Entering the second decade of its existence, the International Centre for Theoretical Sciences of the Tata Institute of Fundamental Research (ICTS-TIFR) is organising a three-day-long celebratory scientific gathering ICTS at Ten starting today.

The speakers include some of the most illustrious researchers from the fields of astrophysics and cosmology, string theory and quantum gravity, mathematics, theoretical computer science, condensed matter and statistical physics, and physical biology, and will deliver broad perspective talks on some of the most exciting questions in these areas.

You can catch the entire event live on Youtube.

Edit: The links to the videos have been updated and can be found here.

What better a state than a state of complete confusion to write this post in? I had plans of writing some (hopefully interesting) stuff on chaos. I mean, on the kind of chaos physicists are interested in. (It’s another thing that you can get them interested in any kind of chaos if you know the tricks well.) But then, I also discovered a new-found love in Shakespeare’s plays yesterday evening, courtesy a charmingly chaotic performance by The HandleBards at Ranga Shankara.

So, to save myself from the trouble of having to make a choice between the two, I’ll just set chaos afoot by talking about both together, with a hope that you’ll end up in a state of complete confusion, too, but one that you’ll be able to find some charm in.

The nice thing about chaos is that it doesn’t take a lot of imagination to think of a system that will display some sort of chaos. The not-so-nice thing about chaos is that it makes it very, very difficult to study such systems.

Try visualising a complicated system. How complicated? Well, let’s say that any system with a large number of interacting components will serve our purpose. Now, you can choose being lazy and not do anything. But you can also choose being not-so-lazy and make a small change to some part of the system. Hold your nerves, wait for a while and behold, the small change made to a part of the system changes the entire system altogether and does so very significantly.

Your system has many interacting components; it’s not surprising then, that with time the itsy-bitsy perturbation you’d created spreads throughout the system. This is what chaos is: small changes at initial times leading to very drastic changes sometime later.

Like many other Shakespeare’s plays, ‘As You Like It’ is a narrative that oscillates between order and disorder, harmony and chaos. It is about the madness and the energy and the whole lot of confusion that the forest of Arden is overflowing with. A perfectly complicated system, right? The bard was a genius, no doubt!

Systematic studies of chaos have helped us realise that it is an important cogwheel in the clockwork that our universe is. It is a common feature of the classical world that we perceive to be inhabiting. And we do understand a great deal about chaos in such classical systems.

But then, we also understand the universe to behave in not-so-classical manner at a more fundamental level. Well, it does not behave classically at all! There are these weird, mystical quantum mechanical laws that dictate the underlying dynamics of all systems. As it turns out, it is even more difficult to make sense of chaos in quantum mechanics. But physics has a knack for providing more of an interesting answer the more difficult the question is.

Alas, this post has its limitations. You’ll have to stay tuned for the next one to read about where this discussion about chaos in quantum systems is heading to. (Hint: What is common to scrambled eggs and a weasel that falls into a black hole?)

The HandleBards are four-strong troupes of cycling actors which perform environmentally sustainable Shakespeare plays the world over. An all-female troupe of The HandleBards is touring India right now. You can join them at different venues across Bengaluru over the next few days.

Postscript: Ranga Shankara, the theatre where The HandleBards performed yesterday was opened in 2004 in the memory of the illustrious actor and director Shankar Nag. Many of you who grew up in India would probably still have fond memories of the television series Malgudi Days based on RK Narayan’s short stories; the series was envisioned and directed by Shankar Nag.

The title of this post has its origins in the title of Paul McCartney’s thirteenth solo studio album released in 2005, Chaos and Creation in the Backyard.

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.

A science outreach initiative of the International Centre for Theoretical Sciences (ICTS-TIFR), Kappi with Kuriosity is a series of monthly public lectures organised in collaboration with the Jawaharlal Nehru Planetarium and the Visvesvaraya Industrial and Technological Museum, Bengaluru.

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.

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!

The title of this post has been borrowed from that of a 1981 single by George Harrison. All Those Years Ago was Harrison’s tribute to John Lennon who had been assassinated in December the previous year.

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].

Thanks are due to Divya Singh and Ramesh Chandra for providing essential feedback for this series of posts on Random Matrices.

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 10^{13 }(!) 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.

Thanks are due to Divya Singh and Ramesh Chandra for providing essential feedback for this series of posts on Random Matrices. The final post can be found here.