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Thesis and Guide details:

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Details of CSIR Fellowship/ Associateship held, if any or from other sources/ agencies.

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Significant foreign assignments:

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(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.

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(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.

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Places where work of last 5 years has been referred/ cited in Books, Reviews:

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Names of the industries in which the technology (ies) has (have) been used :

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The achievements already been recognised by Awards by any learned body:

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The Awardee a fellow of the Indian National Science Academy/Indian Academy of Sciences/National
Academy of Sciences/Others:

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The Awardee delivered invited lecture(s) in India/abroad and/or chaired any scientific
Internatiional Conference Symposium:

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List of Awardee's 10 most significant publications.

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List of Awardee's 5 most significant publications during the last 5 years

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List of Awardee's 5 most significant publications from out of work done in India
during the last five years:

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Complete list of publications in standard refereed journals:

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Complete list of publications with foreign collaborators (indicating your status
as author):

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List of papers published in Conferences /Symposia/ Seminars, etc:

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List of the most outstanding Technical Reports/ Review Articles:

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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.

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Total number of patents granted in last five years.

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Details of Books published: