Day 7, August 25, 2010
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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.