The fun in teaching probability

My last class this semester will be on Monday. Whether you are a student or professor, the end of the semester is always something to look forward to, especially if a four month long summer free of classes awaits. This time, though, I won’t feel the same way – the sentiment is primarily because I've been teaching probability and statistics. Both these topics are so rich and full of startling revelations that I will, for the first time, miss preparing for lectures. The class itself, consisting of ninety undergraduates, was a lot of fun (even if grading was not); I used the personal response system which worked well (I ask students a question and have their answers displayed immediately, enabling instant feedback, like an audience poll in a game show).

The mathematician Steven Strogatz, whose short pieces in the New York Times have thrilled many readers, recently wrote about conditional probability, a notoriously twisted concept. I spent two weeks covering it. Using a common example, I tried to tell my students that “testing positive if you have the disease” is not the same as “having the disease if you test positive”. To some this is mere semantics or subterfuge -- and indeed, much of probability can seem like smoke and mirrors, like the Monty Hall Problem -- but trust me it is not. Strogatz provides another example:

Perhaps the most pulse-quickening topic of all is “conditional probability” — the probability that some event A happens, given (or “conditional” upon) the occurrence of some other event B. It’s a slippery concept, easily conflated with the probability of B given A. They’re not the same, but you have to concentrate to see why. For example, consider the following word problem.

Before going on vacation for a week, you ask your spacey friend to water your ailing plant. Without water, the plant has a 90 percent chance of dying. Even with proper watering, it has a 20 percent chance of dying. And the probability that your friend will forget to water it is 30 percent. (a) What’s the chance that your plant will survive the week? (b) If it’s dead when you return, what’s the chance that your friend forgot to water it? (c) If your friend forgot to water it, what’s the chance it’ll be dead when you return?

I am too tired to give a detailed answer; moreover, conditional probability is just one concept in probability. There are plenty others, and this semester I often felt like I've stumbled upon a treasure trove of delightful ideas. Even this week I was discovering some that I had glossed over as a student. Someday, I will write extended pieces on what I have learned and how they apply to common situations.

Meanwhile, for those of you who would like to be tested, here are a couple of questions I asked in my exams -- up to the challenge? The first one (credit Leonard Mlodinow's The Drunkard's Walk) is easier than the second; it does not need any prior knowledge of probability.

a. Two students were partying in another state the day before their final chemistry exam. They got back only after the exam was over. However, they made up an excuse. They lied to the professor that they had a flat tire while returning and asked if they could take a make-up test. The professor agreed, wrote out a test, and sent the two students to separate rooms to take it. The only question on the test, worth 100 points, was: “Which tire was it?”

What is the probability that both students will give the same answer?

b. Suppose you toss a coin once and roll a die 4 times (these are two independent sets of experiments). Success in a coin-toss is getting a heads, while success in a die roll is getting a 5 or a 6. What is the probability that the number of successes in the coin toss equals the number of successes in the 4 die rolls?

Socialism in ant colonies and its lack in human societies: E.O. Wilson's perspectives

I recently chanced upon an old, 1997 interview of the famous biologist Edward Wilson. Wilson is known for his work on ant societies and in the interview he provides some good insights. Ant societies are well and truly socialist -- why so and why not human societies? The answer might lie in the differences in reproductive abilities. Long excerpts below:

There are about 9.500 known species of ants, many of whom you studied, but there is only one species of Homo. Why?

I think I have the answer for that. That is because we are so big. We are giant animals. The bigger the animal, the larger the territory and home range that the animal needs. Ant-species, consisting of very tiny organisms, can divide the environment up very finely. You can have one species that lives only in hollow twigs at the tops of trees, another species that lives under the bark, and yet another species that lives on the ground. Human beings, being giant animals and particularly being partly carnivorous, cannot divide the environment up finely among different Homo-species. There have been episodes in which there were multiple hominid-species, probably two or three species of Australopithecus, co-existing perhaps with the earliest Homo. But it is evidently the tendency of hominid species and particularly of Homo to eradicate any rivals. It is a widespread idea among anthropologists that when Homo sapiens came out of Africa into southern Europe about a hundred-thousand years ago, it proceeded to eliminate Homo neandertalensis, which was a native European species that had survived very well along the fringe of the advancing glacier.

You write that ants often share food among themselves. Why, and how did you find out?

Back in the fifties Tom Eisner, a colleague of mine, and I did I believe the first experiments tracing radio-active label led sugar-water through colonies of ants. We were able to estimate the rate at which the food was exchanged, and the volume that was exchanged. Not only do many colonies exchange food with fanatic dedication, but in the colonies of many ant species the workers regurgitate food back and forth at an extraordinarily high rate. Now we understand that the result of this is that at any given time, all the workers have roughly the same food-content in their stomach. It is sort of a social stomach. So that an ant is informed of the status of a colony by the content of its own stomach. It therefore knows what it should be doing for the colony. If you only had a small number of extremely well-fed ants and the rest were hungry, the workers would go out hunting for more food, whereas in fact it might be a bad time to hunt for food.

Why doesn't this sort of communism exist among humans?

What I like to say is that Karl Marx was right, socialism works, it is just that he had the wrong species. Why doesn't it work in humans? Because we have reproductive independence, and we get maximum Darwinian fitness by looking after our own survival and having our own offspring. The great success of the social insects is that the success of the individual genes are invested in the success of the colony as a whole, and especially in the reproduction of the queen, and thus through her the reproduction of new colonies.

This was I think one of the main contributions of the idea of kin-selection. We now understand quite well why most species of social insects have sterile workers, and therefore can have communist-like systems. In which the colony is all, the individual is only a part of the colony, and the success of the whole community is what counts far above the success of the individual. The behavior of the individual social insect evolved with reference to what it contributes to the community, whereas the genetic fitness of a human being depends on how well it can individually use the society. We have become insect-like only by extreme contractual arrangements.

You write that a major difference between humans and ants is that we send our young men to war, while they send their old females. Why is that?

Well first of all, all the worker-ants are female. In the bee, ant and wasp-societies sisters are extremely closely related to one another, and therefore it pays to be altruistic toward sisters, whereas brothers do not benefit by giving anything to sisters. So the females are the ones who are fanatically devoted to one another.

Why are they old? Once again it comes down to this matter of what is best for the colony. As the workers grow older, they put more and more of their time outside, and as they become quite old or injured or sick, they spend their time either outside of the colony or right at the edge. The advantage of this is that the individuals that are going to die soon anyway, having already performed a lot of services, are the individuals that sacrifice themselves. It is the cheapest for the colony.

Whereas in humans, not only are the young males the strongest, but by being mammals in a competitive society young males tend to be greater risk-takers, braver and more adventurous. They are moving up in the ladder of status, rank, recognition, and power. And to be a member of the warrior-class when it is needed, has always been a rapid way of moving up. So that appears to be the main reason why we send young men out, and they are willing to go.

Chance masquerading as skill: A personal take on T20 Cricket

For those who don’t follow cricket, the second part might not be of interest.

1. Not quite chess

As an only child, I had many ways of keeping myself occupied. I couldn’t play a game of chess by myself, so I had to create artificial teams that competed against each other. But the game could not be played the regular way – how can you outwit the opponent when you yourself are the opponent? So I created “dice chess”; I worked the specific rules out slowly, but the idea was simple. You moved a piece only as many squares as the outcome of a die roll. You could choose which piece to move, but there were restrictions. A pawn couldn’t be moved unless you rolled a one; a horse couldn't be moved unless you rolled a three or a six. More radically, you might have checkmated your opponent, but the game wasn’t over until you rolled the number that actually finished the job – the king could therefore escape even from a hopeless situation. This resulted in some very thrilling comebacks and unexpected results.

The dice meant that I could play by creating two artificial teams; I could now think for both teams, since the outcome of the die roll decided my moves. Given a roll, I did the best for each team. In all other respects, I was just a passive observer and commentator (yes, I did actually speak out loud when the games were going on). In my commentary, I put up the pretence that dice chess was a serious game and that there was serious skill involved in winning. My teams were typically based on the books and comics I read. Walt Disney had a team; Indrajal Comics had one; the Hardy Boys were in what I called the Franklin W Dixon (FWD) team; as I started reading more novels (around the eighth grade), Agatha Christie had hers. I would design elaborate tournaments, with the usual league round robins, semi finals and the finals. There was also an imaginary audience that cheered the teams (yes, I mimicked the roar of the audience too, when a something momentous happened on the board).

To be sure, there was some skill involved. For instance, a bishop might be two squares away from the opponent’s king: rolling a two would finish the game; the chance of winning was one in six. But if you had a bishop within two squares of the king and a horse within striking distance (the horse in fact turned out be an unusually powerful player in dice chess), it would mean rolling a 2, 3 or 6 could finish the game – you now had fifty percent chance of winning. It was all about maximizing your chances.

I conceived the game in fifth grade, and for fifteen years I kept it to myself-- until graduate school, when I introduced it to my lab mates. They were excited about it, and for a while, we were playing mini-tournaments in the afternoons and avoiding research.

A few key aspects I had not thought carefully about became evident as I played with others. First, the game was so dependent on the die roll that it was no longer chess. Yes, it was based on the rules of chess, but you had to think less. You reacted to the die roll rather anticipating what the opponent would do. Those who did not like chess because it is a mentally daunting game – I include myself in this group – found dice chess a much more relaxing option. Second, there was much less skill involved than I had thought; the victories were victories of chance and very little of skill.

Viewed less charitably, dice chess was a colossal insult to and a diminution of the original game. Chess champions like V Anand would laugh at it, and yet if they were paid millions of dollars – because large numbers of people liked it – it would be no laughing matter; they might feel compelled to play the significantly diminished game. And if separate schedule of a few months were to be carved out for dice chess, they might have less energy for the intense concentration and powers of extrapolation that the original game demands.

2. Not quite cricket

The analogy is obvious, even if not precise. T20 Cricket is not all chance, but most of it is – and this is precisely the reason it is “unpredictable” and “exciting”. It opens up the possibility that any team can win. There is no question that there some skilled players and they do their best within the limitations of the format; and some new strategies are now being used. But to claim that it is cricketing skill that decides T20 outcomes more than chance is to delude oneself. Most viewers, in fact, are aware of this, but just as dice chess thrilled me into excited commentary as I conducted my “solo” tournaments, so too are viewers, myself included, knowingly lured by the temporal pleasures of T20 cricket.

That randomness plays a greater role in T20 than in cricket’s longer forms can be inferred from the simplest statistical ideas. Every game of sport, whether cricket or soccer or tennis, is a random experiment, even if one team or player is stronger and more likely to win. In statistics, you conduct the same random experiment many times to make sure that the data you are collecting has validity. It matters how many times you do the experiment. But equally important is the length of each experiment. If the length is short (think, equivalently, of a one set tennis game or a twenty minute soccer game, or a one day golf tournament), then you cannot be confident of the outcome you are measuring, no matter how many times you measure it. So although a lot of T20 games are played – as in the IPL – there is little meaning in the outcome of each individual game.

In contrast, test matches are played over five days; the game is too long in the view of many but the length is precisely why we are able to infer something concrete (and some of the most exciting test games have been played over the last few years). The length is also why the format is the hardest; if you are skilled, it will eventually show. If you are not skilled, you will not survive test cricket. It is almost impossible for a really weak team (say Zimbabwe) to beat a strong team (say Australia). There are good reasons why New Zealand, a 2009 T20 World Cup finalist, has performed poorly, home and away, in the longest form of the game.

In T20s, the skilled will be successful over time – as Tendulkar and Kallis have demonstrated in this year's IPL – but the short format allows those without skill (read a few lusty blows, or a few freak wickets) to have an unusually high influence in determining a positive outcome for their team. Chance, in other words, masquerades as skill in T20 cricket. It is not a simplification to say that the IPL is a nationwide lottery, a kind of frenzy that urban Indians and the diaspora revel in. It is an exhibit for the worst excesses of capitalism and celebrity worship. It leaves little room for nuance.

At the moment, the murky financial side of IPL is unraveling. One hopes also that a more objective assessment of the format will be made. Even with twenty overs, there are more intelligent ways to design the game. Have 11 players in your team, but the batting team can lose only five wickets (a total of six batsmen), rather than ten; the remaining five will not bat, but are specialist bowlers. It won’t necessarily solve all problems, but at least, the contest between the bat and ball will be more even. Risks will have a high cost. Sixes and fours, which have been devalued in the current format, will gain some meaning.

Just as dice chess is not really chess, so T20 as it is now is not really cricket. Think, in contrast, the epic innings that the serene Hashim Amla played in vain in the second test at the Eden Gardens. That innings, played just before the T20 frenzy began this year, is the perfect antidote to the IPL’s numbing onslaught.

Perelman and the Poincaré Conjecture

A riveting account of the life and mathematical achievements of the recluse Grigory (Grisha) Perelman, who recently cracked the Poincaré Conjecture:

Masha Gessen’s Perfct Rigor is a fascinating biography of Grigory (Grisha) Perelman, the fearsomely brilliant and notoriously antisocial Russian mathematician. Perelman proved the Poincaré Conjecture, one of mathematics’ most important and intractable problems, in 2002—almost a century after it was first posed, and just two years after the Clay Mathematics Institute offered a one-million-dollar prize for its solution.

[...]

Up until March of this year, there remained one more chapter to the Perelman saga. Would he accept the one-million-dollar prize promised by the Clay Mathematics Institute for solving one of the seven so-called Millennium Problems? While the rules say that a proof must appear in a peer-reviewed mathematics journal (not just in an Internet posting), the mathematicians mentioned above have published papers in such journals expounding and amplifying the proof. Surely Perelman deserves the prize, which he was finally and officially offered on March 18.

Five days later, on March 23, Perelman rejected the Clay prize. He reportedly said through the closed door to his spartan apartment, “I have all I want.” The comments he made after rejecting the Fields Medal probably reflect his present state of mind as well:

"I don’t want to be on display like an animal in a zoo. I’m not a hero of mathematics. I’m not even that successful. That is why I don’t want to have everybody looking at me."

Some might argue that monetary awards for mathematical work are inappropriate, or that the Poincaré Conjecture is of little practical value and not worth the one-million-dollar prize. The aesthetic and epistemic value of the proof is priceless, however, and it may eventually yield more earthly consequences as well. As for the size of the award—how many no-name hacks are there on Wall Street who make a million dollars or more not just once but every year, and contribute exactly what? Whether Perelman has practical need for the money or not, he could use it to help support his mother or mathematicians of his liking, or to advance the kind of education conceived by Andrei Kolmogorov, or for some purpose only he could imagine. Reconsider your decision, Grisha.

Pictures of Lake Titicaca

Titicaca is located on the border between Peru and Bolivia at an elevation of 12,000 feet. I took these pictures on the Bolivia side. In the second picture, the little town around the hill, facing the lake, is Copacabana.

The neglect of India's classical languages

Last year, the Sanskrit scholar Sheldon Pollock asked why India pays scant attention to its linguistic heritage. The neglect of classical languages means that the vast majority of Indians remain divorced from a deeper understanding of their own past. To put things in perspective, think of how well studied Western classical languages are; it is no surprise that the West understands its own journey to the present really well -- and is constantly reinterpreting it.

I feel proud of India’s Hindu-Buddhist history, but my knowledge of Sanskrit or even my own mother tongue Tamil – yet another classical language – is so poor, any understanding I may claim to have of this heritage can only be superficial. I have not read a single text in an Indian language. Even the realization that Sanskrit has a complex grammar, that novel numerical methods constitute much of ancient Indian mathematics, was recent – I learned of them through the painstaking work of Western scholars such as Nicholas Ostler and Kim Plofker.

In her new essay, Crisis in the Classics, Ananya Vajpeyi stresses the enormity of the Indian problem:

Kusum Pawde, a Maharashtrian woman of non-Brahmin origins, has written a seminal essay titled “The Story of my ‘Sanskrit’,” describing the struggles she faced growing up outside upper caste society in Maharashtra and becoming a Sanskrit scholar and university teacher. Her story took place back when the study of Sanskrit was still confined to Brahmin men; Pawde had to fight discrimination on the grounds of both caste and gender. Yet, when I first read her essay whilst doing my own doctoral work in Maharashtra thirty-five or forty years after she completed her education, even with my rather different social and financial situation, I found it no less difficult to navigate the world of Sanskrit scholarship.

The only place Sanskrit felt accessible was in the classroom at the University of Chicago with American professors and classmates, or on the fifth floor of the Regenstein Library, which housed every text – old or new, classical or vernacular, Indian or European – that one might conceivably need. Between 1998 and 2003, I spent almost five years in the field, travelling to every major centre of Sanskrit, learning throughout the Deccan and southern India. By the time I returned to Chicago to write my dissertation, I had to concede that there was no “there” there for the study of pre-modern India, in India.

In early 2010, Gurcharan Das told me that he was disturbed about having to go to American universities to study or refresh his Sanskrit in preparation for writing his latest book, The Difficulty of Being Good (2009). Das was an undergraduate at Harvard in the 1960s, where he read Sanskrit with Daniel Ingalls. Four decades later, he returned to the University of Chicago to brush up his philological skills with Ingalls’ students, Sheldon Pollock and Wendy Doniger.

As we shared stories across generations, I recounted to Das my own disappointments and difficulties in trying to study Sanskrit in India. I told him about my misadventures at universities, libraries, archives, and traditional schools in multiple states – Maharashtra, Karnataka, Uttar Pradesh, Tamil Nadu, and Bengal. I had spent my twenties in graduate school, searching for Sanskrit. I found it more readily in the U.K. and the U.S.; in India, it consistently eluded me. Das looked at me in amazement; it had not occurred to him that his experience – call it the difficulty of being good at classical studies in India – was not by any means unique to him.

[...]

Arguably, linguistic diversity and literary richness ought to be India’s strongest suit, given its history both as an old civilization and as a diverse and multi-vocal democracy. Alas, we have driven our languages and literatures into the ground. Linguistic chauvinism and language-centred identity politics abound. Yet, not a single political ideology protects and nurtures the languages, which remain orphans in the political process and in the networks of institutional patronage cultivated by different parties.

[...]

But Sanskrit was never merely – and certainly need no longer be – a tool of epistemic violence against vulnerable sections of our society, including women, lower castes, tribals, and Muslims. It is, like it or not, one of a very small number of keys to our entire recorded history; without an ability to be functional in this language, without preserving its texts, its archives, and its material residues, we simply cannot know our own origins. Wilfully destroying and forgetting the historical past, in the manner of Communist Russia and China in the twentieth century, or distancing and censoring it in the manner of other new republics based on old cultures, like Turkey and Iran, is not the way forward for India.

Try to imagine independent India without its founding, fundamental, and inalienable texts, whether ancient or modern, upper caste or outcaste: the sermons of the Buddha, the edicts of Asoka, the epics of Vyasa and Valmiki, the songs of its Sufis and bhakti poets, the teachings of its saints and sages, the lessons of its gurus, the Constitution of its Republic, Gandhi’s letters, Ambedkar’s articles, Nehru’s speeches, Tagore’s national anthem, and the innumerable stories that we continuously recount. Not land, blood, race, religion, or state – language itself is our essence. Without our words, we are nothing.

It's a superb essay, a heartfelt plea that one hopes will be heard -- and soon. Read it here.

Travels in Tamil Nadu: Kumbakonam

Pardon the typos...

1.

The south Indian city of Kumbakonam lives up to its epithet. Temples are everywhere: their elaborate gopurams (towers) rise dramatically along its crowded, narrow streets, and in the villages and towns around the city, a varied pantheon finds representation. The city’s large Mahamanam tank is said to collect India’s holy rivers in miniature, imparting this local and insignificant seeming landmark a grander, pan-Indian connection. Bathing in the tank on the day of the Mahamanam Festival – Tamil Nadu’s own Kumbh Mela – is akin to washing one’s sins in such rivers as the Ganga, Yamuna, Saraswati, and Cauvery (see rare picture from 1905 below of the festival and the tank; picture credit here).

2.

I grew up with little interest in temples and deities. As characters in stories and epics, the gods with their quirks and foibles were fascinating, but as idols to be worshipped in a temple, they were invariably boring. Kumbakonam has the usual – Shiva, Vishnu, Murugan; there’s also a rare Brahma temple. But the majority of the holy places around town pivot on a fascinating theme -- the deification of celestial bodies: the sun, the moon, the planets, better known as navagraha. It made sense to me that these mysterious entities, markers of time, direction and the recurrence of seasons – and, in the case of the sun, indispensable – should be worshipped.

In the temples I visited as a child, the navagraha were personified as black figurines, their details faint; they were typically arranged on a raised platform in three rows. Though I was always instructed to circle them devoutly, they were never the main attraction. In Kumbakonam, there is a temple for each graha. Though they consist of five planets, the grahas generally refer to celestial landmarks – the sun (Suryan), the moon (Chandran), Mars (Chevai), Mercury (Budhan), Jupiter (Guru), Venus (Shukram or Velli) and Saturn (Sani). The remaining two, Raagu and Ketu, are not actual bodies but nodes: “they are the two points of intersection of the paths of the Sun and the Moon as they move on the celestial sphere”[source].

3.

The Gurukovil (or Guru Temple) is in Alangodi; it has the most garish gopuram (tower) I have ever seen. Though striking, I personally prefer the understated elegance of older temples, such as the Brihadishwara. The difference is in the degree of detail: the newer temples tend to exaggerate (dilated eyes, fiery moustaches, tusks and teeth), while the older temples prefer abstraction and are yet noticeably sensuous.

The cool inner sanctum of the Gurukovil hosts a lingam with a Nandi facing it. Guru himself is to the left of the sanctum, almost an afterthought, reached only by walking halfway around. But he receives plenty of attention from worshippers. I caught a glimpse of him right after an abhishekam, bejeweled, garlanded, slits that were eyes marked out from his otherwise ash-covered form. It was also at this temple that I noticed four Muslim women in black chadors lighting candles for him (I mentioned this in an earlier post).

4.

That same evening, I went to the Suryanar Kovil (Sun Temple). Guru was here too, facing a large idol of the Suryan: the heavyweights of the solar system, Jupiter and the Sun, enjoying a tete-a-tete.

The rest of the temple hosted the other seven grahas, each housed in a smaller enclosure, aptly around the sanctum for the Sun. At one of them, an archaka (a priest) was conducting an abhishekam for a large family. This meant stripping the black stone idol that represented the deity, bathing it in water, milk, a mash of chopped bananas, honey, jaggery and dates, washing it again with water, and finally dressing it in new cloth and adorning with flowers -- all this while prayers with a distinctive cadence were chanted.

The priest was stocky, pot-bellied and wore earrings and a sullied sacred thread; his veshti (a white wrap-around) was worn in a complicated, many-layered fashion and sat tight over his wide waist. He did his work earnestly. But the lady of the family – who was clearly in charge – was not satisfied with the material aspects of the abhishekam.

“Why aren’t you adorning more? Shouldn’t you be using more fresh flowers? What about the clothes for the grahas? I paid a lot for this; and I specifically asked for something elaborate.”

The priest was immediately on the defensive.

“Amma, what use is it telling me? I have no control over these things. You should have been more specific when you paid for the abhishekam; you should have told the people there. What use is it coming and telling the Iyer [Brahmin priest] here?”

It was a comical tussle about money and service quality in the middle of a somber religious ritual. As the priest moved from one graha to another (from Raagu to Chandran to Budhan), the lady kept insisting that it wasn’t enough. And the priest kept defending himself in his distinctive Brahmin-accented Tamil. At the end, the lady gave money to the priest as was customary, but he refused it, leaving the family bewildered and dissatisfied (though some of the younger members snickered).

There is a subtler issue at work here and it has to do with political control of temple revenue. The government of Tamil Nadu – currently led by the avowedly anti-Hindu Karunanidhi – has increasingly taken control of many temples. You pay for services, such as an abhishekam, to the temple officials and it is they who decide what you get. I am speculating here, but this probably puts priests at odds with the officials: the former who once had more control now have less say.

On not being a morning person

They say you are either a morning or a night person. I am certainly the latter: it is a pity to miss the freshness of dawn, but it seems an even greater pity to miss a few extra hours of sleep. No matter how early I get to bed, my eyes just won’t open earlier than eight – so much for the old adage about rising early.

I wonder if I ended up this way because as a child I had to be in school only at 11:30 am. This was back in the eighties; my parents lived in Gujarat, in the Naranpura area of Ahmadabad. We were tenants in the small upper section of a house owned by a large family. The living room doubled as the bedroom; attached to it was a small kitchen. My mother would wake me up at eight, well after my father had left, and with eyes barely open, I would begin my journey to the toilet.

I call it a journey because the toilet was downstairs and was shared – we didn’t have our own. I made the descent lazily, leaning against the railing, contemplating a nap every step of the way; so slow was I that on some days it took me thirty minutes to get down. The stairs were out in the open and faced the backyard, most of which was under the canopy of a neem tree. To the left was a narrow pathway, a kind of neutral zone between houses that led to a nearby temple. The pathway was frequented by stray dogs, cats, and especially cows, which came to chew on discarded pieces of paper or rummage through trash. It was during those slow morning descents that I started observing domestic animals – to the point of being mesmerized. Even today, I can watch dogs play and interact for hours on end.

I got back around nine. Then, as I sipped Bournvita or Maltova (chocolate drinks) I felt the texture of my mother’s sari with my fingers. This meant she had to sit on the floor next to me for the entire time it took me to finish. It was a strange ritual, one I find hard to explain today. But I do remember distinctly that the drink tasted heavenly when her sari’s texture was somewhat rough.

Such laid back mornings – minus the chocolate drink and strange ritual – were still the norm during high school in Nagpur. At college in Trichy, though classes started at 8:30 am, I routinely bunked the first two, so I could have a leisurely breakfast of tea, eggs, bread and jam at the mess until 10:00. Early morning classes were harder to miss in grad school, where the quality of education was just too high to be ignored, but once course requirements were done, I reverted to the late start schedule. For a while, my breakfast consisted of enormous quantities of whole wheat bread, almond butter and soy milk.

And the pattern continues to this day. One of the pleasures of being an academic is that I don’t have to be in my office at 9; in fact, I don’t have to be there at all unless necessary. On the days that I don’t teach, I wake up, make my coffee (flavored hot milk really), and settle to read the the blogs listed on this page. It’s only around noon time that I really get going. The flip side, of course, is that I do most of my work at night, which thankfully is capacious enough to accommodate my worst procrastination excesses.