ISI Chapter – PGDBA Blog

Chances of Consequence

Grown-ups love figures. When you tell them that you’ve made a new friend, they never ask you any questions about essential matters. They never say to you, “What does his voice sound like? What games does he love best? Does he collect butterflies?” Instead, they demand: “How old is he? How many brothers has he? How much does he weigh? How much money does his father make?” Only from these figures do they think they have learned anything about him.

-The Little Prince

Our first exams in this first semester at ISI have finally arrived. One of the subjects of the course, and of the more interesting ones I might add, is ‘Probability and Stochastic Processes’. This was also the subject of our first exam. I have been pouring over notes and hundreds of pages of text in preparation of this exam. The intricacies in some of the ideas reminded me of these lines taken from ‘The Little Prince’ and quoted by David Freedman in his book on Statistics. Are such matters as figures and charts always dry and boring? In our first class, the professor of the same subject also remarked, “Maybe our insistence on numbers is the limitation of man”.

‘The Little Prince’ is a famous classic written by Saint Antoine du Exupery. Although it was first written as a children’s book, it has been enjoyed over the years by adults alike as a heart-warming story on friendship, growing-up and the curious fascinations of man. One of the oft used coinages in the book is the phrase ‘matters of consequence’. The Little Prince finds, at one point, a man aimlessly counting stars to the extent that he reaches ‘five hundred and one million’. This man then ironically refers to his counting as a matter of consequence.

In these days leading up to our exams, I began to wonder whether factuality could be made interesting in the context of some intricacies I came across in problems of Probability Theory. Are there questions that I can ask that are not as dry as the character’s occupation of counting in The Little Prince?

Consider the following questions. How many random people would I need to collect in a room, such that there would be half a chance that at least two of them share their birthdays? The answer to this is a surprisingly low number. Let’s look at a different question. How many people would I need to catch hold of and ask birthdays of to have half a chance of finding someone who shares my birthday? It is also interesting to note that this number is different from the first one. Louis CK, the famous stand-up comedian, made an interesting remark in one of his live shows. He said that there were enough people attending his show for there to be a fair chance that a few of them would die in the coming year. Ouch! Let’s pose that as a question on chance. Assume a population with a certain death rate. What is the number of people I would have to randomly collect in a room, such that there would be half a chance that someone would die tomorrow?

Enough with birthdays and death days. Consider coin tosses. Say, I keep tossing a coin and keep getting heads for a hundred tosses. I only know that it is equally likely that my coin is anything between a perfectly rigged coin to a perfectly fair coin. What is the chance that I win if I bet on the next toss to be heads? Let’s stretch this slightly further. Let us assume that when the universe was born, the probability that the sun would rise over planet earth each day was made to depend on a coin toss. We only know that this coin was likely to be anything between a perfectly fair to a perfectly unfair coin (God is playing a cruel game). What is the probability that the sun would rise tomorrow, given that it has been rising each day over all these years since the birth of the universe?

Let’s give some rest to our poor coin and move on to other questions. The Monty Hall Problem is a famous example on how sometimes the idea of chance can stump intuition. Here’s a different example in a similar vein. Suppose there are three suspects in jail who have been cleared of any wrongdoing and are all going to be released soon. It is announced that two of the three would be released the next day but these three don’t know exactly which two of them will be released. One of them decides to go ask the guard which one of the other two prisoners is going to be released. But then he thinks the following to himself, “After I have asked, I have only a one in two chance that I’m the second guy to be released. But before I ask, there is a two in three chance that I will be one of the guys to be released. So should I rather not ask?”

Like one of our professors quips so often, ‘Remember that there’s a chance model somewhere in the background.’ Somebody somewhere is flipping a coin. Maybe someday a story will be written on a travelling mathematician who visits magical planets to ask questions on chances of consequence. It’s a long shot that it would be anything as beautiful as the masterpiece that is the ‘The Little Prince’. But I wager it wouldn’t be as bad as dry pointless counting.

The Best Laid Schemes, of Spiders and Men!

What do entropy, linear programming and Riemann surfaces have in common? Puzzled? Now imagine this connection explained by an eccentric speaker in the attire of a French stage magician, with the charm and virtuosity of a storyteller. Cedric Villani, French mathematician, Fields Medal awardee in the year 2010 and famously called by the NewYorker magazine as the ‘Lady Gaga of Mathematics’ delivered a public lecture titled ‘Of Triangles, Gases, Prices and Men’ at ISI on 26^th August 2016. The second PGDBA batch, currently in its first semester here at ISI, had the opportunity to be present at this intriguing and informative session.

The abstraction and intrigue of Cedric could be assumed from the fact that an introduction to him included a reference to the number of his pets. This abstraction could also be inferred from the title of his talk, which was a play on the title of John Steinbeck’s famous classic ‘Of Mice and Men.’ The first slide of his presentation was taken from Tennyson’s Lady of Shalott. In Cedric’s own interpretation, the Lady of Shalott, accursed to see the world only through a mirror, was actually an allegory to the mathematician forever accursed to look at reality through his equations! Cedric then said that there are many more unsolved mysteries in Mathematics today than there were a hundred years ago. There are ever so many new problems that keep arising. Then there are those age-old ones that lie in famous mathematicians’ lists of unsolved problems. One such famous unproved hypothesis is the Riemann hypothesis. This led Cedric down the path to explaining Riemann’s works and then to the first part of the evening’s presentation – ‘triangles’.

He introduced to the audience Riemann surfaces and how Escher employed curved surfaces in his art. As examples of negative curvatures, he showed images of art installations in museums and models of coral reef. Einstein, with the help of his mathematician colleagues, used Riemann’s ideas to develop his General Theory of Relativity. Cedric went on to add that, the GPS technology so ubiquitous in the world today has its roots in Riemann’s works in topology. In a humorous turn of speech, Cedric noted that Riemann was as oblivious to his work being of practical use in 21^st century devices, as modern day GPS users were to Riemann’s surfaces. An ironic symmetry indeed!

At this turn of his presentation, Cedric spoke about how it is equally important for scientists to pursue inspiration and not just utility. He marked out Riemann as someone who was particularly interested in approaching problems in his own unique way. Cedric quoted Poincare who had once said, “Mathematics is the art of giving the same name to different things.” The second part of his talk on ‘gases’ started with a description of his visit to Vienna and a search for Boltzmann’s grave. He said that he stopped to ask a family for a map not expecting them to know Boltzmann, let alone his grave. To his surprise, he was not only directed to the location of the grave but the person also exclaimed Boltzmann’s equation of entropy, “S=klogW”!

In connection with entropy, he then talked about the Gaussian curve, its ubiquitous nature and its uncanny appearance in many natural systems. He called the study of Probability and Statistics as ‘the extraordinary adventure of mastering of chance.’ As a matter of coincidence, he discussed a famous problem in his presentation called ‘Buffon’s needle’, which was also discussed in class earlier on the same day with the PGDBA students by their lecturer. Experiments such as coin tosses, Cedric said, are best done in the most careless ways! Then he explained how gases are modeled as billiard balls in collision and when there are many such sufficiently small billiard balls, their velocities are accurately modeled as the Gaussian distribution. As a note, Cedric remarked on the power of this distribution by quoting Sir Francis Galton who once called it the ‘supreme law of unreason.’

The next part of his presentation was ‘prices’. Cedric introduced Leonid Kantorovic, the father of linear programming. He explained how math is used to model the optimal allocation of resources. He then strung together ideas from the optimized distribution of resources to the distribution of gas molecules with the least loss of energy. The analogy of prices in linear programming is energy in distribution of gas molecules. This is where Cedric began piecing everything together with the last part of his talk called ‘men’. Cedric described how he had happened to meet his collaborators John Lott and Felix Otto. These men put together the ‘triangles’, ‘gases’ and ‘prices’ and helped Cedric complete his research on how fast gases reach the equilibrium stage described in Boltzmann’s equation. Cedric was awarded the Fields medal in connection with this research.

What would have been an intimidating subject matter coming from volumes over volumes of text, was aptly introduced in a two-hour lecture by Cedric Villani, a master at his trade, a storyteller par excellence, a dinosaur catcher in his childhood dreams and a true ambassador of modern mathematics. In a surprising irony of sorts, apart from the many hidden mysteries in the details of his works, the most apparent mystery is the brooches of spiders that he wears on the lapels of his coat!