मैच न होने से और ऐसे ही कुछ मन की बातों से वह टूट गया। उसका नहीं हुआ। 0.4 से लेक्चरशिप रह गई। किस्मत की ही तो बात है, बहुत हद तक। कभी-कभी लगता है कि Fresh Off the Boat खत्म कर दिया, बहुत अच्छा लगा, अब किताब भी पढ़ूँगा कि मैं एडी हुआंग जैसा हूँ। उसके SAT में ऐसे ही बिना पढ़े 1600 आ गए थे। बचपन से ही वह स्कूल में भी बिना पढ़े कर लेता था। उसका सबसे छोटा भाई इवान् इस बात से चिढ़ जाता है कि उसके लिए कितना आसान है और जीवन कितना अन्यायपूर्ण है। डायलॉग ही देख लेते हैं:

Evan: You're both naturals, and I'm a sucker who has to put in the work.
Eddie: Dude, you're a straight-A student. Who cares?
Evan: I care because I have to earn it. You guys just wake up and it's there.

मेरे जीवन में कोई इवान् नहीं, तो मेरी अंतर्चेतना ही इवान् बन बैठी है। मुझसे पूछती रहती है। पाठक सोचेगा कि क्यों बड़बड़ा रहा है ये, इसका तो सब सेट है। हाँ, शायद सेट है, लेकिन सही नहीं लग रहा। शायद इसी भावना को लेकर पाश्चात्य में “प्रिविलेज” को लेकर इतना खेल होता है। अभी Crazy Ex-Girlfriend (यह भी अच्छा था, पूरा देख लिया, मानसिक स्वास्थ्य का संदेश देने वाला सिटकॉम है) देख रहा था। उसमें भी एक एपिसोड में नाथेनियल प्लिम्प्टन तृतीय को ऐसी चीज़ का एहसास होता है और फिर वह इस बारे में बोलता ही जाता है। क्या मैं भी उसकी तरह हूँ?

लेकिन सब कुछ भी तो नहीं मिला। श्यामा प्रसाद मुखर्जी फ़ेलोशिप लगती है कि मिल सकती है, लेकिन 15 रैंक आकर भी नहीं मिलेगी, क्योंकि जनरल में पहले 14 ही लोगों को बुलाते हैं। चलो, सब तो सही नहीं है। और हाँ, एग्ज़ाम आ रहे हैं। पहले भी लिखा था, जिसमें ऑपरेटर थ्योरी का कुछ अता-पता नहीं था, आज भी स्थिति बदली नहीं है।

 बरहामपुर वाले में चयन हो गया वैसे। GATE में आजकल AI से उत्तर मिला सकते हैं, उसका भी देखकर लग रहा है कि शायद 100 के अंदर रैंक आएगी। और लोगों की मानें तो 25 के अंदर। पहले एक बार निकल गया था, 616 रैंक। लेकिन मुझे एहसास ही नहीं कि यह अच्छा है या खराब। इन सबकी सभ्यता से अलग ही हो गया हूँ मैं। इंजीनियरिंग करनी तो थी नहीं मुझे, लेकिन JEE कल्चर से बहुत हद तक जुड़ा हुआ था मैं। Kota Factory
 भी देखी थी। पहला शो था मेरा, और उस समय सोचा था यही है। 

खैर, ये बात तो पुरानी है। फिलहाल इस हफ्ते के एग्ज़ाम के लिए ही पढ़ना चाहिए मुझे, कम से कम कोशिश तो करें।

Departmental Picnics, Operator Algebras and why Calrson wins over Calculus at this stage

 Next week will hopefully be the last mid sems of my life, soon after will be the last end sems and in the months following that will be my Comprehensive exams, marking the end of my exam-taking career. This looks so exciting, much life before the class 12 boards, which marked the end of my school career, but exactly like the last time my mind has raced ahead and my motivation to study for these exams is zero. Haven't yet even tried to open the textbooks. I usually spend my time reading for my research or for pleasure these days. Some Wizards of Waverly Place thrown in between, or recommendation (see balckmal) of DD, for good measure.

This is just as the post exam phrase I had previously described. Mind is a poet, and as they say, "जहाँ न पहुँचे रवि, वहाँ पहुँचे कवि. And indeed, the Sun is bound by the laws of time and is scheduled to rise about 200 times to meet the temporal place my mind has occupied as of present, 

I think despite all the warnings from seniors, Algebraic Topology is easier than expected. Currently, we are following Munkres for the Homotopy part, and after middsems another text will be followed for the Homology part. Perhaps it is so because we learnt a lot more than required stuff about fundamental groups in our first topology course. Anyway, one subject being easier is good. As for Operator Theory, while the syllabus for the course looks like this:

Compact operators on Hilbert Spaces. (a) Fredholm Theory (b) Index, C*-algebras - noncommutative states and representations, Gelfand-Neumark representation theorem, Von-Neumann Algebras; Projections, Double Commutant theorem, L∞ functionalCalculus, Toeplitz operators

In reality, due to a combination of department politics and egos of people involved, the course has morphed into some combination of Operator Algebra and von Neumann Algebra and Harmonic Analysis course following the legacy of VS Sundar, being mostly taught from Stratila & Zsido's Lectures on von Neumann Algebras and Javier's Fourier Analysis. This feels like a betrayal. The same politics have led to WCNH attendee list looking like a family picnic of certain academicians. The Douglas clan is evidently not invited. The ugly departmental politics aside, I have been a bad picker of electives anyway. 

On a side note, I had to cancel the prospective Sampablpur trip due to clashing with the exams. Now looking at the Mini workshop: Cantor set & Brownian motion at IISER Berhampur at the end of March. Anyway, ranting over, I do need to look at those vNA notes, but only after a few more hours with Carleson (1962) whose end is now in sight. 

Navier-Stokes-R Sir

 Okay, I am in awe of R Sir, have successfully cleared Advanced PDE after much stress (88 marks nonetheless, 2 short of the perfect A grade) and have gone to a PDE Workshop at CAM with a dubious level of utility extracted. Still, my heart yearns for more PDE and R Sir. Krantz's Partial Differential Equations and Complex Analysis sits proudly on my table and combines three of my favourite things in Mathematics. 

So when R Sir sent a mail announcing that he wants to lecture from Robinson et al's The Three-Dimensional Navier–Stokes Equations, I was perhaps one of the first to reply back to get on the mailing list. Timing is an issue this semester, but still, I can't bring myself to pass up this opportunity.  

He is lecturing in a very Feynman technique sort of way to a mixed audience of postdocs, PhDs, advanced undergraduates and even physicists! In his own words:

I am seeking an audience that is willing to hear me lecture on whatever I have managed to understand from the first four chapters of that book (I haven't attempted anything beyond that.......trying to understand the first four chapters is a sufficiently ambitious goal). 

My goal is to try to unravel the proof of the existence of a weak solution for the Navier-Stokes equation. 

If you are interested in listening to what I have figured out....

The thing is, I don't particularly care about the Navier-Stokes equation or Fluid Mechanics for that matter.  First, I am not that interested in physics myself. Second, despite the mythicalization and the omnipowerful status granted to the problem in online discourse due to the Millennium Prize, it fails to account for many things and doesn't even correctly model water at very small scales, which is not a criticism so much as a reminder of what the model is and what it is not ought to be evaluated to. Nor do I care much for the classical solutions, I am much more interested in Weak Solutions and the techniques of manipulating a nonlinear PDE. My point of view seems to be shared by R Sir too, for when talking of some boundary constraints inthe case of the Torus, he clearly stated:

Although I personally don't care about the Physics.

Literally my words. He then went out to start from the very start, the Geometer's definition of Laplacian:

$$ \Delta := -\left( \frac{\partial^2}{\partial x_1^2} + \frac{\partial^2}{\partial x_2^2} + \frac{\partial^2}{\partial x_3^2} \right) $$

 Not that PDE persons', not analysts', not physicists', but of his own tribe. No wonder we got assignment problems from Warner's Foundations of Differentiable Manifolds and Lie Groups in the first (mind you, first, not the Advanced course) PDE course. Like me, to him, PDE is a tool and a delightful one at that. I, too, have started to prefer his definition of Laplacian, it creates some notational inconsistencies between my thesis and the bibliography, but the results about Subharmonic functions make much more sense with this definition. 

I am sure a lot many people will drop starting today, but I aim to complete the lectures with perfect attendence. LB, with our firendship levels up on the sky, will also be attending so at the very least we will have fun tackling stuff we didn't understand. I can't wait till my COmprehencive exams after which I will be beyond the cycle of mid sems and end sems तस्य हि ध्रुवो मृत्युर्ध्रुवं जन्म मृतस्य च. Passing the comprehencive is the Nirvana moment of the PhD life. When one can just audit courses witjhout caring about exams. R Sir have just granted me this boon early, and I will make the most of it. 

Pythagoras, Baudhayana or Meera Nanda?

 I recently started reading Meera Nanda's Science in Saffron: Skeptical Essays on History of Science. This is about some pseudoscientific claims made routinely by people on the Hindutva side. While our side is indeed suffering from many quacks, something which I myself have noted previously, but still the publisher 'Three essays collective' is a known leftist publisher. Also the book is dedicated to her "comrades". Predictably the preface starts with bashing the Sangh. 

However, while ideologically opposite to us, Nanda is still a scholar. She is not a a humanities type trying to weasel her way into the debates of sciences. She is a PhD in Biotechnology from IITD who have later branched into history of sciences. She is a someone no responsible person should ignore, for निंदक नेड़ा राखिये. Her sharp tounge and witty-to-her comments are more than countered my her scholarship and research. One can disagree with her philosophy that harmonizing science with traditional worldview is not conductive to reasearch (while I agree that radical decolonized relativist "science" say as taught in New Zealand is, but disagree with Nanda on the very thought being wrong), still, her footnotes does not care about our feelings. We can't attack the thesis unless we learn the facts. Perhaps she is not actually anti Hindu and writes against Hindutva side exclusively only for it is now rightfully and joyfully the dominant force politically. Still even in the first 50 or so pages she did give a much softer rapt to African scholars claiming more than their share of mathematical discoveries. This is much relevant when now there have been attempts to portray Yoga as an African practice.

I have read till the end of first chapter only and I did change my views. I read a rejoinder to it ( more accurately a earlier version published as an article) but it is mostly rhetorical and trying to catch her in a gotcha moment when there are none. As facts stand now (or at least in 2016 when the book was written) the first statement as well as proof to what we call the Pythagoras theorem are chinese in origin. Nothing can be done about this fact. It does not make us inferior. She rightly acknowledge that the Suvlasutras tackle a very challenging problem and can be appreciated even if it were not the source of Pythagoras theorem. There is a very nice article in Bhavana (excellent magazine) exploring this. The author however uses the term Baudhāyana-Pythagoras Theorem. To Meera Nanda the mere act of naming is just a childish act of one-upmanship and turing science into a zero sum olympics fought by civilizations. 

Her exposition is very clear, it would be a delight to read a textbook written by her. The citations check out. One has to concede this chapter to Nanda. 

CAM Diaries: Post Lecture Learning

 While the actual learning in the lectures have almost came to a stall, I still have learned a lot here. First of all I was wrong, one can't lump Bengaluru with Delhi, it is an ideal mix of Delhi and Odisha. It is a metro city where the seasons matters. At least Yelahanka is such as area. My threpatic walks have lead me to take steps in just 4 days I normally take in a week. 

It is indeed ironical that walks here are more conductive. Greenery, elegant temples and just the, dare I use that word, vibes in general are unmatched. I should indeed study Kannada temple art, this is fascinating. I have never been into architecture before, but I do need it now. Just another thing I do need to know. 

One thing that I am required to know is that the workshop was about since the last lecture begins in like half an hour. I can hear Ismail bhai locking his door, perhaps to breakfast before going to the lecture hall. It's not that I am totally lost. I have a big picture idea. I just think that this is so divorced from my work that I am not putting on the elbow grease. But I did learn a lot about workshops, grants, reaserch life, paper publishing and collaboration from fellow but senior attendees. Who am I kidding, 90% of my information comes from my roommate who is 5th year PhD in IIT Roorkee. But still, I think it is very useful and I would have never encountered it at NISER. As for the lectures, we still have t actually solved the original SPDE promised. Today, perhaps. But I think the model algebra and trees are similar to something I saw at the IMA Conference on Geometry and Topology in the Honour of Prof. M. S. Raghunathan, F. R. S. I dont exactly recall who, but I do have my notes back at NISER. It was a talk about algebric computation or something should I be remembering correctly. Don't think it's a one one correspondence, but still I think recalling that might be useful. 

While I try to plot these connections of ideas, the streets outside are highly connected. Like too much I think. The neighborhood is semicircular. All the radial streets are called Main Roads. And the one normal to them are called cross roads. Is is indeed a geometric layout, fitting of hosting TIFR CAM. We, LB and I, went to ICTS yesterday. That thing doesn't look Indian. Nah. Truly international. But in an outside locality. Bad connectivity. Yelankha on the other hand is very well connected. 401 bus are like a dense subset of the streets. During our walls, LB and I love to explore random cross roads and look and comment at the houses. They look so peaceful and posh. My working theory is a lot of them are retirees. This is indeed a good area to retire. Langauge barrier will be the only problem. My kannada level is at "Anna eradu ticket beku". English will take me far, but need to learn Kannada should one aspire to retire here. But that is talk of day after tommorow. Today the more pressing matters is end of workshop. 

CAM Diaries: It goes over head now

 Three days of lectures have passed as quickly as these lavish bedroom in my CAM appartments  lulls me to sleep. Faster than Ismail Bhai's (for I can't properly pronounce the Ismaël Bailleul) words flew over our head. The darkness of ignorance is like in which I left NISER.  No तमसो मा ज्योतिर्गमय granted yet. The thing is this workshop is about solving certain singular parabolic heat type Stochastic PDEs with a novel renormalization technique developed by Fields Medalist Hairer. Phew. Long sentance. Knew it was out of my bounds much earlier. Ismail bhai thinks that it is the Hopf algebraic structure of Models and regularity which might be a problem. But that part sits well with me. My problem is the standard PDE ailment. Kesavan avoids in as does R sir, but they are outsiders to the field. So am I. Perhaps that point of view is what suits me. But I think a lot many "natives" are also lost in the workshop. LB most certainly is. This I know from our long walks. Other I talked to are in the same boat too. But even as an outsider, since I work in Corona Problem, PDE is an important tool in my toolbox. I have been in an search of the elusive proof Hall's Lemma.  There is an 1983 paper which derives it using Stochastic methods. Which was one motive to come here. But alas it didn't help directly. But now I feel now confident to actually tackle the '83 paper rather than cowering hearing Stochastic. Anyway random pics from the walk today. 

CAM Diaries: दिग्गज, दोस्त और दिव्यता

एक दिन वर्कशॉप का बीत गया। एक थोड़ा बहुत बस idea सा लगा कि क्या हुआ लेकिन कुछ ठोस नहीं। अब जिस विषय में क ख ग का ही अंदाज़ा ना था, उसमें यदि 2021 के पेपर को पेश किया जाए तो इतना भी बहुत बड़ी बात है। काम cutting edge है एकदम। और बहुत लंबा चौड़ा, तो ऊपर ऊपर से ही बताया जा रहा है। इस्माइल भाई वर्ता है, और कल रात पता चला कि मेरे ही साथ वाले कमरे में ठहराएं गएं हैं। नामी दिग्गज है, उनके पास रुकना ही तो सम्मान की बात है। मुझे भी एक आलिशान अपार्टमेंट मिला है। रूड़की के एक भईया के साथ मिला है लेकिन ये इसकी तो रसोई ही मेरे निजी के कक्ष से बड़ी है। निजी लैट्रिन है। खाना भी एक नम्बर है और सस्ता भी। काश 2023 में इधर मुझे ले लेते, दिल तो बस गया हे इधर। जाते वक़्त दुःख होगा। अब क्या ही कर सकते हैं। आस पास का इलाका भी शान्त साफ़ सुथरा है। सुबह शाम टहलने जाना भी एक Therapy है इधर तो। आज सुबह ही गली में एक सुन्दर का मंदिर मिला।

भव्य मंदिर होते हैं गली गली गंगापट्टी के बाहर। देखकर ही दिल में दिव्यता का अनुभव होता है।

वैसे तो इधर जो है मेरे काम से बाहर है। लेकिन आज और कल इस्माईल भाई बोले हैं कि कुछ tools develop करेंगे जो analysis में काम आएंगे। The mechanics of local expansion और Renormalised continuous models । अभी तो हवा तक नहीं की ये क्या है। देखते है भविष्य में क्या होता है । पलक्कड़ में गए थे और ज़्यादा सीखे नहीं, बुरा लगा था। इधर नहीं लग रहा लेकिन। LB के साथ खाना, चाय, घूमने का ही कार्यक्रम करते रहते हैं। पिछले 6 महीने छोटी सी कक्षा में पढ़े तो दोस्ती तो बढ़ गई थी, अब तो बिलकुल भाई बहिन ही हो गए हैं। अच्छा है। कुछ तो फायदा हुआ। लेकिन नहीं सीखने का प्रयास करना है, बिना टेंशन के लेकिन ।अब जाना होगा दूसरे Lecture के लिए। चलते हैं।

NISER to CAM with Azithromycin

 It's 3 am and it's pitch black inside. Skipped bath. Bath, irrespective of the water temperature - which anyone needs to be regulated by my induction for the geyser works in timming nobody knows and nobody can change - and duration, would have resulted in death due to my cold which have survived 5 Azithromycins. Still I must leave for the SPDEs workshop I have been talking of again and again. The car won't come to the hostel, for there is some nonsensical rule about gate timming which is sporadically implemented, perhaps the workshop will help with this knowledge. Yet in the half dead state I must push on, push on I must. 

Arrived at Bangalore, or Bangluru as one have to say it now. Flight was not at all how I assumed it would be. It was nice. While apparently sinus worsening in flight or Aerosinusitis is a real thing, in my case the pressure difference seemed to have made some nudges in the postive direction. Nah, not nudges, particularly potent proding it was. It am all but almost cured. The Banglore skyline, welcoming me for the third time (and second time when I am headed for CAM) camouflaged itself like the early November Delhi smog, minus the smoke. So yeah just fog. I almost forgot how nice it felt in childhood when we could experience pure(ish) fog. And the excellent BMTC and KIA brings me and the batchmate LB at the gates of TIFR CAM. 

Workshop starts from tomorrow, as does hopefully my learnings too. Now it's just unpack, wait, relax and watch.

Post seminar debriefing

 वर्षों तक वन में घूम घूम, बाधा विघ्नों को चूम चूम
सह धूप घाम पानी पत्थर, पांडव आये कुछ और निखर
सौभाग्य न सब दिन होता है, देखें आगे क्या होता है

-Ramdhari Singh Dinkar in Rashmirathi


If I may be dramatic, and being dramatic is what I suppose I am at least entitled as of now, and quote Dinkar, yes, my luck has woken up. And woke it did, at 1600 on the 25th. Yes, the 25th I called dreaded on the day itself. It has been kinder to me that I had been to it in times of tension. Even when ungrateful, I have somehow still been treated kindly, I have been a lucky man most of my life. In this way, at least. Well by topic was nice too, I got to show an image from a news article titled Solar Eclipse 2019 in India: Best Places to Spot the Annular Solar Eclipse in India   in a maths talk. It does look beautiful. 

Dare, and that I will for I am on my entitlement period as abovementioned, I say it looks better in the slides. 

Ek aur https://t.co/XdKn8B4h3w pic.twitter.com/ml7WF8y3l6

— Aryan Kumar Prasad (@Aryan_AKP) November 27, 2025

Gloating over what I call aesthetics (my lack of taste of which will be clear from the Blog's theme) and success aside, lets go over the talk. There were faces I didn't expect to come. Like last semester, R Sir made an unexpected entry. But this time, like in my Seminar for his class and unlike the RKHS Seminar in April, he was more content and fine rather than bewildered and bored as one normally sees him during talks. This was the highest praise I got from the seminar. This trumps multiple audience members and 2 committeemen letting me know that my talk was good. Even Panchalli, of the-algebraist-taking-advanced-PDE fame, said that it was the most entertaining talk of the session and the only one where she did not feel sleepy, even if I understood nought. My talk did not have elaborate poetry or such, or jokes or gimmicks as some love to add. The most "gimmicky" thing was the photo of the Solar Corona to explain why the Corona problem is called so. But blame Newman for coining the term. This one ain't on me. 

My narrative structure did work, it seems. But one committee member, better known as Fatakeshto for his in-seminar antics, he told at the end of the talk that "you promised to take us to Kashmir, but took us to Daringbadi. I was not bad, but misleading". But I still think the talk would be too bland without the "clickbait" of the Kashmiri Corona Problem.  Estimates are dry, pure estimates might catch the fancy of some, but I think motivation, and mind you I ask of no dastardly "real world" applications, for estimations are needed. And hopefully, next semester, I will be completing reading Carleson (1970) , with of course help from Garnett which unlike Duren has actual complete proofs., and take the committee to Kashmir, but only if  तुने मुझे बुलाया शेरा वालिये, मैं आया मैं आया शेरा वालिये . That's it, hopefully no more seminar talk for six months. 

Pre Seminar briefing

 The dreaded 25th has arrived. There have been no 'gentle reminder' but my guide has already announced my talk before:

Dear all, 
This is an announcement of a math seminar by an integrated MSc-PhD student. The details are as follows.
Speaker: Aryan Kumar Prasad
Date and Time: 25-11-2025, 4:00 PM - 5:00 PM.
Venue: SMS Seminar Room.
Title: Zeros and Factorization of Bounded  Analytic Functions
Abstract: In 1962, Lennart Carleson solved the Corona Conjecture, a major problem concerning the maximal ideal space of the algebra of bounded analytic functions on the unit disk. Carleson proved the result by solving an equivalent analytic formulation known as the Bezout problem, or the reduced corona problem. The proof relies on precise estimates of the moduli of bounded analytic functions, which in turn require an understanding of how these functions vanish. In this talk, we will derive the necessary and sufficient conditions for the zeros of a bounded analytic function and construct the canonical factorization using Blaschke products.
All are cordially invited.

Slides are also made, the report submitted and printed out. Mocks given. What remains is the actual talk. This blog post, the bound on whose maximum readership will not be above 5, is just ranting, venting and procrastinating. 

I have tried a different approach to the seminar this time. Narrative based. Starts with motivation, develops the machinery, and states some estimates without any proof (they are in the report though) and even lays out a roadmap for completing the proof of the Corona Theorem a la Wolfe.

 The meat of the talk that develops the Blasckhe products is the part I found most unsatisfactory. The theory is nothing much advanced, even Rudin and Convey have it (but I will swear they were absent when i was looking for them). The "big brain" proofs are the equivalence of Corona Problem nd Reduced Corona Problem as well as the two estimates, neither of which will I be proving. I can only hope to make "motivation" masaledar enough to suit the bland basic complex analysis I am serving.

The tension this time is unlike before. I might have said before that I was more worried about the presentation in R Sir's class. That turned out well. But I was worried about a last minute change and in aew of R Sir there. For today, I think my work itself is not worth anything. I am not afraid of $\epsilon$ or $\delta$, the problem is, I am not getting much after boring the committee with those. Also, I do think I am rather unprepared to deal with questions if someone asks me more regarding the Gelfand theory implications. 

I should not waste my time here and get back to those functions. I will just leave you with my thoughts, which Nana Patekar perfectly delivers:

Advanced PDE Seminar Advanced

I previously noted the tention regarding the internal assessment for the Advanced PDE course. Today it happened, and it was a job well done, and satisfying enough it was. What was supposed to be a 20 minute disaster became a 2 hour learning cum satisfying experience. 

As noted previously, once again it was Krantz who came to my rescue. And on top of that I even prepared notes. Yes, me, the one who needs a scribe to do rough calculations, wrote down notes myself. Full 12 pages of monstrosity it is. It was, for I need now to rewrite it and submit it to R sir. 

The entire talk was like an actual teaching experience with an argumentive, attentive, eager and open audience. It was unlike previous seminars which were an anxious perfomance to a largely don't-care audience with some troublesome elements sprinkled in between. It was teaching. Teaching like R Sir. Teaching as if the teacher cares if the subject. As do the audience. Board work, matters. Pace of writing, matters. Knowing what to erase, matters. Giving references, matters. Knowing which step is the key, matters. Teaching can be an art as well as a science. To quote Snape on Potions, it is a "subtle science and exact art". Well, it can be. We all know what it can be at its worst.

Chalo, talk done for this sem too pic.twitter.com/wyoKoaZyYq

— Aryan Kumar Prasad (@Aryan_AKP) May 2, 2025

I think it is probably one of the few times R Sir appeared to be almost happy with any student. I made a silly silly mistake regarding some domain, was caught and panicked. Instead of the usual R Sir we all love and fear, he took on the stage and worked it about on the spot with me. That he did. Unbelievable. Even consulted me about it. He has an heart, it's just not open to public. 

Anyway R Sir was satisfied with my work, I have no need to give another attempt or attend other's second and third attempts. All this modulo submission of giving the assignment next day that is.

सेमिनार से बड़ा सेमिनार, Advanced PDE की मार

 हर 6 महीने में सेमिनार देना होता है। 25 को फिर देना है, पहले भी बोले थे। कल उसका ही mock है। तीसरी बार देंगे इस बार तो उतना, यदि बंगालियों की भांति बोले तो, चाप नहीं आ रहा। अच्छा है। शायद आदत सी हो गई है अब। लोगों के सामने बोलने में शायद पांचवीं कक्षा से ही डर लगने लगा था। कारण अज्ञात है। अब ज़हर का नाम जान के क्या ही फायदा? नहीं फायदा तो है, Anti venom शायद सब का अलग होता है। थेरैपी का शायद ये ही फण्डा है। ज्ञान होना चाहिए ये सब का भी। 

लेकिन ये ज्ञान recommended श्रेणी में है, सेमिनार आवश्यक है। उसका मोटा मोटा आ ही गया है। उससे खराब दिक्कत तो लेकिन Advanced PDE में है। हाँ बहुत अच्छा विष्य है, वर्कशॉप भी जा रहे हैं। लेकिन जैसे उस वर्कशॉप की चिन्ता है वैसे ही अब course के लिए भी लग रहा है। 

अधिकांश तो आर सर के विष्य में अधिकतर मेरा काम चल जाता । उनका जीवन बुनियाद में ही समर्पित है । अच्छी बात है। गणित शब्दों या अंकों (Number Theory करने वालों भाइयों के लिए एक सावधानी) को इधर उधर करना और एक equation या वाक्य गढ़ देना नहीं है। क्लर्कगिरि नहीं है ये। रामानुज क्लर्क थे, लेकिन जब ये नौकरी मांगने गए तो Ramanujan: The Man and the Mathematician ( मेरी समीक्षा) में लिखा गया है कि उसने कहा लिया कि

 Ramaswamy Ayyar: It is too bad. If you become a clerk in any of these offices, your mathematical abilities will soon dis- appear. I do not want to sin that way.

अब उनपे देवी नामगिरि की असीम कृपा थी । लेकिन अय्यर साहब की बात हम आम लोगों पे वैध है । आर सर जैसे लोग हम शोधशास्त्रों को इस क्लर्कगिरि के जाल से बचाने में लगे हुए हैं। कई बार इसको इतना श्रेय मिलता नहीं, श्रेय दूर अपयश मिलता है। लेकिन ये सुकर्म है, आवश्यक है। हम 3 ही छात्र है इस विष्य में । तो internal के 13 अंक के लिए सब को एक एक सवाल present करने दिए हैं। अब सवाल सुन ही लें, Integration by parts on   ${H^1 ({\mathbb{R}_+}^2)}$ और Trace map के Image पे है। हमारी किताब में है दोनों। हम Kesavan की Topics in Functional Analysis and Applications ही पढ़ते हैं कक्षा में, इसमें ही 2.8.1 और 2.8.2 देना है। मुझे 2.8.2 का दूसरा भाग मिला। लेकिन इसमें त्रुटि है, और चोटी मोटी नहीं, पूरा का पूरा proof नष्ट कर देने वाली। बहुत ही basic गलती है, जिसका derivative ही नहीं उसे Schwartz कह रहा है। अब इतना ही गलत होगा ये थोड़ी कोई सोचेगा। हम भी सोचे नहीं थे । आज present करना था, उसके घंटे भर पहले जाने की क्या कांड कर रहा है किताब में। घबरा उठे । जल्दबाजी में  Strongly Elliptic Systems and Boundary Integral Equations में जवाब मिला । कक्षा की और जाते हुए ही पीडीएफ फाइल मिली। आज तो गए भईया। मेरा नंबर दूसरा था। लेकिन जब चालू की पांचाली (नाम बदला हुआ) तो बेचारी को 20 मिनिट जो मिले थे, उसमें तो कुछ नहीं हुआ। एक घंटे में भी कुछ हुआ नहीं। फिर पता नहीं किसकी महर थी, shayad मंगल है न आज, नॉनभेज त्यागने का फल है बंजारबली से, लेकिन R सर ने कहा कि तीसरा होगा। उसका तो चालू करने से पूर्ण एक बुनियादी सिद्धांत पे रोक दिया। दरअसल वो, मैं और किताब भी एक छोटी सो चीज़ को नज़रंदाज़ कर रहें थे। लेकिन सर से नहीं छिपता ये, कैसे छिपे? निठरता तो घोल पी गए हैं। आस रखे थे कि हम लोग भी अब तक कुछ चरणामृत की भांति पिए होंगे। नहीं लेकिन खड़े उठे आज तो। नहीं उठे। उनको न वाक्चातुर्य से मतलब है ना हि एकान्तप्रयास से। कई दफ़ा तो बाहर चले गए कहके कि तुम लोग आपस में बात चीत कर लो । अब परसों मेरा है। सेमिनार, जो 300 नंबर का है, उससे ज्यादा इसमें, 13 नम्बर का, से डर लग रहा है। लेकिन सच कहे तो इसमें सीख भी अधिक रहे है। अब चलो 2 दिन ईश्वरीय कृपया से मिला है तो अच्छा है। तो Krantz की Partial Differential Equations and Complex Analysis में भी छान बीन किए, और मिला। ये सबसे मंगनी किताब है मेरी, फेलो बार काम आई अभी। सही है। लेकिन इस आदमी का काम बहुत व्यापक है। मेरे से भी बहुत मिलता जुलता है। एक न एक बार इससे मिलना है। 

अगर PDE की किताबों की बात हो तो महाविद्यालय में बहुत सी पढ़ी । Partial Differential Equations with Fourier Series and Boundary Value Problems ( मेरी समीक्षा), सिंह-शर्मा की Partial Differential Equations for Engineers And Scientists, Sneddon की Elements of Partial Differential Equations ( तस्वीरों ), Amaranath की An Elementary Course In Partial Differential Equations ( मेरी समीक्षा),  लोकनाथ देबनाथ की  Linear Partial Differential Equations for Scientists and Engineers, Raisinghania ये सब तो भाई नहीं। Coleman की  An Introduction to Partial Differential Equations with MATLAB पे प्रतिक्रिया तो अच्छी लिखी थी, लेकिन कुछ याद नहीं आ रहा। छाप तो कोई नहीं है। और तब तो अपरिपाक्क था। फिर से देखना होगा, ही था क्या। डिस्ट्रीब्यूशन Theory से तो था नहीं, मतलब असली गणित के तौर पे तो निकम्मी ही है। अब खैर, हर किताब का एक पाठक होता है। इन सब का मैं न था। कोई न कोई होगा। अभी तो चिंतना मेरे इस सेमिनार को हे, और उसके बाद अलसी वाले सेमिनार की ।

Review: Theory of $H^P$ spaces

Theory of H[superscript p] spaces by Unknown Author
My rating: 2 of 5 stars

This book is cited as the only other (non-probabilistic) proof of Hall's Lemma from his 1937 paper. It was also perhaps the first book to contain Carleson's proof of the Corona Conjecture. The book however is very lax in places, harmonic measures are not even defined explicitly. It makes no use of Distribution theory and Greens' Function is treated like an actual function.


For Carleson's Proof, one should instead refer to Garnett's Bounded Analytic FunctionsBounded Analytic Functions. He follows the latter approach of Carleson and ditched harmonic measure for an alternative proof from the lecture in Proceedings of the 15th Scandinavian Congress, Oslo, 1968. Garnett have two books on harmoic measures- Applications of Harmonic Measure (The University of Arkansas Lecture Notes in the Mathematical Sciences) and Harmonic Measure , but both are lacking the Hall's Lemma. As for the other content in the book, Introduction to Hp Spaces and Banach Spaces Of Analytic Functions have much better exposition.

View all my reviews

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.