Editor’s note: I don’t want to preface every sentence hereafter with various versions of ‘I am no historian but…’ so I am going to assume that every sentence henceforth that deserves that qualification already starts with it.
I find Ancient Roman history fascinating.
First, the Roman Empire was large. Very, very large. At its greatest extent, it covered the entire Mediterranean coast, stretching from the British Isles to Egypt. In a pre-industrial society, with communication taking weeks, this was a massive stretch of land for a single political entity to control, but even so, it was not totally unprecedented. When Alexander the Great died, his empire controlled more land, and there were brief periods coinciding with Roman rule during which various empires united large parts of the Indian subcontinent and China. But none of them lasted for as long. Alexander’s, for example, fell apart almost immediately following his death.
To get an idea about what communication would have meant in the Roman era, check out ORBIS.
Rome’s giant empire lasted far longer. If we concentrate only on the Mediterranean coast, the Romans already controlled most of it by 50 BC, and remained in control of that territory for the next four hundred years. Given the slow communication, the Romans needed to be experts in military and administration, and also had to be able to find leaders who could first conquer and then govern the far-reaching provinces with some level of independence from the central authority, continued for multiple generations. To put this longevity into perspective, 400 years back from today puts us in the 1620s, when Galileo was a middle-aged man, the Taj Mahal hadn’t been built, and… It’s so long ago that I couldn’t find any other examples to contextualize it.
The political systems underlying this are therefore interesting simply because of the outcome: a large entity that actually lasted for more than a fleeting moment.
Starting around 500 BC, Rome had four major political eras. The first, lasting until 27 BC, was the Roman Republic. Next was the Roman Empire, which was effectively divided into the Western and Eastern Roman Empire in 286 AD. The Western Roman Empire lasted until 476 AD, while the Eastern Roman Empire survived in some form until the 1300s.
I am personally most interested in the Republic, for two main reasons.
The first is the system itself. While not unprecedented (republican governments existed in Ancient Greece) the Roman Republic was the first republic to control territory that went beyond just a city state. By the end of Julius Caesar’s conquests, the Republic controlled most of modern France, nearly all the Iberian Peninsula, all of Italy and Greece, the Mediterranean coast of Turkey and much of North Africa. It would not be outlandish to claim that the stability of the Republic was what made much of this possible, and resulted in a long line of capable leaders who led it through several crises and eventually turned it into a superpower. The systems that provided this stability lasted for several generations, and this was actually unprecedented. Further, even after the Empire replaced the Republic, the administrative backbone continued to be a reformed version of the system from the Republic, which makes this era worth studying.
But that skips a few steps, because stability itself is a double-edged sword, and might have been a strong contributor to the end of the Republic in the first place; the second interesting thing. By Caesar’s time (around 50 BC), the Republic was in trouble. Since stable systems are hard to change, all of Rome’s vast territory was still being administered as if it only controlled the city of Rome, the riches from the conquests had fed widespread corruption, and concentration of power in the resulting rich families had turned it into an oligarchy. Further, the Roman senate, which was the seat of the Republic, had been paralyzed for decades by partisan politics fueled by class divides, and a huge effort would have been needed to fix these issues. Caesar attempted some fixes, but was assassinated before he could finish and within twenty years, the Republic would be dead with Caesars nephew (and adopted son-heir) Octavian as the first Emperor ). The fall of the Roman Republic, to me, is a historical example of what happens when a political system that works well for a time slowly falls apart, and how much work needs to be done to keep these systems running.
Caesar’s untimely death left has left historians debating if Caesar’s actions killed the Republic for his selfish interests, or if it was close to dying anyway and the Empire was an improvement; like all things, the truth is probably in the middle.
Apart from conquering and administrating, Romans were also adept and prolific writers and their writings have survived over the years, to the extent that we have a decent idea about conversations between historical figures from two millennia back. For example, Caesar’s own accounts of his Gallic Wars and the civil war with Pompey have survived, and while these are of course embellished, they give a very personal insight into his eventful life. Even stories of political intrigue have survived. We can read and think about the machinations of Cicero in the aftermath of the Caesar’s assassination and analyze his words when he worries about the end of the Republic; such an opportunity is rare for a person who lived so long back (example: this video).
All of this meant that I have spent an inordinately long time learning about Rome and Romans. Sufficiently abstracted, there’s often a lot to learn. The tenacity of Rome that allowed them to survive the Punic Wars, winning the war despite losing several devastating battles. Julius Caesar, the genius on the battlefield and the way he waited for the smallest advantages, exploiting them fully when he got one. How the people he trusted the most were among the ones who (literally and figuratively) stabbed him in the back (looking at you, Decimus Brutus). Bibulus, a passive politician who always managed to find himself at the wrong place at the wrong time. Labienus, who stood up for his ideals over his friendship with Caesar. Cicero, who tried his best to save the Republic, but whose kind of politics was the reason why the Republic was malfunctioning to begin with. Octavian, and how he used his intelligence and ‘ambition that outshone the sun’ (quote from this video) to overcome a sickly life and multiple serious illnesses to became Augustus Caesar, the first Emperor. His friendship with Agrippa, and how important Agrippa was to Augustus reaching the heights that he did.
Given all of this, when I visited Rome in July of this year and sat in front of the Roman Pantheon, reading Agrippa’s name on the facade made me feel a connection to the place and the long dead person that is hard to describe. Rome is a city that feels like it is a continuation of its ancient past, a modern metropolis with one foot firmly rooted in history. I could almost imagine what the narrow cobblestoned roads must have been like in Roman times. I could enter the Colosseum and imagine being part of a cheering Roman crowd around me cheering, though admittedly helped a lot by the portrayal of the place in Ridley Scott’s Gladiator. I could walk on the Roman Forum surrounded by ruins of ancient buildings, and picture some famous senator giving a rousing speech to the citizens. In more somber reflective times, I could also put myself in the shoes sandals of a Roman citizen, perhaps right before one of the several sackings of the city when it fell in the 400s AD. Would I think of the (until that point) 900-year history of the city until that point and be confident that this crisis, like the many, many crises that the city had survived, would also pass? Or would I be able to see the (with the benefit of hindsight) clear signs of decline and drop everything to move somewhere else? What is it truly like when an institution like that falls? Maybe the Romans were good enough at writing things down that someone recorded their thoughts regarding this…
If the problems faced by the Roman Republic that led to its fall seem somewhat familiar to you, you are not alone. Just before publishing this, I saw this answer on Quora which is an interesting read even though I don’t agree with the conclusion in that answer.
Diocletian’s story is all the more funny to me because of a book that added the observation ‘Maybe he had a cabbage for a brain’, but looking up his full quote while writing this made me respect him a little more: “If you could show the cabbage that I planted with my own hands to your emperor, he definitely wouldn’t dare suggest that I replace the peace and happiness of this place with the storms of a never-satisfied greed.”
The sacking of Rome in 455 AD by the Vandals, a Germanic tribe, left its mark on the language: that’s where the word ‘vandalism’ comes from.
Let me (re?)introduce you to relative deprivation theory. I first heard of it in Malcolm Gladwell’s talk at Google Zeitgeist several years back, which you can see here - Why Did I Say “Yes” to Speak Here? | Malcolm Gladwell | Google Zeitgeist. Very basically, relative deprivation theory suggests that (quoting Mr. Gladwell)
As human beings, we do not form out self-assessment based on our standing in the world. We form our self-assessments based on our standing in our immediate circle, on those in the same boat as ourselves […]
So when I found myself as the only one in my circle with no plans for Christmas, relative deprivation theory kicked in HARD. I had seen decorations preparing for Christmas weeks in advance. I recalled all the fun I had had in Christmases past, which I had always spent with family. For a few hours on Christmas Eve I wanted to be anywhere and with anyone rather than spend Christmas alone in my room but some dark part of me also wanted to give in to my overwhelming laziness and spend the day staring into space moping about it. But then I surprised myself…
Over the past few weeks I have realised that even though 2020 has been a soggy-thing-that-bulls-drop-on-grass-fields year, it has also taught me some valuable lessons that go beyond the year itself, and one of those is to be aware of my own mental health and discover ways to face any challenges that come up. At some point I decided that spending Christmas within my room or outside it was entirely up to me, and I ended up planning a solo ‘trip’ amidst the natural beauty that is conveniently situated close to the College Park area but that I hadn’t (and haven’t) yet fully explored. I walked to places that I would normally take an Uber for, went on trails that I had never been on, saw some deer, watched some streams (filled with water to an unusual extent because it had rained the night before), did some bird-watching (now I know what a Northern Cardinal looks like), and in the end, rounded up the day with a nice dinner from a restaurant that I hadn’t tried before. Not bad, at least according to me.
I imagine this won’t be the last time I need to do something like this, but thanks to this word-you-get-if-you-say-sheet-quickly year I believe I will know how. Thanks but no thanks.
]]>It has been about a year since I received my undergraduate degree, which was also my last in-person meeting with all but one of my college friends. Many of us are now scattered all over the world, and our lives have changed massively over the course of the past year. Much of our social interaction is now mediated purely through screens and texts, even before the pandemic increased the extent. I don’t have to travel to and from my workplace now, and I have used some of the free time to wonder about how friendship interacts with distance. 1
There are two ways to interpret that question, of which one is the subject of this essay. I will not consider here if friendships can happen over distance, but rather what happens when previously co-located friends now have to worry about the literal distance between them, and whether that translates to figurative distance too.
The first thing to realise is that change is inevitable. If the friendship grew at a common location, when the people involved saw each other every day, it inevitably gets tied to the prevailing circumstances. When we do not need to work for it, we tend to underestimate things like non-verbal communication or interactions happening by accident and not needing to be explicitly set up. The shared location itself is a ready source of conversation that is of interest to both people, and there’s also the effective information transfer that comes from being part of the same social network.
Consciously making time for a friendship and taking into account personal schedules and time zones feels very different from the relative spontaneity of the past. Finding topics to talk about can also become hard - because small talk about mundane details might not be interesting to both people any more.
Without belabouring the point further - maintaining exactly the same kind of relationship with a friend that existed in the past when both people were co-located might not logistically feasible, and, if not, we need to find a new equilibrium. That’s what will concern us for the rest of this essay.
The first thing is to define a realistic goal. This may vary from person to person, but we can find some general guidelines. Talking every day, for example, might not be a realistic goal because of the logistical issues involved, but maintaining the same level of comfort might be. This comfort level is a more abstract notion that is not easily described by easy metrics like how often two people talk. As an often reshared post on social media reads, the best friends are those with whom you may not have talked for a month and yet when you do talk it feels like nothing has changed. What this is talking about is that the comfort level hasn’t changed in that month, and frequency of interaction might be a lousy metric to judge this.
It is also important to recognise that people themselves change, not just the friendship. The person you’re interacting with now may be quite different from the person from a year or two back - adjusting to a new location and its challenges can have its effect. In case two people are spending the effort to actually maintain a long distance friendship, it is safe to assume that they are invested in it; if so, living out a version that one has grown out of just for the sake of the friendship not only discredits the investment but also puts an unnecessary strain on the friendship, and should be avoided.
Now that we have some idea of what a new equilibrium might look like, the question becomes how we reach it.
A stable equilibrium is one that both people are okay with, an ideal situation which might not be easy to reach. But the only hope of reaching anywhere close is a deliberate commitment to clear communication.
Interacting via texts means missing important information from tone of voice and body language that provides context, and therefore our words must be taken at face value in such interactions. This makes misunderstandings easier, and demands more work from both people to avoid them - which can eat into the already limited time for interaction or worse, start feeling like a chore.
Misunderstandings happen even when people are face-to-face, and a medium like texting makes them worse. Letting misunderstandings remain is a sure-shot way of ensuring people will drift apart, and so we must talk about them to the other person. Even when they didn’t exist before and now do, by themselves grievances are not a sign that the friendship is weakening; if anything, it shows that you still care enough. Being able to discuss grievances reflects the strength of the relationship and feeling afraid of voicing them does just the opposite. It is important to have enough trust in the relationship to know that one hard conversation will not kill it.
The question of whether two people have drifted apart is complicated by the simple fact that we don’t perceive our feelings on a continuum. As an example, take any one of your closest friends, and ask yourself when it was that you became close friends. In most cases, even if you can point to a specific time, it will be a time when you realized that you were then close friends, but that doesn’t mean the friendship only became close from that moment on but rather that the continuous process of getting close has been discretized by our minds. I propose that the same holds true for drifting apart - suddenly you’ll wake up one day and think that things aren’t the same as they were before. Also since no one likes drifting apart from people one wants to keep in touch with, one tends to ignore the signs that it is happening, which also results in late detection.
I like to use an analogy here. Issues in a close friendship are like grains of sand in a very fine machine. Their presence is easily noticed, and can easily strain the machine because one is so used to everything running smoothly. The smart thing to do is to recognise the presence of a sand grain and get rid of it. Ignoring it is just going to fray the machine more over time.
This isn’t easy to do, and requires the investment of time and effort, which is a big change from not having to work very hard when co-located. The difficulty isn’t surprising, though - it is never easy to have something change from a point that we are comfortable with.
One other thing to keep in mind is that it is not possible to maintain relationships with everyone we once knew, because the number of close relationships we can maintain is limited. So, some people are bound to drift apart. Agonising over this doesn’t help, and trying way too hard might put an unnecessary strain on that and other relationships. It might help to ask if the relationship with the person is important to you and to be as objective as possible in answering it. If the person is indeed important, and you don’t mind a possibly difficult conversation, then lead it. Speak out about your expectations and grievances. If the other person reciprocates, well and good. If not, you save yourself a lot of long-term heartburn.
Reaching out also has a different kind of utility - sometimes a friend may just be going through a hard time themselves, and might not be in the mood to initiate a conversation. Maybe they’ll appreciate you reaching out.
For the sake of completeness, it is possible that you are much more invested in a friendship than a friend is, or that you are able/willing to expend much more effort than your friend. Asymmetries of this kind can be hard to deal with. At some point the only remaining thing is to let go and find a way to be okay with it. Maybe while you do so, also mentally thank the friend for teaching you a valuable life lesson - that after all, change is the one inevitable thing.
I hope this has been useful to some of you who are in a similar boat as I am. Every friendship is unique, but I hope these are abstract principles that generalise well.
]]>As I was growing up, my grandfather told me many fantasy stories which contained magical protagonists doing fantastic things on winged horses in far-away kingdoms beyond ‘seven seas and thirteen rivers’ involving giants, fairies, demons and whatnot. Many of them were either original stories or adapted from other fantasy tales by Bengali authors, and as such they contained many influences from Bengali and Indian culture. One of these was the concept of the ‘ishaan kon’ - if the story contained a sword that the hero could use to defeat the demon, it would always be hidden in the ‘ishaan kon’ of a lake or garden. I didn’t know then (and never asked) that the ‘ishaan kon’ or the northeast corner is a holy direction in Hinduism, but because of the significance that it carried in so many of the stories, I thought of it as a particularly hard to reach corner of wherever the hero was.
The second is an incident from primary school. The film Ghajini starring Aamir Khan came out when I was in primary school, and you could see the lead actor’s physique plastered all over movie posters everywhere. One of my all-time favourite teachers, who used to teach us maths at the time, told us that just like Aamir Khan had been able to build his muscles by exercising them, we could build our math muscles by exercising those. She added that it was difficult to focus on both at the same time, and that we could choose between being muscle bound and being good at math (a very layered message for a kid in primary schooler, though, with the benefit of hindsight, it’s obvious what normal primary schooler would choose).
In middle school, I encountered the Chetan Bhagat book ‘Three Mistakes of My Life’. It was never my favourite book by Bhagat, let alone being among my favourite books, but the core of the story - about a passionate cricket coach, called Ishaan, who mentored a talented youngster, Ali, to achieve the cricketing dreams he had never himself been able to achieve - was good, dressed as it was in Chetan Bhagat fluff.
A few years later, ‘Three Mistakes of My Life’ was made into a Bollywood movie called ‘Kai Po Che’, and it was the first big screen role for Sushant Singh Rajput who was already a known face on television. When the tabloids wrote about him, we discovered that he had managed to train his math muscles, dance muscles and muscle muscles at the same time; he had qualified for the Indian National Physics Olympiad (which was one of my dreams), and that he had secured a single digit rank in the AIEEE (one of the big national level engineering entrance exams, that I would appear in after a few years). We found out that even after getting into a decent engineering college, he had left all of that midway to pursue his dreams in dancing and acting, and in doing so had come as close as anyone ever did to the Indian middle class tenet of getting a good engineering degree and then pursuing whatever you want. In doing so, he made himself more relatable than the average Bollywood star - most of whom have familial connections in the industry.
I had never wanted to pursue a film career, but his courage in leaving the beaten track and pursuing his passion had an appeal that went beyond the specific passions he pursued. He even played the role of Ishaan in ‘Kai Po Che!’. It must have been a role that he jumped at - playing someone passionate about his goals. With every film he made, he solidified the transition from the familiar path of a studious kid to the ‘ishaan kon’ (the way I thought of it, as a hard to reach corner) of being a Bollywood star.
It was a few years after his debut, after he had already made a name for himself in Bollywood, that I chanced upon his Instagram account. It was unlike any other filmstar Instagram account I had seen, filled with him talking about philosophy, life, ticking off some of his life goals (which, like him, spanned many, many different areas). I remember following him almost instantly. Later, I discovered that he still interacted with his fans on Instagram, replying to many of their comments, which is also something that is unexpected from a filmstar.
This was the person who passed away a few days back - passionate, living a life that many middle class Indian teens dream of while still maintaining his own identity. I first heard the news of his death when I opened Facebook early one morning and looked at a long post talking about how unimaginable it was that someone with a filmography that he had would do what he had done. It took me about half an hour to process it - my newsfeed had several posts about what he had meant to several different people, and even then I had to Google him to confirm the news. Two of my favourite people in the Indian film industry - Irrfan and Rishi Kapoor - passed away not too long back, but this… this felt personal, because this was one person in the industry that I related to. It feels like if this could happen to even a guy like that, it can happen to anyone at all.
Several stories have continued to come out about him and the lives he had touched. One in particular concerns Digvijay Deshmukh, who played the youngster Ali in ‘Kai Po Che!’. A junior cricketer then, he had told Sushant that he would not meet him until he became a decent cricketer. He was picked up by the Mumbai Indians, but the pandemic ensured the meeting did not happen, and it is not going to happen any more.
This and many other unfinished stories that he left behind should be reminders for us to take mental health seriously and for our society, as a whole, to have more empathy than it does right now.
‘Kai Po Che!’, like the book it is based on, is set in Gujarat. The title is a Gujarati phrase meaning ‘I have cut’, used when a competitor uses their kite thread to cut off another competitor’s kite thread during the Makar Sankranti celebrations. The life-thread of the lead actor in the film is now cut, and no one is saying “Kai po che!”
Directly inspired by an Instagram story by Parul Pawaskar
]]>The sound of the quill pen dragging across the paper was like music to him, drowning out the mooing of the cows. The light rain throughout the day meant that he hadn’t needed to take them out of their sheds. He was thankful for that; he didn’t have to run after those cows and get feed for them. He had celebrated by spending his time in his room, thinking and writing.
Such days had been few and far between ever since the university had shut down a couple of months previously. As he dipped his pen into the ink bottle, he glanced out of the window. He had wanted a day when he could work uninterrupted, but now he yearned to go out into the yard. The day had been long, and he had not made much progress. The numbers shuffling around in his head just didn’t arrange themselves in a way that gave him any insight. He looked a little closer at the sky - maybe it did look a little brighter? It would need to clear soon if he wanted to take his walk outside, otherwise it would be dark and it wouldn’t be quite as refreshing.
He wrote a few more lines, but the glance out of the window had diverted his attention permanently. Stretching back in his chair, he looked at what he had written. He had slowly understood some of the principles that make the world run, walking further on the path indicated by Galileo than Galileo himself had. It had not been easy - he had found the math that he had at his disposal to be much too clunky to describe the beauty of nature’s inner workings, and he had developed his own ways to describe how nature itself moves. He often thought of the world around him as some kind of clockwork, much like the pendulum clock at the university, except that a few simple gears weren’t the only things he could explain - his ideas could do much more than that if they were true.
If they were true? Of course there were true! Three small ideas were enough to describe every motion around him. If the sheer elegance wasn’t enough, they were powerful too!
He was still not satisfied though. He had imagined the clockwork nature of the pendulum extending to many things around him and it had worked beautifully. Especially on a farm, he could see it in action everywhere - pebbles rolling, people and animals walking, someone raising water from the well - his ideas were everywhere!
He was itching to see if they extend even further. Could this clockwork structure extend further, to the moon, say? The sun? The stars?
He returned to what he was writing. His thoughts placed him among the stars - what chance did a measly yard stand against the universe?
There was no clock on the farm. It disappointed him, even though he had several sundials, all of which he had made himself, strewn all over the farm. They weren’t so useful on a cloudy, rainy day.
He liked looking at clocks. Especially the precise craftsmanship of the pendulum clocks, and how they were independent of the weather. He had thought of making one - but he had been keeping his craftsmanship to one side. The mathematics was too interesting at the moment.
A sudden brightening attracted his attention back to the yard - the rain had stopped, and the clouds were clearing a little! He quickly set aside his pen and rolled up his parchment and within a few moments was out of the door.
He could tell that the sun had just set even though the clouds hid almost everything. A few stars could be seen in the region of the sky that was devoid of clouds. He felt strangely close to the stars - he thought about them so much, after all. On second thoughts, though, one of them seemed to be a planet. It wasn’t twinkling. Venus, maybe? The evening star.
He felt the wet grass below his feet as he walked to the far corner of the farm. He would sit underneath the trees there and have a good hard think. That should open his mind up.
He reached his preferred spot. The branches of the tree seemed laden with fruit. He considered plucking one, but decided against it. It wasn’t time to eat yet.
He could feel becoming one with the nature he wished so dearly to understand. The plan was to stay there and wait for inspiration to come, the way it had come several times before. He shut his eyes, and just then…
Thud.
A sudden sharp blow to the head broke the spell. As the apple fell to the ground with another thunk, he could almost feel his spirit drop like it had done. Why did it have to fall just then and spoil his thoughts?
Why did it have to fall?
Why did it fall?
He looked at the apple. And then back at the stars in the sky. The moon showed itself for a moment through a crack in the clouds. The clouds were clearing further… and not just in the sky. He could feel realisation and discovery send a shiver down every part of his body.
The numbers were in order now.
He couldn’t even be angry at the voice that came from far away.
“Isaac? Where are you, Isaac? Go right now and see whether the cows have enough food. You should have done that an hour back. Isaac?!”
He took one last look at the apple, before walking slowly towards the cowshed.
He had his answer. The clockwork did extend further.
]]>I remember another ride, on a bus this time, to the Tai O fishing village, and feeling transported to a different world, far away from the tall buildings dressed in glass. Here there were villages and shops selling sea-food that I had never seen before. Was this really less than an hour away from the city? I would later discover that the tall buildings were just one facet of the place - about 70% of the region is forested land!
I remember the first time I went to the CUHK campus, at first loving the fact that I would be spending one-and-a-half months in that green campus on that hill. Almost immediately, though, I remember loathing the fact that I had to trudge up the same hill, dragging a heavy suitcase on steep footpaths on a warm and sweaty summer day. It reminded me of Kolkata, the heat and the sweat.
I also remember how I was looking at a map on the street corner, trying to figure out where the room I was going to stay in was, and the kind elderly lady who appeared, asking me what I was looking for. She then walked with me, in the opposite direction to where she was originally going, into random buildings and even more random elevators, until she put me on the road to iHouse, block 3, and ensuring that I could get back to the place where she had found me. We talked on the way. She said that people new to the campus often get lost. I would learn over time.
In a couple of weeks, I could walk up from the metro station at the bottom of the hill to iHouse, where I lived, and from there to the Ho Sin-Hang Engineering Building, where I worked, taking all the possible short-cuts through buildings and the elevators there-in and not even being out of breath. The campus had adopted me. Revealed to me her secrets. The elderly lady on the first day had made me feel like home; this made the place feel home.
I remember going to one of the many canteens on campus on one of those first few days, walking past the densely forested slopes, accompanied by my parents. I remember looking at a sign that said that the canteen was for students and employees of the university only. I remember asking the person at the counter whether my parents could also come in, but her not understanding English beyond whatever was on the menu, and the man who walked up to me, asked me what the issue was, and when he heard me, who gave me a big smile and said ‘But of course everyone is welcome!’ and then proceeded to explain to me all the meal plans that I could avail at that place.
I remember exploring different kinds of food at the many canteens on campus, trying real Chinese cuisine for the first time. You know I grew up in Kolkata, and my liking for chowmein and chilli chicken or fried rice and chicken manchurian. I had heard that Chinese food in China is nothing like Indian Chinese. Very true, that. What I had also heard was that if you like Indian Chinese, you won’t like Chinese Chinese. I completely disagree! You know how I loved every single dish I had there, including a very hot chicken dish that came with a free litre (!) of Coca Cola to make it bearable. I also learned to use chopsticks, so much so that by the end of my stay, having chowmein was easier using chopsticks as compared to a spoon and fork.
You also know how eagerly I waited for mooncakes to hit the shelves as I was approaching my last few days in the city, and how fulfilled I felt after buying the two cases as they came to the stores just on the day before I left.
I remember being naïve and thinking I could ‘pick up’ some Cantonese while staying there, and my disappointment at discovering how different and therefore complicated the language is. You know how I found out about name seals, and how on one of my weekend trips I sought out the small, telephone-booth like shops along Man Wa Lane, Sheung Wan that sold them, getting my name translated and getting one made for myself.
I remember how, after my parents left after us enjoying the city for a week and also celebrating my mother’s birthday, I left every weekend to explore the city. It was the first city I really roamed around alone. Every weekend I left iHouse, walked down the hill to the metro and travelled to the middle of the city, seeking out something I had read about the day before on Google and found interesting. While coming back my phone would be almost out of charge, and I would need to know which metro stop to get down at and the way back to my room because Google couldn’t help me any more. I can now take my phone and travel fearlessly, cheaply and conveniently to any part of a modern city, having since done so in both Paris and Washington DC.
I somehow also love how I have recounted so many happy memories without even mentioning the research group I was part of, the amazing advisor I had there and what my time in the city means to me academically. You know how I had only recently changed my research area from quantum computation to classical information theory, and how stressed I was about the switch. You know how very happy I was when I finally had an academic internship for the summer of 2018. The time in Hong Kong not only gave me a wonderful advisor whose recommendation letter and general guidance made navigating graduate school applications and finally getting and deciding on an offer a much smoother experience but the papers I wrote directly based on the work I did there helped my applications a lot. You know how I used to be iffy about embarking on a PhD, but how one conversation with Sid gave me the confidence that I wouldn’t feel like a fish out of water while doing my PhD. We went on research meetings on the beaches, enjoyed Chinese food at restaurants recommended by my advisor there and so on - there’s a lot!
I grew up travelling to and from school on the Kolkata metro hearing frequent comments about how I should carry my bag differently, and even how schoolchildren with heavy bags on a crowded metro were an inconvenience. I can now contrast that with the day I left the city, carrying two heavy suitcases on the metro during rush hour, half expecting someone to tell me I couldn’t carry them, or to berate me for doing so. I don’t know whether I didn’t overhear any stray comments simply because I don’t understand the language, but you know what, I will go out on a limb and say that there were none. It seems in line with my other experiences in the city. It felt like a pat on the back, a last bit of understanding, a last memory to cherish.
I now read about the turmoil the city is going through for months now, and I remember all of these memories. I see protesters. I see university campuses under siege. I see pictures of roads that I know blocked and tear gas shells flying over fields that I have walked past. I feel like people I know are in the crowds of people I see on the news. Maybe they are. When I also see videos about people recording goodbye messages to their loved ones, not knowing what the government’s reaction might be, I feel scared. But some part of me also feels proud that some people somewhere in the world are standing up for their rights. I feel concern for the well-being of the friends I made in my few weeks in the city, and the places and institutions I got to know. It feels like another lesson from the city, even though I am now miles away - making me realise, for the first time in my life, that the crowds protesting are made of people, and are not just faraway masses in faraway places.
I hope you are well. I hope the city resolves its problems and remains as welcoming as it was to me when I washed up on its shore for the first time. I hope that I can recognise the city when I get an opportunity to go back there and reconnect with you.
You, the bit of my heart that I lost on the streets of Hong Kong.
]]>I remember the first time I went to the PK Kelkar Library and looked at the vast collection of books, and the goosebumps I had when I first roamed about in the Physics section. It was a few days into my first year, and when I called home later that night, I excitedly declared to my mother that if I read one of those many, many books per month, I’ll have learned a lot before it was time to graduate. Needless to say, like most of the plans I had made in the first few weeks that I had spent on campus, this did not work out. It did not work out to such an extent that the only time in the past year that I have stepped into the library (apart from graduation formalities) has been to get print-outs and one other time to roam in the physics section, trying to recapture the naïve awe from my first year. What happened in between is something I will call the Four-year transform and will try and spend the rest of this essay trying to elucidate the properties of this strange beast.
Here’s how the Four-year transform operates.
On entering the campus, we are faced with a multitude of choices about everything related to campus life. Some will probably not influence much of our future lives, like the question of what to do with the roll of TA101 sheets after the course ends. Others do have some ramifications later on. Questions in the early days like, “Which clubs to join?” and “Which Takneek competitions to participate in?” later change to, “Which courses to take?” and “Which professors to approach for projects?” It is clear that the choices in the second category are not scalar quantities – for example, if one chooses to join the Dance Club and also becomes active in Science Coffeehouse, it’s hard to imagine how they can both directly contribute to the same outcome. Therefore, we can say that the effects of these decisions add like vectors and that it’s only the resultant of the different pulls that matters. What the Four-year transform does is that it changes the magnitude of some of the vectors and rotates some, both of which change the resultant. So to understand the resultant, we need to understand the properties of the Four-year transform.
Now that we have some idea of what the transform is, it is instructive to try and see what it doesn’t transform, namely, its fixed points. For me, it didn’t change my love for science or the motivation to do research. It has matured, sure, and I certainly understand what research involves far better than I did then, but the essence of my fondness remained the same. It should be a valuable exercise to try and figure out what your fixed points are, or, what you want them to be because it can reveal what is most important to you. I am sure everyone has some fixed point(s), even though the specifics may be highly variable from person to person.
Coming to the things that it does transform, we can try and find some properties of the transformation. Briefly, they are
Integration property – The only way to judge the effect of the transform is to observe the person at the end of the four years.
Time-scaling – The amount of time spent in a particular activity has some effect on the future impact of that activity.
Strong non-linearity – More effort doesn’t always lead to proportionately more payoff, and
Time-shifting – Time shifting of events changes outcomes in unpredictable ways.
Now, let’s try and tackle these one by one, breaking down the aspects and its effects, trying to build a parallel with real world scenarios.
The integration property tries to capture the differences between the way your life in IITK will be interpreted by you yourself on one side and the rest of the world on the other. It is very unlikely that anyone will care about the specifics of your education at IIT Kanpur after you graduate. All that you study in the courses, all that you learn in the clubs will only leave its mark in what you know and can apply. It is too early for me to say whether your CPI will matter in the long run, but even if it does, it is likely that people will care about only the number, not how you got it. In some sense, what you do on campus will be represented in life as a few lines or some number on your resume – no one will be interested in how they got there. For example, my fifth semester was by far my busiest and craziest semester, and at the end of it, I got two of the grades that I am personally most proud of getting. One of them was a B in EE370, Digital Electronics. Due to circumstances, I was not able to study for it at all, and my preparation for the midsem and endsem consisted of watching the lecture videos that Prof Mazhari so kindly sent us, three hours before the respective exams. I think that, given the situation, I have grounds to be happy about that grade. The other one was a B in PHY690G, which was a course on coherence and quantum entanglement meant for advanced Physics undergraduates, MSc and PhD students before they embarked on real research. I am rather proud of the fact that I, as a third year UG in a different department, was able to do enough in that course to get a B. But, both of these will be one among many B’s (among many A’s and a sprinkling of C’s) on my transcript. Given the fact that many PhD applications are judged in around 5 minutes, I don’t think anyone even noticed the B’s, let alone my personal story behind them.
In contrast, everything that you do on campus will leave its imprint on you, because your own personal story is the sum total of all of these little things. The wide world might not care about the specifics of the story, but you do – for example, the two courses above taught me how to take on challenges that might have felt insurmountable, what to focus on while having to learn something quickly and so on, skills that came in useful later in life. The balance, therefore, must be between optimising both what other people will care about and what will be important to you as a person. And, as usual, the balance is hard to reach.
The best illustration of time scaling comes from the following. There are two ways to approach a course and its exams – either attend most of the classes, do assignments on time and go to the exam well prepared or mug up three days before the exam and trust your brilliance (if any). The difference is that in the latter case, you have compressed a semester’s worth of studying into three days, and that means that the impact that the course will leave on your knowledge, long term, will be scaled down by an appropriate factor. The more time you spend on a particular topic, the more comfortable you’ll be with it (and the more the long term impact) – importantly, and vice versa. Take a wild guess what the long term impact of EE370 (above) on me was.
Before moving on to the last two properties, let me narrate another small story. The first seeds of me studying information theory were sown in the middle of my second year during a series of talks organised to celebrate the birth centenary of Claude Shannon, the father of information theory, but, before the third year summer when I went to the Chinese University of Hong Kong (CUHK) for a research internship in information theory, I had only studied the subject for a cumulative time of around a month. At that point, by comparison, I had studied physics for more than two and a half years – so much so that I was able to handle the PHY690G course mentioned above. Fast forward a year, and I have a couple of publications in information theory and in a few months, I will be starting a PhD in the same area. I stopped doing Physics about a year back.
The points to note are the following. One, the timing of Shannon Centenary Day was crucial – a few months earlier, and all the talks would have gone right over my head. A few months later, and I would likely have already chosen another topic to work on. Either way, my life could have been very different. The impact of an event, therefore, depends crucially on when and how it occurs – which is the time-shifting property. Also, the pay-off per unit time spent in physics and information theory were very, very different for me – which is the non-linearity. The irritating bit is that the only way to judge the impact or pay-off of a decision is with the benefit of hindsight, the importance of which becomes apparent in the next paragraph.
The non-linearity and the time-shifting property together imply that it is difficult to predict the eventual impact of your choices and actions. This lack of control and predictability is in sharp contrast to the life in the JEE days, where the relationship between effort and outcome was much less messy. The following, I feel, is the most important thing that I learned during my own Four-year transform: Most of the time, the decisions that you must take cannot be fully justified at the point of time at which they must be taken and each of these must then constitute leaps of faith, big or small. The most effective way to handle this randomness, I feel, is to try and take the best possible decision given the data available to you at that point of time, along with the willingness to take responsibility for the results and the ability to handle the outcome, good or bad. If the outcome isn’t good, at least you’ll have the satisfaction of knowing that at every point leading up to the outcome, you took what appeared to be the best available path.
The next natural question, therefore, is that given the unpredictability, how to decide what the best choice is at a given point of time. To answer this, I must invoke the keen insight of Prannay (one of my closest friends and in general a person worth listening to) and point the reader to the birthday paradox. Briefly, the birthday paradox is the following (perhaps non-intuitive) statement – in a group of 23 people, there’s a 50% chance that two people have the same birthday. The idea is that the probability that none out of several low probability events happen decreases pretty quickly when the number of events increases. Therefore, it would perhaps help to choose, early on, a target and a sufficiently large neighbourhood set of outcomes that you want to reach when you graduate, and then work hard enough and smartly enough such that your probability of achieving something in that neighbourhood increases. With a smart choice of neighbourhood (such that a sufficient number of events exist in it) and an honest assessment of your own skills (which will tell you the probability that you will achieve something in that set of events), you can try and maximise the chances that you’ll be satisfied with where you are when you graduate.
We only discussed the birthday paradox in this context when we were both deep into our final years, so I have no idea whether this works. But, it does share some characteristics of a good theory. First of all, it’s elegant. Second, it does explain some things well. For example, someone asked me how much CPI matters for an MS/PhD offer. I said that a higher CPI doesn’t hurt, and if you can build a great research profile and also maintain a high CPI, then go ahead and do it, also adding that no-one knows the precise trade-off. This could be interpreted as an aspect of increasing the probability of the target outcomes. The theory also explains the fact that those with very strong research/job backgrounds or very high CPI’s or both get good offers for higher studies/jobs – essentially they make the probability of some of the outcomes so very high that the chances are low that they’ll miss that. If you are in such a utopia, great! If not, invoke the birthday paradox to come to your rescue.
Well, that’s my two cents. Maybe some of you would be kind enough to apply this in your lives and tell me what the outcome is so that there is some data to judge its efficacy. Good luck, and may the Four-year transform be with you!
]]>I walked up until I was standing just behind him and cleared my throat in an attempt to attract his attention. He didn’t do so much as glance at me! On looking at what he was doing, I failed to understand a word for it was definitely not any language I knew, even though the script was the Latin script. I could rule out English and German for sure, but there were too many other options. As he continued to write, I was a little surprised to see some known algebraic symbols. Even though the notation was archaic, it was something related to solutions of some equations that I had seen in my college math classes! This man was clearly a mathematician with what I thought were impressive powers of concentration, for even after I cleared my throat more loudly, I received no response.
“Excuse me?” I said in English, loudly enough for him to hear, but still there was no reaction. I tried to poke him, or push his chair, or pick up one of the many papers, but my fingers seemed to pass right through everything. I could move nothing, change nothing, and he took no notice of me. Wherever I was, it appeared as though I couldn’t interact at all with physical objects in this world, and from what I could see, no noise I made reached the man in front of me.
“Don’t try to attract his attention, you won’t be able to do it!”
My initial shock was quickly replaced with relief, for the voice was of my guide! She had been standing very close to where I had originally been, but the long shadows had hidden her too well. I could still only see her silhouette and she made no attempt to move into the light.
“Welcome to Paris,” she continued, “You seem disoriented. That’s natural, happens to everyone the first time they experience it, though I must say you are taking it rather well! Until you remember how you ended up here, let me fill you in. This is the year 1832, and it is the month of May.”
Even after everything that I had seen or heard, I still couldn’t believe that this was possible. But what choice did I have?
I started putting two and two together. We were in Paris and there was a young mathematician probing the secrets of algebra. It really wasn’t very difficult.
“So that… that’s Galois?”
“Yes, that’s him.”
“We can’t affect anything in this world right? I tried to move stuff, but my hand seemed to pass through solid bodies! He doesn’t even seem to know we’re here!”
“We can’t change events in this world, we can only observe. if you travel back in time, you cannot change anything that happened. You can’t interact with objects or people and no one will hear any sound you make, except other time-travellers like yourself. Imagine everything here is a hologram, you will just pass right through any object.”
“That’s exactly Stephen Hawking’s chronology protection conjecture, right?” I asked, while observing my leg which seemed to appear directly out of a table. It was a strange feeling! I hadn’t even noticed that there was a table where I was standing. The question was of course rhetorical. I had only recently read about that conjecture. My companion must have known this, for she didn’t answer.
Before I could ask any of the many questions in my mind, we were interrupted by a groan from Galois, as the candle went out. He quickly got up and relit the candle. There was a picture of a woman on his desk that I hadn’t noticed before, and he seemed very upset that in the darkness he had caused the picture to fall on its face. After correcting it almost reverentially, he gathered some sheets together and continued to work. He seemed to grow more restless as time went by, repeatedly glancing at the clock that hung on the wall. It was four in the morning.
“You can go closer, like you did before. He can’t sense your presence, remember?”
I did. There were sheets of paper all over the floor of the room, every piece of paper filled from corner to corner with mathematical symbols only some of which I could vaguely recognise. There was a smaller desk just beside the desk on which Galois continued to work, and on it was something that seemed strangely out of place in what was definitely the abode of a young mathematician – an old fashioned gun, recently oiled and ready for use. It was now that I fully understood what I was about to witness in a few hours and why I had been brought to this particular day in the life of a mathematician I admire so much.
My thoughts were interrupted by the sound of breaking glass. Somehow a bottle of ink had been overturned and was now lying in pieces on the floor, and there was ink all over some of the sheets. Galois looked close to tears but that only stayed for a moment before his steely determination came back. After quickly cleaning the mess and opening a new ink bottle, he started writing again. I could now totally understand his urgency and a feeling of dread was spreading slowly throughout my heart. I wished I had never come here, for it would be difficult to witness what I would surely witness.
I decided to explore the room a little bit more. Everything around it had an air of neglect which is difficult to describe. It was as if the room was frequently uninhabited. There was stale food in unlikely places. The books, shelves and papers in the room were all almost uniformly dusty. I couldn’t pick anything up, of course, which was a shame, for there must have been documents there that could’ve given more of an insight into the great mind which I could see at work in front of me.
My guide had been entirely silent for the entire time I was roaming around the room. Now I heard movement from her side. She was looking out of the window and from the amount of light it appeared as though dawn was approaching. For the first time I could see her in some sort of illumination and she looked remarkably human, except that there was no hair on her head and she had larger than usual eyes which she had slowly turned towards me. Not for the first time, I felt like I was being X-rayed.
“You are surprised to see what I look like,” she observed. Correctly, of course.
“Yes. I have read articles on astrobiology that say that the portrayal of extra-terrestrial life forms in films is sort of wrong because it is unlikely that aliens would have human like forms. Yet you look so remarkably human!”
“What you read is true, but we have no fixed form. We can imitate any physical object or life-form. We can’t do it on a fine enough level to imitate hair, though, the result of which is there for you to see.”
Before my already bewildered mind could process this completely, I became aware that the continuous sound of writing that had stayed with us throughout was now gone, and the only noises to be heard were coming from the city outside that was slowly waking up. I looked at the man and he seemed to be looking at the picture on his desk. He had a piece of paper on his desk and he seemed at a loss for words.
Presently I heard another kind of sound entirely. Footsteps.
“His seconds must have arrived,” said my guide. Galois must have heard the steps too, for his manner became decidedly more rigid. He glanced at the clock, and let out a small sigh. The steps had now reached what I imagined would be the staircase outside the door. Soon, there was a knock on the door and he got up to answer it. There were a few words in French, and I looked at my guide, nonplussed.
“You don’t know French, do you?” she asked, half-mockingly. “Let me translate. They told him that it is now time to go, and he asked for five more minutes.”
Galois proceeded to arrange all the papers into piles and neatly packaged them. After sealing the packages with a little bit of wax from the candle and his own ring, he addressed each of them separately. There was one package that was much larger than the rest and this he addressed to one Auguste Chevalier. The rest appeared to be individual letters. After putting all the work of the long night aside, he picked up the last piece of paper and quickly wrote a few lines, keeping it very close to himself while he wrote it. His repeated glances at the picture in front of him were enough to tell me that it was she that the letter was meant for. He finished writing it at almost the same time as the next knock on the door. He sealed this last letter too, and then went to answer the door again, returning with a man slightly older than him. He showed this man the results of his night of labour.
I hadn’t noticed when my guide had left the window to stand beside me.
Galois finally spoke.
“I have written addresses on all these packages, please get them delivered, won’t you?”
He was speaking English all of a sudden!
“No, he’s not. I am translating for you in real time,” whispered my guide, strengthening my assumption that she could read minds. “But listen to what they are saying.”
“This letter is the most important of the lot,” Galois continued speaking, handing the last letter to the other man. “Get it to her and ask her to remember me, please?”
“I will give it to her with my own hands, Evariste. There is the one big package, can I ask who will receive that?”
“Monsieur Chevalier. It contains my life’s mathematical work condensed to a few hundred pages. I do hope he is able to understand the work that I have done and ensure that it reaches the proper people.”
“Don’t worry about that, Evariste. I will personally ensure that your work isn’t lost, and I will help M. Chevalier do full justice to it. The world shall know about your contributions to mathematics.”
He didn’t say anything in reply, but the look in his eyes showed that this statement meant a lot to him. He started going around the room, pausing at various points and trying to take in as much of it as possible. His air was that of a man who was seeing his own home for the last time before going away on a long journey.
“I tried to write everything down, as far as possible. I don’t know if I did a good enough job. I hope there’s someone who profits from all the mess,” he said.
“Let’s go. It’s already time,” replied the other man. I thought his voice shook just a little.
“You might think a man who is going to his death could enjoy one last look at his quarters without being asked to hurry up!” replied Galois, simultaneously picking up the gun. There was no humour in the voice.
“Are you sure you want to do this?”
“We have had this conversation before, Emmanuel. It is a question of my honour, and I shall go and face that crooked fool even if it means my death, which it probably will for he has unfortunately always been a much better shot than I could ever be.”
He took one last glance at the room before speaking again.
“There’s nothing of value here… except that picture. Send it to her along with the letter please.”
“As you wish, Evariste.”
With that they left the room, locking it as they went. Perhaps fittingly, the candle went out again just as they left the room.
“Now what?” I asked my guide.
“Do you want to see how it finally ended?”
“I don’t want to, but I am too far in to not see it.”
“Come on then.”
A horse carriage had stopped just below the window. I guessed they were going to the rendezvous on that carriage, and evidently my guide thought so too, for sje jumped onto the roof of the same and I followed suit. After some time on the cobbled streets of 18th century Paris, we arrived at a small clearing just outside the city, where there were already five men waiting. Galois and his friends quickly got out of the carriage and walked towards the other people while we followed closely behind. There were hardly any words exchanged. The man who appeared to be the leader of the other group first shook hands with Galois, and the other people shook hands with each other. Then they marked off two lines about thirty metres from each other and this was double checked by one person from each group. Galois and the other leader took their positions at the two marks. They both had guns in their hands.
The man who appeared the oldest in the small congregation then spoke.
“Are the contestants ready?”
They both nodded their heads.
“Bow to each other, please.”
They bowed.
“Let the duel begin.”
The other man fired first, and missed by a whisker. Galois fired next, but missed by a fair margin. From the look on his face, he knew at that very moment that he had lost his chance.
One more shot was all it took and suddenly Galois was buckled over clutching his abdomen. Blood dripped from the wound, and it wasn’t long before he collapsed. His friends rushed to him and tried to block the wound, but it was pretty clear that all attempts to save him were in vain – the shot was too good and there was no hope of saving him using the medical knowledge of the century I was now in.
It wasn’t long before Evariste Galois departed this world.
While his friends carried the body away, my guide and I remained at the spot where he was shot. None of us spoke for a very long time. I couldn’t read her mind, of course, but I guess we were both wondering about the tantalizing possibilities of what could have been – thoughts that invariably come when someone that brilliant dies that early.
“You know what, I always wanted to visit this day because I thought I could maybe change the outcome by, I don’t know, somehow stop him from reaching the appointed place. Watching while the events happened made me feel so helpless!”
“Yes, it is heartbreaking! To see it happen in front of your eyes and yet being powerless to stop it from happening!”
No one spoke for a few minutes, until she broke the silence again.
“This is enough for today I guess. Let me get you back to your timeline. Next time let’s choose something less bloody.”
The infinite possibilities that this opened up was enough to take mind away somewhat from the brutally sad event that I had just witnessed.
“So I will be seeing you again.”
“Yes, of course. I will get in touch. But now, close your eyes and hold my hand and let’s go back to where we came from. You shouldn’t be here longer in a different timeline.”
I complied, and for the second time in my life her long fingers closed around mine. There was again the impenetrable darkness and feathery lightness that I had felt while starting my extraordinary trip to Paris. Soon, however, I found that I had reached familiar surroundings, the same secluded spot in the grounds of my building from where we had left. I was alone and my hand was clutching only air – my guide was gone. As I absentmindedly made my way back to my room, deep in thought about Galois and the events that I had just witnessed, it occured to me that he had been younger than me when he died. He was barely old enough to be called a man! If it is true that at the point of death our whole life flashes in front of us, it must have been a brief but brilliant flash for the tragic figure of Evariste Galois.
]]>Why would it be an interesting problem to talk about the Lee metric and bounds of the kind that we were looking for? It turns out that there are quite a few applications for results of this sort.
A bound on the volume of a sphere in a discrete metric is used to find bounds on the limits of reliable communication over a channel (more on that later). We actually see this in the paper itself - the moment we obtain an upper bound on the volume of a sphere in the Lee metric, we could use known results to immediately find analogues of the binary Hamming, Elias-Bassalygo and Gilbert-Varshamov bounds in the case of the Lee metric (more on all of these later). The basic utility of bounds on the limits of reliable communication over a channel should be clear - it gives people who design communication protocols something to aim at, and if such a design hits the upper bound, we know that one cannot do better, at least from the rate point of view (one could still improve things like encoding-decoding efficiency etc.). There are other examples too - consider the result of another paper that I was a co-author on that was also accepted to ISIT, where we showed that if the capacity under vanishing average error of a channel does not equal the zero-error capacity (both of these terms will also be talked about later in slightly more detail) of the channel, we need at least \(\log(n)\) bits of common randomness, where \(n\) is the blocklength of the code, to communicate reliably at capacity over the channel. Common randomness being a valuable resource (it is hard to generate and distribute such randomness), such a bound is good to know. How to show such a gap exists? We need an upper bound on the capacity of the kind that our results help us find.
This is the thing that brought our attention to this problem in the first place. We already knew the common randomness result, and we knew that the gap exists for the Hamming metric, but we thought it would be interesting to show that the gap exists for a more general channel - enter the Lee metric.
Before proceeding further, let me quickly talk about metrics and why they are useful in information and coding theory. So a major subfield of information theory involves finding the limits of reliable communication over a particular channel. Now reliable can be defined in various ways - two of which are
In most cases, the maximum rate at which we can transmit information in the first case will be higher than the maximum rate for the second case. Now, any transmission is done using an encoder-transmitter pair and a receiver-decoder pair, with a (possibly noisy) channel in between. In recovering the transmitted data from the noisy data at the receiver, it makes intuitive sense that the decoder should choose the most probable transmitted codeword given the model of the channel noise and the received data, an idea called Maximum Likelihood (ML) decoding. This is where the metric comes in. If the channel noise model is sufficiently ‘nice’ we can replace the ML decoding at the decoder by using a metric (which is just a measure of distance) such that choosing the nearest neighbour in the set of all possible transmitted codewords to the received data, we get the same result as what we would get if we were to use ML decoding. When this happens, we say that the metric is matched to the channel under consideration. Note that the metric that will be used depends crucially on the encoder, noise model of the channel, and the decoder. This idea will become clearer in the next section.
Two metrics that have been studied relatively deeply in the literature are the Hamming metric and the Lee metric. The Hamming metric is a very simple metric - given two \(n\)-length words \(a\) and \(b\), it counts the number of differences between them. It is useful in cases where, given any two distinct symbols (not words) \(a\) and \(b\), the probability that the channel corrupts \(a\) to \(b\) is the same, and this probability remains the same if the pair \((a, b)\) is replaced by any other distinct pair \((a',b')\). Even though this metric is really simple, it can be used to describe a large class of encoder, decoder and channel combinations, and thus has been the most well-studied metrics in information and coding theory. The Lee metric is useful whenever symbols closer together are more probable to corruption than symbols further apart, and the decision of the ‘closeness’ of two symbols is made by arranging the symbols in a circle (for example, the first symbol and the last symbol are considered adjacent, which wouldn’t be the case if they were on a line).1
Another interesting concept is that of a rate-distance trade-off. It again makes intuitive sense to say that if two codewords are far apart in a suitable metric, then the probability that the channel will confuse the two is low. So, to protect against as many errors as possible, we want our codewords to be as far off from each other as possible. However, in a finite space, we can only pack so many codewords that are all pair-wise more than a given distance away. So, given the number of errors a code can handle, we get a bound on the number of codewords we can use. If we use \(n\)-length code, the number of such codewords grows exponentially in \(n\), and therefore it makes sense to define the rate as \(\log \mathcal{M}(n)/n\), where \(\mathcal{M}(n)\) is the number of possible \(n\)-length messages. In the case of the error being exactly zero, we want there to be no codewords in a sphere around a given codeword. For average error going to zero, we can allow small intersections between the spheres. Note that this means that finding how many \(n\)-letter words are there in a neighbourhood of a given radius around a point is going to be important because we have to ensure that none of these is part of the code that we are going to use.
In the rest of this post, we will be concerned with only zero-error information theory, the second error criteria used above. In this case, we know a lot for the Hamming metric - given a particular rate, we know that a particular rate is achievable (that is, we can construct a code that achieves that rate), and we know that some rates are not achievable. The achievability result is via the Gilbert-Varshamov bound, while there are several upper bounds known (like the Hamming, singleton and Elias-Bassalygo bounds). Unfortunately, these do not match, and a gap exists 2 which has been an open problem for quite some time now.
In the case of the Lee metric, far less is known. We saw above why it is important to bound the number of words that are within a given distance away from a point, a quantity that we will call the size of a ball in the metric. The main difficulty in the case of the Lee metric is that it is much harder to find the size of a ball in the Lee metric than it is in the Hamming metric. Assuming we know the size of the ball in the Lee metric, a formula for each of the rate-distance bounds mentioned above appears in a wonderful book - Algebraic Coding Theory by Elwyn Berlekamp.
So now, the issue is to figure out what the size of a ball in the Lee metric is. This is the main content of our paper. Once we had this result, we were able to use Berlekamp’s work to immediately obtain the above-mentioned bounds for the Lee metric.
The first step was essentially a rediscovery of results that were known when Berlekamp wrote his book in the 60s. The idea is that one can think of this problem as estimating a particular coefficient in the expansion of a particular generating polynomial for the metric. To gain some intuition for this, consider the simple case of the binary Hamming metric, and suppose we need to find all the \(n\)-letter words that are a Hamming distance \(d\) away from the word \((0, \ldots, 0)\)3. The Hamming metric just says that if there are \(d\) locations where two equal length words differ, the Hamming distance is \(d\). The required number of words is given by the number of choices of \(d\) positions out of \(n\) that will have \(1\)’s, which is simply \(\binom{n}{d}\). This is also the coefficient of \(x^d\) in the polynomial \((1+x)^d\). One simple way to understand the equivalence is that this coefficient also involves the choice of \(d\) \(1\)’s from \(n\) choices.
This idea generalises to the Lee metric. Say the alphabet size in the Lee metric is \(5\). Then the generating polynomial is given by \((1 + 2x + 2x^2)\), and the problem of finding the size of the Lee ball of radius \(d\) reduces to finding the coefficient of \(x^d\) in \((1 + 2x + 2x^2)^n\). Now, this is not an especially easy problem, because no such simple idea as the one in the case of the Hamming metric works in this case. There exist formulae for this (also known from the 60s), but they are a) difficult to use and b) do not give much insight to the problem. I was at this time doing an online course on analytic combinatorics on Coursera, and there they talked about problems of this form and how to find analytic solutions for the same. I learned a lot of interesting new approaches to solving problems like this (in fact the first time I thought about the generating polynomial method of solving the problem was while doing the course), but ultimately in turned out that the techniques in the course weren’t well suited for this particular application (in particular because here we need to fix \(n\) to be a particular finite value and there one could take the limit as \(n \rightarrow \infty\), and the forms of the polynomials that we were working with could not be handled very easily via the techniques proposed in the course).
While spending a few weeks trying to use techniques from analytic combinatorics and ruling them out one by one, I noticed that even the Hamming metric calculations are done using assuming that we are finding the coefficient of some \(x^d\) where \(d\) is then the location of the largest coefficient. I figured it would be a start if we could at least figure out where the maximum coefficient would be, we would at least know the parameter range where we would be working. After a few days wrestling with this problem, I seemed to have guessed a solution to it but was not able to prove it myself. So I posted the following question on the Mathematics StackExchange - Largest coefficient in the power of a polynomial. A couple of answers could be used to get a proof of the maximum coefficient, and I finally hit upon some path of progress - I could use the central limit theorem to find the required coefficients for large enough \(n\). Using Python’s numpy.polynomial
, I was able to get some graphs that showed me how the coefficients were behaving and how the normal approximation looked. I tried to use the results to obtain the bounds that we want, but the results were rather bad - they didn’t even match the known results for the Hamming case, leave alone the Lee metric. When I went to see what the problem was, it was quite apparent - I had been using the log scale to look at the graphs, and the normal approximation was really good close to the peak but was really bad away from it. The log scale hid the fact that a couple of standard deviations away from the peak, the approximation was giving a value many orders of magnitude more than the actual value (say \(10^{10}\) times higher).
This was a problem that reeks of large deviation theory - but it took me some time to recognise that because I had not used it before. Large deviation theory, as the name says, gives good bounds when one is working more than a few standard deviations away from the central peak.
One of the basic results in large deviation theory is something called Sanov’s theorem. It gives a powerful bound on the size of a set of type classes drawn from a probability distribution. The idea is as follows - suppose we have a discrete probability distribution, like the one that governs what face of a die comes up when it is thrown. Now, if we sample the probability distribution \(n\) times (throw the die \(n\) times), the outcomes will probably not be distributed exactly like the underlying distribution, but would be something called the empirical distribution - in the case of a die, this would consist of the number of times each number came up when the die was thrown \(100\) times. Empirical distributions are also called types, a powerful concept in information theory, with several very nice properties. A type class is the set of all outcomes which have the same type 4. Now, given a set of types, Sanov’s theorem allows us to bound the size of that set of types - the number of sequences such that their type is in the set. In our case, the class is just the set of all types such that the expected value with that probability distribution is less than \(d/n\). This could be calculated using the normal approximation, but as mentioned earlier, the bounds obtained from it are pretty poor away from the peak.
Sanov’s theorem requires us to find the type in the set that has the minimum relative entropy with the underlying probability distribution. This is a convex optimisation problem, and one could take its dual. The dual problem was just a complicated single variable optimisation that became something that I could handle without any of the sophisticated methods used to solve convex optimisation problems, using some Python code that ran reasonably quickly. This was part of the problem solved - numerically finding this maximising parameter gave some upper bound on the required coefficient. I wanted something better.
By the duality properties of convex optimisation it was clear that any positive value of the single parameter that the dual was over would work, so what remained was to choose what the positive value would be as a function of the expected value that was used to define the type class. I plotted out what the optimal value looked like, and it seemed an odd kind of curve. Now the first thing that entered my head was that the negative of the curve looked like a \(q\)-th root function, appropriately scaled and shifted. Now, using curve_fit
from scipy.optimize
in Python, I was able to fit it to other functions, like polynomials or exponentials, but either they were very sensitive to parameters or were not very good fits - something that can be quantified using the output of the curve_fit
function. Anyway, it turned out that the best fit among all the functions I tested was indeed given by that weird function, and that’s what found its way into the paper. We also found that perturbing the optimal values for the constants in this fit did not affect the final result much - so this fit was robust in some sense. If you look at the curve and some other functional form comes to your mind, I would love to know it!
Once we obtained the formula for the size of the Lee ball, all that remained to do was to substitute the result in the formulae from Berlekamp’s book and plot the results. The results for \(q = 6\) are shown below. We also verified that \(q = 2\) with our method gives the same results as the binary Hamming case pointed out above - one way in which we were able to be somewhat sure that the method works.
If you are wondering what we used to make the graphs, there’s a very nice way of using Latex in matplotlib
in Python, and matplotlib
can also be made to output the graphs in SVG, for the rather good looking graphs that you see both in this post and the paper.
If you are interested in looking at out code, it is all reasonably well documented and uploaded on Github. If you have any thoughts about this work, use the comment section below or get in touch via email. Provided things work out, I might be presenting this at ISIT 2019, so this might not be the last time that this paper is featured on this blog.
Brief side note here. One might wonder about the metric that describes the situation where ‘closeness’ is defined by arranging on a line and not a circle. In this case, it can be shown that there are no channels that are matched to this metric, and so, even though the metric is mathematically interesting. it doesn’t concern us that much here (it’s not just mathematically interesting, but a situation where it would be of interest to an information/communication theorist would be too big a detour for this blog post.) ↩
In the Hamming case the gap exists for alphabet sizes less than \(49\), above which there is a construction based on algebraic geometry that closes the gap. ↩
Point to note - for ‘nice’ metrics, one can always translate in this way to calculate the size of the neighbourhood around zero. ↩
One of the cool properties of types is that all members of the type class have the same probability. ↩
You note the date, and it is April 1, 2019. You realise that if you find what the thing in front of you means, you’ll be able to cause a new renaissance in your own world when you wake up. It’s not long before you are snatched out of this utopia, this Garden of Euler - so think fast!
Oh yes, this world is non-ideal in one way - every collision is completely inelastic.