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Jennifer Shyamala Sayaka Balakrishnan is an American
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
known for leading a team that solved the problem of the "cursed curve", a
Diophantine equation ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real n ...
that was known for being "famously difficult". More generally, Balakrishnan specializes in
algorithmic number theory In mathematics and computer science, computational number theory, also known as algorithmic number theory, is the study of computational methods for investigating and solving problems in number theory and arithmetic geometry, including algorithms ...
and
arithmetic geometry In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties. ...
. She is a
Clare Boothe Luce Clare Boothe Luce (; March 10, 1903 – October 9, 1987) was an American writer, politician, diplomat, and public conservative figure. A versatile author, she is best known for her 1936 hit play '' The Women'', which had an all-female cast. He ...
Professor at
Boston University Boston University (BU) is a Private university, private research university in Boston, Massachusetts, United States. BU was founded in 1839 by a group of Boston Methodism, Methodists with its original campus in Newbury (town), Vermont, Newbur ...
.


Education and career

Balakrishnan was born in
Mangilao, Guam Mangilao is a Villages of Guam, village on the eastern shore of the United States territory of Guam. The village's population has decreased slightly since the island's 2010 census. Cliffs lie along much of the village's shoreline provide dramatic ...
to Narayana and Shizuko Balakrishnan; her father is a professor of chemistry at the
University of Guam University of Guam () (U.O.G.) is a public university, public land-grant university in Mangilao, Guam. It is accredited by the Western Association of Schools and Colleges and offers thirty-four degree programs at the undergraduate level and ele ...
. As a junior at Harvest Christian Academy, Balakrishnan won an honorable mention in the 2001
Karl Menger Karl Menger (; January 13, 1902 – October 5, 1985) was an Austrian-born American mathematician, the son of the economist Carl Menger. In mathematics, Menger studied the theory of algebra over a field, algebras and the dimension theory of low-r ...
Memorial Award competition, for the best mathematical project in the
Intel International Science and Engineering Fair The Regeneron International Science and Engineering Fair (ISEF) is an annual science fair in the United States. It is owned and administered by the Society for Science, a 501(c)(3) non-profit organization based in Washington, D.C. Each May, more ...
. Her project concerned
elliptic coordinate system In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci F_ and F_ are generally taken to be fixed at -a and +a, respectively ...
s. In the following year, she won the National High School Student Calculus Competition, given as part of the
United States of America Mathematical Olympiad The United States of America Mathematical Olympiad (USAMO) is a highly selective high school mathematics competition held annually in the United States. Since its debut in 1972, it has served as the final round of the American Mathematics Compe ...
. Balakrishnan graduated from
Harvard University Harvard University is a Private university, private Ivy League research university in Cambridge, Massachusetts, United States. Founded in 1636 and named for its first benefactor, the History of the Puritans in North America, Puritan clergyma ...
in 2006, with both a ''magna cum laude'' bachelor's degree and a master's degree in mathematics. She moved to the
Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a Private university, private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of moder ...
for her doctoral studies, completing her Ph.D. in 2011. Her dissertation, ''Coleman integration for hyperelliptic curves: algorithms and applications'', was supervised by
Kiran Kedlaya Kiran Sridhara Kedlaya ( ; born July 1974) is an American mathematician. He currently is a Professor of Mathematics and the Stefan E. Warschawski Chair in Mathematics at the University of California, San Diego. Biography Kiran Kedlaya was born i ...
. She returned to Harvard for her postdoctoral studies from 2011 to 2013, and then moved to the
University of Oxford The University of Oxford is a collegiate university, collegiate research university in Oxford, England. There is evidence of teaching as early as 1096, making it the oldest university in the English-speaking world and the List of oldest un ...
from 2013 to 2016, where she was a Junior Research Fellow in
Balliol College Balliol College () is a constituent college of the University of Oxford. Founded in 1263 by nobleman John I de Balliol, it has a claim to be the oldest college in Oxford and the English-speaking world. With a governing body of a master and ar ...
and a Titchmarsh Research Fellow in the Mathematical Institute. She became Clare Booth Luce Assistant Professor at Boston University in 2016, Clare Booth Luce Associate Professor in 2021, and Clare Booth Luce Professor in 2023. Balakrishnan is one of the principal investigators in the Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation, a large multi-university collaboration involving Boston University, Harvard, MIT,
Brown University Brown University is a Private university, private Ivy League research university in Providence, Rhode Island, United States. It is the List of colonial colleges, seventh-oldest institution of higher education in the US, founded in 1764 as the ' ...
, and
Dartmouth College Dartmouth College ( ) is a Private university, private Ivy League research university in Hanover, New Hampshire, United States. Established in 1769 by Eleazar Wheelock, Dartmouth is one of the nine colonial colleges chartered before the America ...
, with additional collaborators from other universities in the US, England, Australia, the Netherlands, and Canada.. She also serves on the board of directors of the Number Theory Foundation and the editorial boards of
Research in Number Theory ''Research in Number Theory'' is a peer-reviewed mathematics journal covering number theory and arithmetic geometry. The editors-in-chief are Jennifer Balakrishnan (Boston University), Florian Luca (University of Witwatersrand), Ken Ono (University ...
and
Mathematics of Computation ''Mathematics of Computation'' is a bimonthly mathematics journal focused on computational mathematics. It was established in 1943 as ''Mathematical Tables and Other Aids to Computation'', obtaining its current name in 1960. Articles older than f ...
. She serves on the Scientific Advisory Board for the
Institute for Computational and Experimental Research in Mathematics The Institute for Computational and Experimental Research in Mathematics (ICERM), founded in 2011, is an American research institute in mathematics at Brown University, funded since 2010 by a grant from the National Science Foundation. About A ...
(ICERM).


Contributions

In 2017, Balakrishnan led a team of mathematicians in settling the problem of the "cursed curve" X_s(13). This curve is modeled by the equation :y^4+5x^4-6x^2y^2+6x^3z+26x^2yz+10xy^2z-10y^3z-32x^2z^2-40xyz^2+24y^2z^2+32xz^3-16yz^3=0 and, as a
Diophantine equation ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real n ...
, the problem is to determine all rational solutions, i.e., assignments of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s to the variables x, y, and z for which the equation is true. Although as an explicit equation this curve has a complicated form, it is natural and conceptually significant in the number theory of
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...
s. The equation describes a
modular curve In number theory and algebraic geometry, a modular curve ''Y''(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular g ...
whose solutions characterize the one remaining unsolved case of a theorem of on the
Galois representation In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring ...
s of elliptic curves without
complex multiplication In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible wh ...
. Computations by and had previously identified seven solutions on the cursed curve (six corresponding to elliptic curves with complex multiplication, and one
cusp A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth. Cusp or CUSP may also refer to: Mathematics * Cusp (singularity), a singular point of a curve * Cusp catastrophe, a branch of bifu ...
), but their computational methods were unable to show that the list of solutions was complete. Following a suggestion of Oxford mathematician
Minhyong Kim Minhyong Kim () is a South Korean mathematician who specialises in arithmetic geometry and anabelian geometry. Biography Kim received his PhD at Yale University in 1990 under the supervision of Serge Lang and Barry Mazur, going on to work in a ...
, Balakrishnan and her co-authors constructed a "Selmer variety" associated with the curve, such that the rational points of the curve all lie on the Selmer variety as well, and such that the number of points of intersection of the curve and the variety can be computed. Using this method, they proved that the seven known solutions to the cursed curve are the only ones possible. This work was initially reported in a 2017
arXiv arXiv (pronounced as "archive"—the X represents the Chi (letter), Greek letter chi ⟨χ⟩) is an open-access repository of electronic preprints and postprints (known as e-prints) approved for posting after moderation, but not Scholarly pee ...
preprint and was published in the journal ''
Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...
'' in 2019. Balakrishnan has researched, with
Ken Ono Ken Ono (born March 20, 1968) is an American mathematician with fields of study in number theory. He is the STEM Advisor to the Provost and the Marvin Rosenblum Professor of Mathematics at the University of Virginia. Early life and education Ono ...
and others, Lehmer's question on whether the
Ramanujan tau function The Ramanujan tau function, studied by , is the function \tau : \mathbb\to\mathbb defined by the following identity: :\sum_\tau(n)q^n=q\prod_\left(1-q^n\right)^ = q\phi(q)^ = \eta(z)^=\Delta(z), where q=\exp(2\pi iz) with \mathrm(z)>0, \phi is t ...
\tau(n) is ever zero for a positive integer n. As well as for her work in number theory, Balakrishnan is known for her work implementing number-theoretical algorithms as part of the
SageMath SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation") is a computer algebra system (CAS) with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, group theory, differentia ...
computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The de ...
.


Recognition

Balakrishnan received the Clare Boothe Luce Assistant Professorship in 2016. In 2018, Balakrishnan was selected as a Sloan Research Fellow. In 2020, she was selected for a
National Science Foundation The U.S. National Science Foundation (NSF) is an Independent agencies of the United States government#Examples of independent agencies, independent agency of the Federal government of the United States, United States federal government that su ...
CAREER Award The National Science Foundation CAREER award is the most prestigious award presented by the National Science Foundation (NSF) of the United States Federal Government to support junior faculty who exemplify the role of teacher-scholars through rese ...
. She was named a Fellow of the
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
, in the 2022 class of fellows, "for contributions to arithmetic geometry and computational number theory and service to the profession". She earned the 2022
AWM–Microsoft Research Prize in Algebra and Number Theory The AWM–Microsoft Research Prize in Algebra and Number Theory and is a prize given every other year by the Association for Women in Mathematics to an outstanding young female researcher in algebra or number theory. It was funded in 2012 by Micros ...
in recognition of her "outstanding contributions to explicit methods in number theory, particularly her advances in computing rational points on algebraic curves over number fields". She was selected as a Fellow of the
Association for Women in Mathematics The Association for Women in Mathematics (AWM) is a professional society whose mission is to encourage women and girls to study and to have active careers in the mathematical sciences, and to promote equal opportunity for and the equal treatment o ...
in the class of 2023 "for her support of women in mathematics through mentoring and advising; for organizing and supporting programs for women and girls, especially Women in Sage and Women in Numbers; for her work in outreach and education, including GirlsGetMath; and for working to improve diversity, equity, and inclusion in research communities. In 2023 she was awarded the 2023-2024 AMS-Birman Fellowship.


Selected publications

* Balakrishnan, Jennifer S.; Bradshaw, Robert W.; Kedlaya, Kiran S (2010). "Explicit Coleman integration for hyperelliptic curves". ''Algorithmic number theory,'' 16–31, Lecture Notes in Comput. Sci., 6197, ''Springer, Berlin''
MR2721410
* Balakrishnan, Jennifer S.; Besser, Amnon (2012). "Computing local ''p''-adic height pairings on hyperelliptic curves". '' Int. Math. Res. Not.'', no. 11, 2405–2444
MR2926986
* Balakrishnan, Jennifer S.; Besser, Amnon; Müller, J. Steffen (2016). "Quadratic Chabauty: ''p''-adic heights and integral points on hyperelliptic curves". ''
J. Reine Angew. Math. ''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics''). History The journal was founded by Aug ...
'' 720, 51–79
MR3565969
* Balakrishnan, Jennifer S.; Dogra, Netan (2018). "Quadratic Chabauty and rational points I: ''p''-adic heights". With an appendix by J. Steffen Müller. '' Duke Math. J.'' 167, no. 11, 1981-2038
MR3843370
* Balakrishnan, Jennifer S.; Dogra, Netan; Müller, J. Steffen; Tuitman, Jan; Vonk, Jan (2019). "Explicit Chabauty-Kim for the split Cartan modular curve of level 13". '' Ann. of Math.'' 189, no. 3, 885-944
MR3961086


References


External links


Home page
{{DEFAULTSORT:Balakrishnan, Jennifer Year of birth missing (living people) Living people Guamanian academics 21st-century American mathematicians Harvard College alumni Massachusetts Institute of Technology School of Science alumni Boston University faculty Sloan Research Fellows 21st-century American women mathematicians 21st-century American non-fiction writers 21st-century American women writers Fellows of the American Mathematical Society American academics of Indian descent