August 25, 2010 Michel the Anand and 2560 Concert: Ustad Missionary 2 Squares 3 Rashid Khan 4 Sound Approach to Quantum Unique Ergodicity Photo :Rahul V Pisharody annan Soundararajan is a Professor of yet have. This is re- KMathematics at Stanford University. He lated to the famous works in number theory, especially L-functions Riemann Hypothesis and multiplicative number theory. He won the which is one of the Morgan Prize in 1995 for his work in analytic Clay Prize problems. number theory while still an undergraduate at the University of Michigan. He also won the There are many con- Salem Prize (with Lindenstrauss in 2003) and jectures concerning the SASTRA Ramanujan Prize (shared with the values of zeta and Manjul Bhargava in 2005). other L-functions on the critical line - the When he was an undergraduate student at line of symmetry for the University of Michigan, Soundararajan their functional equa- proved two outstanding results. Firstly, he tions - the Riemann proved a conjecture of Graham in combinato- Hypothesis and Linde- rial number theory in collaboration with R. Bal- lof Conjecture, for in- asubramanian. Then, he proved some deep stance. Among them results on the distribution of zeros of the Rie- there are conjectures mann zeta function. about the growth of Kannan Soundararajan these L-functions on In his PhD thesis, Soundararajan proved that the critical line. There are some simple esti- bined with a parallel develpoment in the sub- more than 7/8-ths of quadratic Dirichlet L- mates, called the convexity bounds about the ject by Roman Holowinsky, it yields marvel- functions have no zeros at the critical point values of these L-functions on the critical line. lous results. They each prove theorems s=1/2. He proved with Ken Ono, a conjecture In several questions, `breaking' this convexity towards QUE, neither settling it completely, of Ramanujan under Generalized Riemann bound is of paramount importance, and is but in the end one realizes that together they Hypothesis (GRH). Soundararajan is an ex- known for a class of L-functions, and in each have completely proven the QUE conjecture! pert on random matrix theory where he case it is rather nontrivial. showed with Hugh Montgomery that the dis- In this context, one should note that Elon Lin- tribution of prime numbers in short intervals is Soundararajan has achieved such a theorem denstrauss proved the QUE when one con- surprisingly different from what classical for what is known as the triple product L-func- siders Maass forms instead of holomorphic heuristics imply. Soundararajan, considered tions, and as a result proved what has been modular forms. This is part of the work cited to be one of the most creative young minds to called the Quantum Unique Ergodicity (QUE) while awarding Lindenstrauss the Fields emerge in the last decade.On August 22, he conjecture of Rudnick and Sarnak. medal. Incidentally, Soundararajan and Lin- delivered a talk at the ICM 2010 on `Quantum denstrauss won the Salem prize in the same Unique Ergodicity and Number Theory'. The QUE conjecture asserts that the so-called year. newforms, which are holomorphic modular The Riemann zeta function encodes many forms on the Poincare upper half plane, and Soundararajan's work is considered by the properties about prime numbers. The prime are eigenfunctions of Hecke operators, give experts in the subject as one of the deepest number theorem which addresses the asymp- rise to measures on the upper half plane contributions to the subject in the last few totics of the number of primes, was conjec- which converges to the invariant measure on years. Given the interest in this subject, it is tured by the young Gauss in the late the upper half plane as you vary either the in a state of flux, with many mathematicians eighteenth century. It was proved about a cen- weight of the modular form keeping the level trying their hand at simplifying and unifying tury later independently by Hadamard, and de fixed, or vary the level keeping the weight the arguments. La Vallee Poussin. One expects to have good fixed. What has been particularly spectacular R.Balasubramanian on Soundararajan, Page 3 estimates on error terms too which we do not about Soundarajan's work is that if it is com-

"The result of Holowinsky and Soundarara‐ “A part of it is complicated; that’s the beau‐ “One of the things that Elon Lindenstrauss jan comes from their separate a*acks on )ful thing, in fact. Our approaches [to the got his Fields Medal this )me for is se*ling a the holomorphic QUE problem and de‐ problem of QUE] are completely different. case of the Quantum Unique Ergodicity pends on the Ramanujan conjecture Actually I got some help from him many (QUE) Conjecture. He se*led it for some ob‐ proved for certain holomorphic forms by P. years ago His approach is purely number jects which are called Maass forms. These Deligne in the 1970’s. Each method theore)c. He does not use any dynamics. are certain non‐holomorphic forms on the achieves the desired result up to a small The difficul)es I have are different from the space. There is also the analogous case of number of excep)ons. Remarkably their difficul)es he has in his approach. It’s one of holomorphic forms which is not addressed approaches are sufficiently different so that the beau)es of the problem. One of the by his theory. They are kind of two sides of the set of excep)ons to both approaches, things I am trying to think about using some the problem but the methods are com‐ is empty! " ingredients from his and combine with my pletely different. Our approach is number kind of techniques. It will be quite an inter‐ theore)c. Elon’s approach comes from dy‐ ‐Peter Sarnak, Princeton University es)ng direc)on. “ namical systems and ergodic theory. “ ‐ Elon Bruno Lindenstrauss ‐K.Soundararajan

REFLEXIONS August 25,Wednesday Mathematical Visits to Developing Countries - A Passion !"#"$%&'!()%*% +,!'-"./ ichel Waldschmidt is a Pro- on whether you talk of the near Mfessor at the Faculté de future or in the long term. Tao has Mathématiques Pierre et Marie spoken about whether computers Curie. He works in Transcenden- will replace our activity as mathe- tal Number Theory. Waldschmidt maticians. My field is number the- has been actively involved in the ory – in particular, Diophantine mathematics cooperation be- equations. There is Hilbert's 10th tween France and several coun- problem which says in a sense tries in Asia, Africa and South that we can never find a complete America. He is passionately inter- theory for all Diophantine equa- ested in running and moun- tions. taineering. Let us hear what he says when he talks to Richa Mal- It is good for mathematicians be- hotra and B. Sury cause it is our experience that Prof. Michel Waldschmidt whenever we close one problem, You have been actively associ- was good to have two activities Yes, well I will tell you the truth, it we may create a new field or ated for several years with co- which were completely different. is not working very easily. After whenever we prove a new theo- operations in several countries In order to do mathematics, we she passed away some people rem we ask some more ques- including India. What inspires need to do something else also. gave us some money, which usu- tions. This is good for the future you to involve yourself with But, the extent I went to is too far ally in our culture is used for flow- and we expect that the number of these activities? because I wanted to run a ers. But we thought that it would problems will be large. In my ca- From a very young age, I wanted marathon. I have run now more be a waste, so we said we would reer I saw very big problems to teach mathematics in develop- than 20 marathons and, now I like to use it for Mali because one which are solved now. I think that ing countries and my first oppor- cannot improve my speed. So, year before she passed away in the near future there will be tunity was in 1976 when I was last year I went to 100 kilometres when she was in Mali in a training tremendous results which solve invited by Prof. K. Ramachandra. and was for 24 hours. session in a hospital, we thought what we consider as difficult now. I visited the Tata Institute for 3 we would send the money to the Some new ideas will be needed months and this was a very I am hesitant to ask this ques- hospital and then donate it to the and when these will come, no- tremendous and impressive ex- tion. Did personal losses like medical centre which was just body can tell. When we are asked perience for me. I was not speak- passing away of your daughter created at the same time. to research we can’t say next ing good English at that time; so I in 2004, motivate you to do year I will prove this and in two also learnt English by being with more for the deprived sections In fact we got a lot of money but it years I will do this. Prof. Ramachandra. of society? How do you find was very difficult to use this outlet of emotions leading from money. But, to help people, for Do you feel the need to moti- This was an experience which such an event? example, when they want to build vate yourself to do mathemat- was important for me. After that, It is always something which is some building, we say we are ics research? I felt that I needed to be more ef- emotional for me.I have put some ready to give you the money but This is a personal question in the fective; I was picked by my Gov- information on my web page and you have to give the plan and get sense that people react differ- ernment for that. I think what I do this means I am willing to speak the authorization of the govern- ently. For me it is important be- is more useful - helping students about this. ment. They do not do that and we cause it motivates the research in developing countries to do have to insist. We have now subject when I work on an open mathematics. When I teach in a People react differently, for exam- found one centre which should problem. I ask myself if it was place like India, in many places, I ple my wife does not like to work in another hospital. For me worth spending much time on it. If meet a lot of students who are speak on that and she often says it was an experience to see how I spend one year or two years, it very eager to learn much more she wants to forget but it is not my difficult it is when we want to help may not be worth it unless I can than in France. When I go to case, I don’t want to forget. Yes- people. It is a part of my life. In see some applications. It may be places like Bhubaneswar (India), terday I met a colleague who mathematics, we are doing work worth spending just one or two Abdus Salam Institute (Lahore), I spoke of my daughter because in Africa; things are quite different months. But, if I spend all this meet students who are very en- he saw my web page, and I told there. time to solve the problem and no- thusiastic about mathematics and him that when I speak of her, it body is interested (including my- ask a number of questions. This was as if she were here. But, I like Where do you see the future of self!), it is not worth it. I need to is what I like. For me teaching is to speak of my daughter although mathematics heading in the know that the problem is motivat- a pleasure. she doesn’t exist. Whatever I do, near future? Do you see it get- ing in the sense that it has some I wish that she would be proud of ting related to computer sci- applications. You have a passion for running me; this has a lot of influence in ence or both? and mountain climbing. So that way. I read this text by Terrence Tao. My motivation is inside mathe- what is the extent to which you He explained very well the future matics, but that is sufficient. I have gone into these things? You instituted a foundation as of mathematics and it is explained would not work on the problem if I was running since I was young. a tribute to your daughter. so well that it is difficult for me to I don’t see that this problem is in- It always gave me great pleasure. Would you like to highlight say something after he said it so teresting for me or for other ques- When I was running I was forget- about the activities of this beautifully. It is really very difficult tions. ting mathematics; so, for me it foundation? to know what to say; it depends ...continued on page 4

REFLEXIONS August 25, Wednesday ‘I am really proud to have played some role in his formative years’

annan Soundararajan or me in the office with a simple Journal of Number Theory. An- KSound as he is affectionately proof of an improvement of this other problem which I suggested called met me in 1986/87, when result. This later appeared in the to him was a conjecture of Gra- he was just finishing middle Journal of Number Theory. While ham on the greatest common di- school and entering high school. still in high school, he spent a visor (gcd) and at that time When he first started visiting me month at TIFR Bombay to attend Zaharescu had just proved the at Matscience, the Institute would a course of lectures given by conjecture for all large values. have a regular working day on Alexander Ivic on Riemann zeta Sound came up with some beau- Saturdays, and he could come function. This kindled his interest tiful ideas which unfortunately did then without missing school. in zeta function and we used to not lead us anywhere. Then I lost Some time later, Saturdays be- discuss a lot on this. interest in the problem but Sound came holidays, but I invited him persisted with the problem and then to come to my house on Sat- He went to the United States for got a complete solution in three urdays, and we would then dis- a six week summer programme years. cuss mathematics. I can still called the Research Science In- recall the days when he used to stitute (RSI), which used to be When I was visiting him in Michi- R.Balasubramanian come to meet me. held in Washington DC. It may be gan for a couple of days , we set- interesting to note that the other tled a conjecture of Erdos on I am really proud to have played I remember, one Friday evening students working with Lawrence distinct divisors and submitted it some role in the formative years when he met me while I was look- Washington then were Terrence to a journal. When the referee of a mathematician who rose to ing at apaper of Alladi, Erdos and Tao and Lenny Ng. Washington pointed out that our method is great heights proving results on Valer on the summation of arith- suggested to Soundararajan a likely to yield stronger results, diverse topics like Quantum metic functions. I suggested to problem on the number of primes Sound was not ready to publish unique ergodicity, upper bound Sound to study the paper in the in certain sequences. Sound the paper in that form but went far for the mean values of zeta func- weekend. Two days later on Mon- solved the problem using Selberg beyond Erdos’s Conjecture. tion on the critical line, structure day morning, he was waiting for sieves and the paper appeared in of multiplicative functions etc. Viswanathan Anand and 2560 Squares! !"#$#%&%'(")*%+%!,-"(.#/ of a rollercoaster ride while playing against one person is controlled”. He adds, “I was completely busted. A lot of people here played very well.” During the question and answer round, he exclaimed that there are periods when one does well and when one does badly. Anand drew a correlation of chess with mathematics and remarked, “Mathematics is similar to chess. If kids get fascinated in math- ematics early, it stays for long just like in chess. Chess is one thing you can teach kids very early. Some countries and communities tend to expose them to chess more. But we need more information to prove this.” The game was drawn after 46 moves.

Vishy Anand playing simultaneous chess with delegates at the ICM 2010. Anand played with white pieces, while the others played with black. Richa Malhotra games!” Srikar’s father was indeed glad to !"#$#%&%0,/")1%'(2%3 iswanathan Anand played chess against see his son perform neck-to-neck with Vishy V40 people simultaneously on 24 August Anand. 2010. All lost except for a 14-year whiz kid who drew with him. This is Srikar Varadaraj Srikar studies in Venkat International Public from Bangalore who also has the distinction School, Bangalore, and likes listening to rap of being one of the youngest to present a music, playing chess and solving Mathemat- paper at the ICM. ics Olympiad problems. While Anand learnt chess from his mother at the age of six, the Srikar says he wants to be a mathematician 14-year old Srikar learnt it from his father. after growing up. He was among the top 30 selected from Karnataka to participate in the About Srikar, Viswanathan Anand said, “Oh! Indian National Mathematical Olympiad in He was great”. Anand felt that the level of the 2009. About the game, Srikar later said, “I game was extremely high and the last 10 traded the Queen and thought I will catch up players held on very tenaciously. He said, in the end game, but Anand is too good at end “Playing with the participants here was more Srikar Varadaraj

REFLEXIONS August 25,Wednesday ‘Mathematics is growing at a speed which is exponential’ ...continued from page 2 ‘’I would consider that mathe‐ ficult. But I think in India, you have a strong I work on it only if I see this problem is inside support from the government. This is some- some theory. It makes some sense to people macians come together to thing extremely good and very important for - this is the kind of motivation that I want. fight against conjectures and the development of mathematics in India. The fact that the ICM is taking place in India is a Who inspired you to pursue mathematics? to make conjectures into the‐ proof of that. Also, we have the NBHM which Well, among the great mathematicians, one is orems. That should be done is doing such a good job and they deserve to Serre. He is a Fields medallist. The way he get more support. does mathematics is very impressive; the way together and not against one he writes mathematics is great. He is a very another.’’ Do you think mathematicians of the last nice man also, I did not anticipate this ques- century did deeper work than those in the tion and I did not know the answer before you tion. I think competition can be good; it can be previos century, in the sense that big con- asked. This is the first name I have, Wolfgang a challenge but it should not be a challenge jectures got solved now than ever before? Schmidt. He proved something which is called of one against the other. On the whole, what It is a continuous process. When you look at the subspace theorem, which is one of the I think is that to work together should be en- the history and development of mathematics, most important results of the twentieth cen- couraged. the development is growing quite a lot. The tury. Now the number of applications of the . mathematics which was done in the previous subspace theorem is large. What are the things you would like to be centuries is not destroyed and we use the changed in mathematics? whole of what was done then. I think lot of Does professional rivalry have a negative Diversity is something which should be pre- things have been done, in the last century, es- impact on the development of the subject served. We should have some great mathe- pecially in the last half century from 1950 to of mathematics or is it sometimes good maticians and also some mathematicians who 2000. It’s growing at a speed which is expo- also? do not make great researchers but good nential and it is really amazing to see. When I was a student, and when we write a teachers. dissertation, we are told to look at the positive How do you think we can enhance the side as well as the negative side and so, I will In a country like India, there is space for more communication of mathematics to the do that. On the negative side, I often meet mathematicians. In a global setting, there public? people who say they are working on a partic- should be more support for mathematics be- This is a very important subject. This is some- ular problem and don’t want to tell others what cause, as a subject, mathematics is not so ex- thing in France which was not good. A good they are doing because someone could steal pensive to pursue. It is not true that mathematician would not explain mathemat- their work. I think this is very bad for the de- mathematics needs only paper and pencil, as ics to a layman, it was not his job. It has com- velopment of mathematics. people say. This is not completely correct; pletely changed now. Many good they need more. For example, one needs mathematicians give some popular lectures, I would think of mathematicians all working to- computers, books and journals and one for e.g. few months ago I attended a public gether to fight against conjectures and to con- needs funding for travel. But, this still costs talk by Villani meant for students of high vert conjectures into theorems. That should less than what the expenses are for pursuing school. When I was the President of the be done together and not against one an- experimental physics or biology or chemistry. French Mathematical Society we organised a other. Whenever I hear the question of prior- It is true that if funds for mathematics are in- meeting for the public, titled “The hidden face ity, I think it is wrong. People should not insist creased by a little, this would have tremen- of mathematics”. Just like the hidden face of that they had the idea first as it is bad for the dous consequences. To convince the the moon, there is a part of mathematics that development of mathematics. On the other government is something which is usually dif- you don’t see. hand, I am also a runner and I like competi-

Use of Metric In Evaluating Research Mathaloon

anellists: László Lovász (Chair), Douglas Arnold, José-Antonio de Pla Peña, Malcolm MacCallum,Frank Pacard.

The use of metrics for evaluating research is a hotly debated issue. The IMU/ICIAM/IMS report on Citation Statistics1 highlighted the dan- gers of uncritical use of impact factors, which play an increasing role in funding, promotions and library purchases. Are impact factors and other such indices good measures of journal quality, and should they be used to evaluate research andindividuals? What can be done about unethical practices like impact factor manipulation? Is there a role for -Arghya Mondal metrics in evaluating research? Are there better alternatives? -Sayan Das

The Citation Statistics report can be found at +,,-.,/!*!,% !"#$!%%!&'(!)* http://www.mathnion.org/fileadmin/IMU/Report/CitationStatistics.pdf Ustad Rashid Khan (Vocal) R.Ramachandran Hindustani Classical Music B.Sury Panel Discussion Geethanjali Monto Communicating mathematics to society Concert at 19:00, Hall 2 Richa Malhotra Midhun Raj U.R at large Simon Singh Mohammed Anvar T Rahul V Pisharody Chair: G. M. Ziegler, Technische Universität, Berlin, Germany Popular lecture, 17:00 - 18:00 Sidharth Varma Hall 4 Nikhil MG 15:00 - 17:00, Hall 2