ISI Kolkata
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en Mathematician Dr Neena Gupta shines as the youngest Shanti Swarup Bhatnagar awardee
https://researchmatters.in/news/mathematician-dr-neena-gupta-shines-youngest-shanti-swarup-bhatnagar-awardee
<div class="field field-name-field-op-author field-type-node-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/people/arunita-banerjee">Arunita Banerjee</a></div></div></div><span class="read-time">Read time: 3 mins<br /></span><span class="submitted-by"></span><div class="field field-name-field-graphic field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img typeof="foaf:Image" src="https://researchmatters.in/sites/default/files/styles/large_800w_scale/public/dr_neena_gupta.jpg?itok=PBQSInsY" width="800" height="469" alt="Mathematician Dr Neena Gupta shines as the youngest Shanti Swarup Bhatnagar awardee" title="Dr Neena Gupta [Photo Credit: Arunita Banerjee]" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><p style="text-align: justify;">Dr Neena Gupta, Associate Professor at the Theoretical Statistics and Mathematics Unit of the Indian Statistical Institute (ISI), Kolkata, has been awarded the Shanti Swarup Bhatnagar Prize 2019, in the field of Mathematical Sciences. This prize, one of the most prestigious awards in the country for research in Science, honours scientists for significant and cumulative contribution to their area of research. Dr Gupta, the youngest person in Mathematical Sciences to receive this award till date, has been recognised for her contributions to affine algebraic geometry, especially in proposing a solution to the Zariski Cancellation Problem.</p>
<p style="text-align: justify;">Elated about winning the award, Dr. Gupta attributes her success to her strong conceptual foundation and the guidance of her teachers.</p>
<blockquote><p style="text-align: justify;">"Behind a successful person, there are many people, not just one. You need support from the whole system. My parents were very keen on getting me higher education. My PhD supervisor, Prof Amartya Kumar Dutta, has been very encouraging. Also, I am fortunate to have a very supportive husband and in-laws," she shares in an interview with <em>Research Matters</em>.</p>
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<p style="text-align: justify;">Dr. Gupta's field of research is Commutative Algebra and Affine Algebraic Geometry. Commutative algebra, apart from being a beautiful subject, provides a base over which a vast body of pure mathematics develops, Algebraic Geometry being one of the primary ones.</p>
<p style="text-align: justify;">A quick recap of high school geometry reminds us of polynomial equations which govern geometric shapes, like x2 + y2 = r2 for a circle or x2/a2 – y2/b2 = 1 for a hyperbola. But these shapes start getting complicated when the number of variables and the number and the degrees of the equations involved increase. Affine Algebraic Geometry, the research area of Dr. Gupta, deals with the understanding of the properties of geometric objects that arise as solutions of systems of polynomial equations. Her natural strength being in Algebra, Dr. Gupta approaches these problems using algebraic methods.</p>
<p style="text-align: justify;">In the last few years, Dr Gupta has provided solutions to two open problems, one of which was posed by Oscar Zariski (1899-1986), one of the founders of modern Algebraic Geometry. She describes these open mathematical conjectures as problems which can be easily explained to mathematicians but are very difficult to solve. The 'Zariski Cancellation Problem' has intrigued mathematicians around the globe, since a version of it was proposed by O. Zariski in 1949.</p>
<blockquote><p style="text-align: justify;">“The cancellation problem asks that if you have cylinders over two geometric structures, and they have similar forms, can one conclude that the original base structures have similar forms?" explains Dr Gupta.</p>
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<p style="text-align: justify;">Another problem solved by Dr. Gupta was posed by Masayoshi Miyanishi, who is now revered as a father figure among the present affine algebraic geometers.</p>
<p style="text-align: justify;">During the later half of the 20th century and early 21st century, eminent mathematicians have tried to work out a solution for the Zariski Cancellation Problem. This particular problem had remained open for about 70 years, before Dr. Gupta finally provided a complete <a href="https://www.insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/Vol46_2015_6_ART08.pdf" target="_blank">solution</a> to it in positive characteristic, in 2014.</p>
<blockquote><p style="text-align: justify;">"I knew this problem right from my PhD days, but I never imagined that I will be able to solve it," she says, pleasantly surprised. The solutions provided by Dr Gupta have given both insights and inspiration to young researchers as they can initiate research into other associated conjectures, which remain open.</p>
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<p style="text-align: justify;">Till 2019, there have been 547 <a href="http://ssbprize.gov.in/Content/AwardeeList.aspx" target="_blank">Bhatnagar awardees</a>, out of which, only 17 are women. "There needs to be social awareness so that people start sending their girl child for higher education," she says in response to the question about gender disparity in the scientific community.</p>
<p style="text-align: justify;">The thrill of solving problems, based on mathematical theories, is her greatest motivation. Ground-breaking results do not come overnight and are a result of patience, perseverance and continuity of efforts put into scientific research.</p>
<blockquote><p style="text-align: justify;">"Maths is for somebody who can solve the problems on their own. The pleasure which I get in solving problems in mathematics is much more than any award," signs off Dr Gupta.</p>
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</div></div></div><div class="field field-name-field-tags field-type-taxonomy-term-reference field-label-inline clearfix"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/tags/dr-neena-gupta" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Dr Neena Gupta</a></div><div class="field-item odd"><a href="/tags/isi-kolkata" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">ISI Kolkata</a></div><div class="field-item even"><a href="/tags/affine-algebraic-geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Affine Algebraic Geometry</a></div><div class="field-item odd"><a href="/tags/zariski-cancellation-problem" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Zariski Cancellation Problem</a></div></div></div><span property="dc:title" content=" Mathematician Dr Neena Gupta shines as the youngest Shanti Swarup Bhatnagar awardee" class="rdf-meta element-hidden"></span><ul class="links inline"><li class="statistics_counter first last"><span>40435 reads</span></li>
</ul>Mon, 09 Dec 2019 06:01:00 +0000Research Matters1911 at https://researchmatters.inUnderstanding the Riemann Hypothesis—the most crucial unsolved problem in mathematics
https://researchmatters.in/news/understanding-riemann-hypothesis%E2%80%94-most-crucial-unsolved-problem-mathematics
<span class="read-time">Read time: 4 mins<br /></span><span class="submitted-by"></span><div class="field field-name-field-graphic field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img typeof="foaf:Image" src="https://researchmatters.in/sites/default/files/styles/large_800w_scale/public/math.jpg?itok=lMnPQMQz" width="800" height="450" alt="Understanding the Riemann Hypothesis—the most crucial unsolved problem in mathematics" title="This image is not a true representation of Reimann Hypothesis" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><p style="text-align: justify;">In the last one hundred and sixty years, in spite of hundreds of claims, some of them from first-class mathematicians, the Riemann Hypothesis, or the holy grail of mathematics, remains as elusive as ever. The conjecture, which originated from the work of Bernhard Riemann on the distribution of prime numbers, has now been extended and generalised into a monstrous beast. Its cunning and long arms now encompass almost all areas of mathematics, far beyond its site of origin. From a technical point of view, the Riemann Hypothesis is a prediction about the solutions of an equation involving 'L-functions', which, at best, can be described as esoteric and abstruse.</p>
<p style="text-align: justify;">Why then are we interested in this conjecture? Well, mathematicians look for symmetries which are both meaningful and beautiful. They try to understand nature by unveiling symmetries and patterns of the underlying design, which though are invisible to our naked eyes (senses), nevertheless are visible to the eyes of mathematics. This effort has been the leading force in humankind's search for the unification of the laws of physics. For example, our understanding that the magnetic field and the electric field are related stems from this.</p>
<p style="text-align: justify;">Why then can't we find a pattern in the distribution of prime numbers - the basic building blocks of our counting system? Apparently, primes do not follow any rules. They stick up wherever they want in the infinite number line. Riemann showed that the primes and the zeros of the zeta function—a special L-function—are related, and though primes fail to show any respect for rules and discipline, the zeros exhibit a pattern; they all line up on 'the critical line'. But, failing to come up with a proof of this 'fact', Riemann wrote it down as a plausible hypothesis in his famous 1859 memoir.</p>
<p style="text-align: justify;">The twentieth century witnessed a tremendous flux in mathematical ideas, which blended and mixed branches of mathematics as never before. Representation theory and geometry became the two central pillars on which mathematicians tried to lay the grand edifice of number theory. L-functions, arising from geometry and representation, played a crucial role to bridge distant concepts.</p>
<p style="text-align: justify;">Halfway into the century, a geometric rendition of the Riemann hypothesis was formulated. This progress immediately occupied the centre stage of mathematics and became one of the driving forces in the development of modern algebraic geometry. Some twenty-five years later, when this beast was finally tamed, half of the mathematics had been revolutionised in the endeavour. The range of mathematical applications of this result can hardly be undermined. Among other things, this settled a long-standing conjecture of Ramanujan regarding the growth of sequences of numbers arising from certain highly symmetric functions, called modular forms. Ramanujan conjecture plays a central role in the construction of optimal graphs, which are proposed to be the basis of post-quantum elliptic curve cryptosystems. In other words, these will keep us secured when the machines become too intelligent!</p>
<p style="text-align: center;"><img alt="" src="/sites/default/files/1000px-vignetteriemannhypothesis.svg.png" style="width: 25%; height: auto;" /></p>
<p style="text-align: center;"><strong>A representation of an almost correct statement of the Riemann hypothesis</strong></p>
<p style="text-align: center;"><em><a href="https://commons.wikimedia.org/wiki/Category:Riemann_hypothesis#/media/File:VignetteRiemannHypothesis.svg" target="_blank">Image credits</a>: AstroOgier, via Wikimedia commons, under license CC BY-SA 4.0</em></p>
<p style="text-align: justify;">The original Riemann hypothesis, however, is a far cry. To make any headway in this problem, we need to analyse the behaviour of these L-functions inside a region called the 'critical strip'. Curiously, our understanding of the objects outside this region is quite clear, but once we cross the 'wall' and get inside, we are as good as blind. Even our present-day highly sophisticated algebraic-geometric tools fail to make any significant dent in the wall to shed any light. Our only successful approach, which only scratches some information from the wall, is more than a hundred years old.</p>
<p style="text-align: justify;">Perhaps a new approach can be extracted from the analytic theory of those mysterious highly symmetric functions called automorphic forms on higher rank groups - the grand generalisation of modular forms. After all, these functions are the very incarnation of symmetry which the Riemann hypothesis is all about. A study in this direction can become an exciting field of research.</p>
<p style="text-align: justify;"><strong><em>This article is the first of a three-part series from leading Indian researchers on the unsolved challenges in the field of mathematics and science as they find solutions to them. This article is attributed to Prof. Ritabrata Munshi, Professor, School of Mathematics, Tata Institute of Fundamental Research, Mumbai and Statistics and Mathematics Unit, Indian Statistical Institute, Kolkata. Prof. Munshi won Infosys Prize 2017 in Mathematics by Infosys Science Foundation. </em><em>The article series has been facilitated by Infosys Science Foundation.</em></strong></p>
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<p style="text-align: justify;"><em>This article has been minorly edited to correspond to the editorial guidelines of Research Matters. </em></p>
</div></div></div><div class="field field-name-field-tags field-type-taxonomy-term-reference field-label-inline clearfix"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/tags/reimann-hypothesis" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Reimann Hypothesis</a></div><div class="field-item odd"><a href="/tags/infosys-prize" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Infosys Prize</a></div><div class="field-item even"><a href="/tags/ritabrata-munshi" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Ritabrata Munshi</a></div><div class="field-item odd"><a href="/tags/tifr" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">TIFR</a></div><div class="field-item even"><a href="/tags/isi-kolkata" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">ISI Kolkata</a></div><div class="field-item odd"><a href="/tags/infosys-science-foundation" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Infosys Science Foundation</a></div></div></div><span property="dc:title" content="Understanding the Riemann Hypothesis—the most crucial unsolved problem in mathematics" class="rdf-meta element-hidden"></span><ul class="links inline"><li class="statistics_counter first last"><span>8235 reads</span></li>
</ul>Fri, 01 Nov 2019 06:22:20 +0000Research Matters1851 at https://researchmatters.in