• Home
  • Brief Profile of the Awardee

Brief Profile of the Awardee

awardee

Dr Ritabrata Munshi

  • 2015
  • Mathematical Sciences
  • 14/09/1976
  • Number Theory
Award Citation:

Dr Munshi has developed circle method to prove subconvexity bound for central values of twists of a degree three L-function and made significant contributions in studying rational points of varieties.

Academic Qualifications:
NA
Thesis and Guide details:
NA
Details of CSIR Fellowship/ Associateship held, if any or from other sources/ agencies.
NA
Significant foreign assignments:
NA
(a) Significant contributions to science and/ or technology development by the nominee based on the work done in India during most part of last 5 years:
Ritabrata has done ground-breaking work in (i) the analytic theory of L-functions and (ii) the theory of rational points on varieties. In analytic theory of L-functions his main focus has been the subconvexity problem. This problem has been the centre of intensive research since 1990’s, due to its relevance in various questions of arithmetic importance (like QUE) and because of its connection with the Riemann hypothesis. Various subconvex bounds have been established in the cases of degree one and degree two L-functions, wherein the “amplified moment method” of Iwaniec has been instrumental. But so far, this approach has been ineffective in the case of degree three L-functions. In his initial years at TIFR, using the moment method, Ritabrata proved two significant subconvexity results for twists of symmetric-square L-functions (“Bounds for twisted symmetric square L-functions” Crelle’s Journal 682 (2013) and “Bounds for twisted symmetric square L-functions III” Advances of Math 235 (2013)). These were the first instances of subconvexity for degree three L-functions which are not self-dual. Certain self-dual degree three L-functions were previously tackled in works of Li (Annals of Math. 173(2011)) and Blomer (American J. Math. 134 (2012)). The moment method, however, could not be adopted to solve the general case of twists of degree three L-functions. Ritabrata abandoned the idea of computing moment, and started to develop a new approach for subconvexity. Ritabrata has written four papers titled “The circle method and bounds for L-functions I-IV” elaborating his idea. In the first paper (Math. Annalen 358 (2014)), he proved hybrid subconvex bounds in the case of degree two L-functions. This was a known case of subconvexity, but Ritabrata produces a much stronger result. In his second paper (American Journal of Math. 137 (2015)) he proved subconvex bounds for special twists of generic degree three L-functions. This is the first instance of subconvexity for any generic degree three L-function. The third and fourth papers in this series are major breakthroughs in the subject. In the third paper (submitted to journal) the taspect subconvexity problem is solved for general degree three L-functions. In the fourth paper (Annals of Math. 182 (2015)), Ritabrata solves the twist aspect subconvexity for degree three L-functions. In a related work Ritabrata proved non-trivial bounds for shifted convolutions sums for GL(3)xGL(2) (Duke Math Journal 162 (2013)). The case of GL(2)xGL(2) is well-known, whereas the case of GL(3)xGL(3) has proved to be well out of reach. The mixed case of GL(3)xGL(2) has been studied previously by Pitt (Duke Math Journal 77 (1995)) in the special case of divisor function. Using his new approach Ritabrata solves the case of GL(3) cusp forms and establishes a much stronger bound. In “Pairs of quadrics in 11 variables” (Compositio Math. (2015)) , Ritabrata employed his “level lowering tricks” to study rational points on smooth intersections of two quadrics. The result of Birch from 1962 remains the benchmark in this subject. Any improvement over Birch, for even a subclass of varieties is considered to be outstanding. Heath-Brown (Invent. Math. 170 (2007)) has improved Birch’s result for cubics. Ritabrata has gone beyond Birch for intersections of two quadrics.
(b) Impact of the contributions in the field concerned:
Ritabrata's new delta method (circle method), together with his systematic usage of “level lowering tricks”, has opened new doors in the subject of subconvexity for higher degree L-functions, and has reinvigorated research in this important field. The circle method has gone through several modifications since its inception in the work of Hardy and Ramanujan in 1910’s. All of them had one thing in common – they only involved representations of GL(1). Ritabrata’s new derivation, based on the Petersson trace formula, involves representations of both GL(1) and GL(2). This is more involved than the usual circle method but is more flexible. The GL(1) part (the off-diagonal) of the formula is comparable with the usual formula. However in Ritabrata’s formula there is more freedom with choice of the parameters. Moreover he has employed his formula to solve a very important open problem. One of the referees of the Annals notes - “This paper is one of the best piece of analytic number theory I have read in recent years (and there have been a number of recent breakthrough some published or to appear in the annals) and it is certainly the most technically demanding one. Beyond the result itself the method of proof although very involved open new avenues to the analytic theory of automorphic forms of higher rank and their associated L-functions...” In the last two years the only papers in analytic number theory that appeared in Annals are Zhang’s ground-breaking “Bounded gaps between primes” 179 (2014), where he made fantastic progress towards the thousand year old twin prime conjecture, and the follow-up paper of Maynard titled “Small gaps between primes” 181 (2015). The referee of Annals chooses to compare Ritabrata’s work with the above work of Zhang and even puts his work ahead from the perspective of technicality. This clearly indicates the impact of the work. In the subject of rational points on varieties, Ritabrata’s work on intersection of quadrics has given new hope of bypassing the half-a-century old result of Birch. A referee from Compositio writes - “This paper improves a long-standing result of Birch (1962) on asymptotics for representations of zero by a pair of quadratic forms…” and then goes on to add “…within the circle method community a reduction of variables on this scale is commendable, but what particularly recommends this paper to a journal of this caliber is the innovation of its method…” Ritabrata’s new forms of the circle method have now become a topic of further research. Several doctoral students of Holowinsky, Michel and others have started investigating the possibilities with these new ideas. Ritabrata himself is working on this and he believes that his ideas hold the crucial key for unravelling the mysteries of the higher degree L-functions.
Places where work of last 5 years has been referred/ cited in Books, Reviews:
(i). Paper Cited
NA
(ii). Book Cited
NA
Names of the industries in which the technology (ies) has (have) been used :
NA
The achievements already been recognised by Awards by any learned body:
NA
The Awardee a fellow of the Indian National Science Academy/Indian Academy of Sciences/National Academy of Sciences/Others:
The Awardee delivered invited lecture(s) in India/abroad and/or chaired any scientific Internatiional Conference Symposium:
NA
List of Awardee's 10 most significant publications.
NA
List of Awardee's 5 most significant publications during the last 5 years
NA
List of Awardee's 5 most significant publications from out of work done in India during the last five years:
NA
Complete list of publications in standard refereed journals:
NA
Complete list of publications with foreign collaborators (indicating your status as author):
NA
List of papers published in Conferences /Symposia/ Seminars, etc:
NA
List of the most outstanding Technical Reports/ Review Articles:
NA
Statement regarding collaboration with scientists abroad:
Ritabrata has co-authored papers with Henryk Iwaniec during his stay at Rutgers University (2006-2009). Currently he has three foreign collaborators – Roman Holowinsky, Zhi Qi (student of R. Holowinsky) and Tim Browning. But most of the breakthrough papers (see the tables 16.a and 16.b) of Ritabrata are single authored and are based on his own ideas. All of them are written by him during his stay at TIFR, Mumbai.
List of Patents taken
NA
Total number of patents granted in last five years.
Details of Books published:
NA

Contact Details


  • School of Mathematics
    Tata Institute of Fundamental Research
    1 Homi Bhabha Road, Colaba
    Mumbai - 400005
    Maharashtra INDIA
  • rmunshi[at]math[dot]tifr[dot]res[dot]in
19 Nov 2018, http://ssbprize.gov.in/Content/Detail.aspx?AID=511