## Why I Loved Fluid Dynamics as a Kid (and Still Do)

Like many other Physics undergraduate/graduate students, my first introduction to Prof. Julia Yeomans was as the author of the brilliantly written textbook Statistical Mechanics of Phase Transitions. Prof. Yeomans is a theoretical physicist at the University of Oxford who does some pretty cool stuff involving bacterial swimmers and water drops on hydrophobic surfaces among many other things.

It was indeed a delight, then, for the seven-year-old fluid dynamics-loving kid in me to listen to Prof. Yeomans speaking about the science of fluids yesterday at Kappi with Kuriosity.

Hold on. When I say the seven-year-old fluid dynamics-loving kid, I do not mean a seven-year-old crossing-out-the-time-derivative-term-in-the-Navier-Stokes-equation kid but rather a seven-year-old intrigued-with-his-toy-steamboat kid.

It was a lot fun, I remember, watching the noisy steamboat moving around in a tub of water. A couple of years later, it was Janice VanCleave’s Physics for Every Kid that added more substance to the intrigue and delight. A remarkable book in many ways, Physics for Every Kid was where I got my first introduction to the physics behind the swinging of a ball and the upward push or lift on an aircraft. A number of years and physics courses later, I do understand more of fluid dynamics than I did then, or so I think. Though, in any case, the love for the science of fluids remains the same.

Well, as I said, yesterday’s talk was a delight. You should watch it online when it’s up; you’ll find it on the ICTS YouTube channel.

#### To Maryam Mirzakhani, one of the finest mathematicians of our times, who passed away yesterday at the age of 40.

Dear Maryam

Your going away is, indeed, a loss, to mathematics, and to all of us over the world for whom you were no less than an icon. But in this loss, I believe, many more young people will find hopes, hopes to achieve greatness and to contribute to the world in whatever way they can.

When, at present, we are busy building walls and spewing hatred and animosity, I wish that your life and work will stand exemplary of noble human pursuits, of human creativity and the universality of human endeavours.

I believe that very much like me, there are numerous students and young mathematicians and scientists across the globe, who, even though, barely understand the intricacies of your work, realise the importance of the role you played as a mathematician and as a citizen of the world.

I believe that you will forever remain a wonderful example telling us how shallow and meaningless are the stereotypes that we fabricate. And I hope that in your going away, we will all find motivation to work wonders, for ourselves, and for the world.

A Young Fan

## All Those Years Ago

#### What’s the decade dearest to a Beatles-loving physics enthusiast? The 60s, of course!

Now yesterday and today, our theatre’s been jammed with newspapermen and hundreds of photographers from all over the nation. And these veterans agreed with me that this city never has witnessed the excitement stirred by these youngsters from Liverpool, who call themselves The Beatles. Now tonight, you’re gonna twice be entertained by them. Right now, and again in the second half of our show. Ladies and gentlemen.., The Beatles.

This was the iconic television show host Ed Sullivan’s introduction to the Fab Four in New York City in February 1964. George, John, Paul and Ringo, the four twenty-something English lads who had started playing together less than four years ago, were already a sensation on both sides of the Atlantic. They were to sweep the entire of the decade, a decade that is dear to me for one more reason – particle physics!

I’m a second generation Beatles fan (or the third, it doesn’t matter, anyway). And for about last five years, I have this bug called The Beatles. (Sorry, couldn’t resist the pun.) Born too late to attend a live Beatles performance or to buy vinyl records of Beatles albums, I fell in love with The Beatles when I discovered them on the internet.

It was impossible not to fall for them; in less than a decade, they’d influenced music like no band or artist had ever done. They were winning hearts and earning lots of money (they are the most commercially successful band of all time).

Fifty years ago, in the summer of ’67, they released their eighth studio album Sgt. Pepper’s Lonely Hearts Club Band. An experiment in a number of ways, the album became an instant commercial and critical hit (like almost all of their albums and singles). (Sgt. Pepper’s ranks number one in the Rolling Stones magazine list of the 500 Greatest Albums of All Time.)

But 1967 was phenomenal for another important reason. That very year, Abdus Salam and Steven Weinberg, two scientists working in the United States, produced their seminal work on the unification of the electromagnetic and the weak nuclear force.

Now a little something on what this means to help you understand and appreciate its importance. All objects in this universe interact with each other through one or more of four forces – what we call the fundamental interactions. Now, each of these forces has a very specific characteristic; and obtaining a proper and complete description of each of them is quite non-trivial, enough to have kept physicists busy all these years.

Though understanding the behaviour of all these forces is difficult, one smart thing to do is to think of them as interactions between particles mediated by, well, some other particles. So, for example, you’ve electromagnetic interactions between charged particles which are mediated by photons, the corpuscles of light. And gravity which can be understood as a manifestation of interactions mediated by what are very creatively (?) called gravitons.

Now, this is where Salam and Weinberg come into the picture. What these two gentlemen were able to show was that the weak and electromagnetic interactions, which are mediated by different particles, and of course, have different behaviours, are two different manifestations of one fundamental electroweak interaction. First proposed by Sheldon Glashow in 1961, the electroweak unification, as it is called, marks an important milestone in our understanding of nature at the most fundamental levels. (Glashow, Salam and Weinberg were awarded the Nobel Prize in Physics in 1979 for their contributions to the theory of electroweak interaction.)

In fact, the entire of the sixth decade of the last century saw numerous contributions coming from theoreticians and experimentalists alike – all these culminating into what can unarguably be called a triumph of human endeavours – the Standard Model of particle physics. Efforts of countless individuals have given us this fine theory which not only classifies all the elementary particles but also explains how the electromagnetic, strong and weak interactions are related to one another. (As for gravity, it still is a hard to nut to crack.)

The sixties were rather strange, though; the silliest and longest of wars was going on in Vietnam, there were successful lunar missions, but there were assassinations, too – JFK, Martin Luther King Jr. were killed; it was like a win some, lose some kind of thing, perhaps it has always been. But then, there were The Beatles – young, energetic and innovative. And thinking about why a twenty-something guy in India, over forty years after the last Beatles performance, is fascinated by them, I realised something.

It is not just about the music, it is about the themes as well. So, you have this boy band, singing beautiful songs about love and friendship. Four twenty-something English lads generating admiration with their songs and charming personas, captivating an entire generation (and more). Listen to this to get a feel of what I mean (and possibly, to get a break from this tedious read as well).

Following The Beatles album by album made me realise that all the while these guys were growing up, too. Their music was maturing, and so were their themes. I couldn’t help but admire the variety and the depth in their themes. They were not just talking of love and friendship now, there were also talking of nostalgia and were spinning yarns about the lives of regular people and at the same time, using their songs to express their worldviews. They were experimenting with music and songwriting, and in the process, producing a sheer volume of invention. And if I am able to connect to a band that played all those years ago, it is because of the wonderful music, for sure, but probably, it is also because their songs represent what I feel; they cover an entire spectrum of emotions, all the essential themes.

Sheldon Glashow, Abdus Salam, Steven Weinberg, George Harrison, John Lennon, Paul McCartney and Ringo Starr, they all symbolise how important ingredients creativity and innovation are in human endeavours. We, as a species, have come quite far, learning and evolving, but probably sometimes repeating the same mistakes again and again.

I’m not sure if the world today is better than that in the sixties or not, but for sure, not everything is fine at present. I’ll leave you with something which I believe is important for all of us to understand, a 1968 Beatles song called Revolution. (And as I tell everyone whenever suggesting a Beatles song, read the lyrics of the song as well, especially when it is as meaningful a song as this one.) Because, though these are troubled times, don’t you know it’s going to be…alright!

## Random Matrices in Three Short Stories – III

#### This is the third and final part of the series. For those who haven’t already, going through the first two posts, which can be found here and here, will be a good idea.

Three

Why do we want a quantum theory of gravity? We just want it, okay?

Hold on, this isn’t me. This was the American theoretical physicist John Preskill at a conference earlier this year [1]. And though, in almost all probability, he was trying to be funny, this does give an idea about how difficult (and often, frustrating) it is for theoretical physicists to answer why-do-we-want-this or what-use-doing-that questions. (By the way, apart from his seminal contributions to the fields of quantum information and quantum gravity, Preskill is also famous for winning a black hole bet against Stephen Hawking which Hawking conceded by offering him a baseball encyclopedia.)

But something about the notion of universality in random matrix theory before I go on to talk about the theory of quantum gravity, cosmological inflation, black holes, wormholes and time travel. (Okay, I was kidding about the last two – no wormholes and time travel here.)

Large matrices with random entries have some very intricate (and interesting) statistical properties. And the reason they come in so handy in our attempts to answer questions of different sorts is the applicability of the statistical laws of random matrix ensembles to all those systems which have the same symmetries as those of the ensemble.

These statistical laws often involve eigenvalues of the matrices. Eigenvalues are certain quantities associated with matrices. In fact, all square matrices (arrays of numbers with the same number rows and columns) can be characterised by their eigenvalues. And in most cases, computing the eigenvalues is not a very difficult task.

##### For a matrix H, all the possible values of λ satisfying the above equation are its eigenvalues. I is a square matrix the size of H with zeros along the diagonal going from left to right. And det() represents computing the determinant which again is a number associated with a matrix.

For a set of random numbers, it is very natural and useful to talk about the probability distribution – how probable the occurrence of each of the numbers is. Now, if your matrices have random elements, its eigenvalues will be random as well, which makes it useful to talk about the eigenvalue distributions of the ensembles.

These distributions in random matrix theory are universal in the sense that they don’t depend on the underlying structures as long as they have a common overall symmetry. In the context of a physical system, these eigenvalues correspond to the energy levels of the system and, in essence, contain the information about its dynamical properties. Superficially, this is what a lot of random matrix analyses is about. The energy spectra of systems which are difficult to interpret otherwise find meaning in the language of random matrix ensembles.

But understanding the underlying dynamics of many systems is not easy. And as we have come to realise in the case of black holes – it is certainly not.

Black holes are interesting; though calling them interesting might be an understatement. They are formed in a number of different ways all over the universe. Within the framework of the theory of general relativity, one can, in quite a straightforward manner, show that a black hole has a gravitational pull so huge that nothing can escape it, not even light.

But everything is not that straightforward. Though general relativity explains the force of gravitation, we believe our universe to be inherently quantum mechanical, that is, we expect every object in the universe to follow the laws of quantum theory. And taking quantum mechanical laws into consideration, one can show (as Hawking did for the first time in the early 1970s) that black holes radiate stuff [2]. Now, this may seem puzzling; in fact, it is puzzling. But what this radiation also indicates is that black holes are thermal objects – they have thermodynamic properties (say, temperature, for example) in a manner similar to how your daily cup of coffee does.

It is evident that black holes demand a better understanding than the one at present. And this can possibly be achieved by constructing a unified framework incorporating both gravity and quantum mechanical principles. This is somewhere random matrix analysis turns out to be useful – understanding energy spectra of black holes.

Depending on the theoretical framework in which the calculations are being done, black holes can be studied using different models. In principle, the details of the energy spectra can be worked out for each of these models. But as it turns out, not all black hole models are soluble. The trick is to then use random matrix models which can possibly mimic the expected properties.

Black holes have posed some of the most interesting challenges to those working in physics for the last hundred years. They have also motivated a quest for quantum gravity. However, a theory of quantum gravity will possibly also explain many other mysteries. One of them is cosmological inflation. Based on a large amount of astronomical data, it has been conjectured that our universe underwent a phase of extremely rapid expansion for a small fraction of second just after the big bang. This conjecture also explains the origin of the large scale structures in the universe pretty well. However, what is missing is a concrete theoretical underpinning of inflation itself. Among various proposals to explain inflation, there are a few which employ random matrix techniques as well [3]. But again, a complete understanding of inflation, like that of black holes, still belongs to the large set of open problems waiting to be solved!

#### References

[1] John Preskill, Quantum Information and Spacetime (I)https://youtu.be/td1fz5NLjQs, Tutorial at the 20th Annual Conference on Quantum Information Processing, 2017.
[2] Leonard Susskind, Black Holes and the Information Paradox, Scientific American, April 1997.
[3] M.C. David Marsh, Liam McAllister, Enrico Pajer and Timm Wrase, Charting an Inflationary Landscape with Random Matrix Theory, JCAP 11​, 2013, ​ [arXiv:hep-th/1307.3559]​.

## Random Matrices in Three Short Stories – II

#### This is the second part of the series – the first one can be found here.

Two

If you ever happen to be in a conversation with someone with a devout love for number theory, it won’t be long before the conversation will evolve into one about prime numbers – about how fascinating these elementary, yet mystical creatures are and about how despite their apparent randomness, there is an intriguing rhythm, a captivating music in their distribution.

Prime numbers are the prime ingredients of natural numbers (and of interesting conversations, too). Take any natural number (greater than one) – you can always write it as a product of certain primes. And these primes themselves cannot be expressed as the product of smaller numbers. Simple as they sound from this definition, prime numbers have a very surprising and inexplicable manner of showing up on the number line. Much of folklore and mathematical literature alike have their origin in this mystery surrounding primes.

But hold on, why are we talking about primes in a story about random matrices?

We wouldn’t have been, had it not been for a chance teatime conversation between Freeman Dyson and mathematician Hugh Montgomery (interesting conversations, remember) [1].

In the spring of 1972, Montgomery was visiting the Institute for Advanced Study at Princeton, New Jersey to discuss his recent work on the zeros of the Riemann zeta function with fellow mathematician Atle Selberg. Selberg happened to be a leading figure of the time on the Riemann zeta function and the much fabled Riemann Hypothesis.

First formulated by Georg Friedrich Bernhard Riemann in his 1859 paper, the Riemann hypothesis is a hugely famous and celebrated conjecture yet to be (dis)proved. (And guess what, this was the only number theory paper the mathematician extraordinaire Riemann wrote in his entire lifetime!)

Bernhard Riemann made an important observation that the distribution of primes was intricately related to the properties of a function that now bears his name (it was first introduced by the Swiss mathematician Leonhard Euler, though). Now, as it turns out (and as you’ll see if you happen to delve more into abstract mathematics), mathematical functions and objects have personalities of their own. The zeta function, as Riemann observed, appeared to have an interesting one.

##### Behold the mighty Riemann zeta function! The Riemann Hypothesis states that all the interesting values of s for which ζ(s) = 0 lie on a straight line in the complex plane.

Numbers as you probably know, can have two parts – a real part and an imaginary one (and you’ll probably realize later that the imaginary part is not so imaginary after all).

##### A complex number has two parts – real and imaginary (here, a and b respectively). The tiny i hanging alongside the imaginary part, b is what makes itthe imaginary part. (i is the imaginary unit, the square root of 1.)

The Riemann zeta function has some non-interesting zeros (values of s for which ζ(s) = 0) at s = -2, -4, -6 and so on. What Riemann conjectured was that all the other zeros of the function will always have their real parts equal to 1/2 – and hence, when you plot them on the complex plane, they’ll all lie on a vertical line [2].

Now, over a century and a half later, all we know is that this hypothesis appears to be true. We know it to be true for the first 1013 (!) zeros we’ve found till now, but have no idea whether it holds true in general or not. (For those of you who refuse to take abstract concepts arising in pure mathematics seriously, the zeta function will keep appearing in your life even if you restrict yourself to more concrete (?) areas of applied mathematics and/or physics.)

As has been the norm with challenging problems since the beginning of the previous century, the Riemann Hypothesis along with five other open problems are listed as the Millennium prize problems by the Clay Mathematics Institute – each with a bounty of a million dollars.

But this story is not about the million dollars – mathematicians don’t care much about it anyway (or so I am guessing). In the early 70s, number theorist Hugh Montgomery was working on the statistical distribution of the interesting zeros of the Riemann zeta function on the critical line – the vertical line where all of them are conjectured to lie on.

When, during their teatime conversation at the Institute for Advanced Study, Montgomery mentioned his recent results to Freeman Dyson, both Dyson and Montgomery were up for a surprise. Dyson realized that the statistical distribution of the Riemann zeros that Montgomery had worked out had a lot in similar with the statistical properties of a certain class of Random Matrices Dyson had earlier looked at while working on the physics of heavy atoms. And more importantly, the theory of random matrices had, by then, pretty well established results that could be applied to Montgomery’s problem [1].

Dyson then wrote a letter to Selberg referring Madan Lal Mehta’s book on random matrices to be looked up for the results that were needed by Montgomery. (You must read this article published in the IAS Spring 2013 newsletter; the article also has a scanned image of Dyson’s handwritten note to Selberg!)

This striking similarity in the statistics of the Riemann zeros and the spectra of heavy atoms points towards an universality in the underlying structures. Stronger results coming from probability theory and mathematical statistics appear to give a clearer picture; though much of it still appears to be an outright miracle. More on this notion of universality in the last part of the series when I’ll narrate the story of one of the greatest quests in present day science – the quest for a theory of quantum gravity.

#### References

[1] Kelly Devine Thomas, From Prime Numbers to Nuclear Physics and Beyond, The Institute Letter Spring 2013.
[2] Peter Sarnak, Problems of the Millennium: The Riemann Hypothesis, 2005.

## Random Matrices in Three Short Stories

#### This post is a slightly modified version of a student talk I gave for the IISER Pune Science Club earlier this year. All the three stories (this one and the two to follow) will be accessible to anyone with some exposure to high school physics and mathematics. Readers with a formal training in advanced level physics and/or mathematics have all the rights to criticize the author for an over-simplistic presentation.

One

The first story begins with that of my personal hero, Freeman John Dyson. Growing up as a kid in England in the period between the two of the most disastrous wars this planet has seen, Dyson developed a strong interest for everything numbers. This interest, quite naturally, evolved into a passion for physics and mathematics. When the eighteen-year-old Dyson arrived at Cambridge in 1941 as a student, there were few physicists around – a constant phenomenon at the universities during the war years; physicists were perhaps the most suitable people to be sent away with war-related responsibilities.

As it happened, the greatest influence on Dyson, while at Cambridge, was the famous mathematician duo, Hardy and Littlewood [1]. After working for a few years on number theory problems (he published a couple of influential papers in this period), Dyson moved to the United States where he was appointed a professor at the Cornell University; he didn’t have a Ph.D., though (and never got one).

With the brightest of physicists around (Richard Feynman and Eugene Wigner to name just two from a pretty illustrious list), Dyson’s focus shifted towards problems from quantum physics. (Number theory, however, was to appear in his life again, albeit for a small period, as we’ll see in the second part of this series of stories.)

Quantum mechanics, one of the two greatest triumphs of the twentieth century physics (General Relativity being the other one, of course), reformulates the study of physical systems in the language of the Hamiltonian. If you were reading a chapter from a textbook on Quantum Physics (which this post is not), you’d be told that this Hamiltonian is a Hermitian operator. Now, a Hermitian operator, to put in rather simple words, is a matrix with some special properties. And a matrix is nothing more (?) than an array of numbers. But what has a matrix got to do with a physical system?

As it turns out, the way a system evolves can be mathematically expressed in terms of the product of certain matrices – and the Hamiltonian of a system, which itself is a matrix, determines what matrices you should be multiplying. Say, you want to study a Hydrogen atom. You’ll have to begin with its Hamiltonian and see what all you can say about the energy of its components. And then you can check how well you did your job by comparing your results with a Hydrogen spectrum. (A quick Google Images search for a Hydrogen spectrum at this point is strongly recommended for those who haven’t seen one.)

As seems intuitive, the Hamiltonian of a Hydrogen atom should be a lot simpler than that of a more complicated atom. A more complicated atom would also mean a heavier atom; consider, for example, a Uranium atom, which is over 200 times heavier than the Hydrogen atom. But with increasing weight comes increasing complexity (sic); heavier atoms have a larger number of interacting components. This essentially renders it impossible to write down a Hamiltonian that you can use to predict the spectrum of your complex atom. Poof! All the powers quantum mechanics bestowed upon you go awry.

Not really. What Freeman Dyson and Eugene Wigner showed that you can make a pretty smart guess about the Hamiltonian of such a complex, heavy atom using a Random Matrix – a matrix which contains random numbers. It’s like having your usual matrix except for that the entries are drawn randomly from a set of numbers.

##### H, which can describe the Hamiltonian of a system, is a matrix with elementsH11, H12, H21 and H22. You can call H a Random Matrix if the entries of H are random variables. Here, H has four elements – in practice, you’ll have to take much larger matrices for your computations.

Now, the essential idea here is to realize that to make predictions about the system you are trying to model, you need to consider an ensemble of all the random matrices which would give rise to the properties you’re expecting your system to have. These properties are manifested in terms of certain symmetries of the system under consideration. (The notion of symmetry is both utterly important and extremely fascinating in the physical sciences. If you’re ever in a need to spot a theoretical physicist in a large crowd, a passing mention of symmetry will do the job.)

The next step in your analysis of complex atoms will then be to study the statistical properties of the matrix ensemble with the appropriate symmetries. And that’s pretty much all. There is this entirety of machinery coming from the theory of random matrices that gives you the freedom to treat the complex atom as a black box with a very large number of interacting components and to still be able to extract out the essential information with great accuracy [2].

A point that must not go without a mention here is the practical importance of studying heavy atoms. Heavy atoms act as the source of nuclear energy, the foremost candidate for being the chief contributor to the fulfilment of our energy needs in the future.

If you’re feeling bewildered and fascinated by the fact that such an apparently unrelated notion of random matrices can help you predict the spectra of complex atoms, you’re not only in the company of some of the greatest minds working in this area, but also stand a chance of being a part of the wonderful discoveries yet to be made. Our understanding is very limited at present – we know that things work but have very little idea why. There appears to be some statistical law of large numbers working behind the scenes. Interestingly, it was in mathematical statistics where random matrices had first made an appearance in the 1930s with the work of the agricultural statistician John Wishart.

Talking of statistics and numbers, the theory of random matrices have also led great insights into the solutions to challenging problems in number theory, which will be the theme of the second of our stories. As you’ll see, Random Matrix Theory has made its way far beyond statistics and atomic physics. A comprehensive and insightful reference for those interested is the classic book Random Matrices by Madan Lal Mehta. (Mehta, who had a very fruitful collaboration with Dyson and Wigner, was one of the leading contributors to the subject.) However, you’ll need to have some background in linear algebra to follow the text; but it is surely worth the efforts.

#### References

[1] Freeman Dyson, Selected Papers of Freeman Dyson with Commentary, American Mathematical Society, 1996.
[2] Madan Lal Mehta, Random Matrices, Vol. 142, Pure and Applied Mathematics, Academic Press, Ed. 3, 2004.

## Mimamsa: Stories from Behind the Scenes

The past weekend saw a gruelling but very exhilarating contest for the coveted winner’s trophy of what we at IISER Pune love to call the toughest undergraduate Science quiz in India. Mimamsa, in its ninth edition in 2017, had teams from IISc Bengaluru, NISER Bhubaneswar, IIT Bombay and IIT Madras in the finals, selected after a preliminary round held earlier this year.

Undergraduates from across the country who have participated in Mimamsa over the years have called it “intellectually stimulating” and “enjoyable” – quite aptly so, given the ideology behind its conceptualisation in 2009 by Dr. Sutirth Dey.

However, this post is not about the Mimamsa presented to the participants, but the one students at IISER Pune spend time creating – and I’d argue why these two are not the same. But of course, the arguments I present here are (almost) entirely based on my personal experiences, and I don’t expect everyone to agree with them.

Perhaps the most significant element that makes Mimamsa unique is the flavour of the questions. The questions are non-trivial to begin with, and in fact take a form that makes them seem impenetrable, until, of course, one gets to know the solution. Then the participants feel awestruck if they were not been able to solve a question or otherwise feel elated to have grabbed some essential points to add up to their tally in the contest. That’s it, right? No.

Remember when I said that the Mimamsa presented to the participants is not the Mimamsa students at IISER Pune spend time creating? The questions (on most occasions) evolve from being raw ideas to taking the final forms they’re presented in. And in the course of this evolution, the students involved in the making of these questions evolve too; learning a lot in the process – new ideas, new methods of enquiry, ways to come up with smart solutions, the ability to gauge the level of difficulty of problems, the intricacies of posing questions. The process is long and tiring, and like any other venture, the students make numerous mistakes in the process, but then they get to learn from these mistakes, too. Exactly the things we expect ourselves to become extremely good at as students of Science (and Mathematics).

On the front-end what appears to be a nice, sophisticated quizzing event, is, behind the scenes, a very dense, sometimes exhausting but almost always rewarding process.

Mimamsa is surely about the spirit of quizzing and about motivating enquiry, but it is also about the enormous efforts that are put in by the students on all fronts, and it goes without saying that the teams involved in the organisational aspects over the years must be given an equal credit for what Mimamsa has come to be today.

It might be a bit too early to call Mimamsa a phenomenon – it certainly appears to have the potential to become one in the time to come – but it has already impacted the lives of many of us who have been associated with it, and this does make Mimamsa a phenomenon in our lives.