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Microsoft Research Prize in Algebra and Number Theory of the AWM
The Association for Women in Mathematics established the Microsoft Research Prize in Algebra and Number Theory in 2012. The first presentation was made in 2014, and it has subsequently been made in even years to a woman early in her career who has published exceptional research in algebra. The area of algebra will be broadly interpreted to include number theory, cryptography, combinatorics and other applications, as well as more traditional areas of algebra. Candidates must be women working at a US institution either within ten years of being awarded a Ph.D., or not being tenured at the time of application. The website of the Association for Women in Mathematics states:-
Winners of the Microsoft Research Prize in Algebra and Number Theory:
2014 Sophie Morel, Princeton University.
Citation: The 2014 AWM-Microsoft Research Prize in Algebra and Number Theory is presented to Professor Sophie Morel, in recognition of her exceptional research in number theory. Professor Morel received her doctorate in 2005 from l'Université Paris-Sud. After appointments at the Institute for Advanced Studies, the Clay Mathematics Institute and Harvard University, she is currently a Professor of Mathematics at Princeton University. Professor Morel is a powerful arithmetic algebraic geometer who has made fundamental contributions to the Langlands program. Her research has been called "spectacularly original, and technically very demanding." Her research program has been favorably compared to that of several Fields medalists. She accomplished one of the main goals of the Langlands program by calculating the zeta functions of unitary and symplectic Shimura varieties in terms of the L-functions of the appropriate automorphic forms. To achieve this, she introduced an innovative t-structure on derived categories which had been missed by many experts. Her book 'On the cohomology of certain noncompact Shimura varieties' published in the Annals of Mathematics Studies Series is described as a tour-de-force. Professor Morel found another remarkable application of her results on weighted cohomology. She gave a new geometric interpretation and conceptual proof of Brenti's celebrated but mysterious combinatorial formula for Kazhdan-Lusztig polynomials, which are of central importance in representation theory. We would like to congratulate Professor Morel for her substantial achievements.
2016 Lauren Williams, University of California, Berkeley.
Citation: The 2016 AWM-Microsoft Research Prize in Algebra and Number Theory is presented to Professor Lauren Williams, in recognition of her exceptional research in algebraic combinatorics. Professor Williams received her doctorate in 2005 from the Massachusetts Institute of Technology. After appointments at MSRI, Berkeley, and Harvard, she is currently an Associate Professor of Mathematics at the University of California, Berkeley. Professor Williams is a powerful and broad combinatorialist, whose scientific reach extends into representation theory, algebraic geometry and physics. Her early work on the totally nonnegative Grassmannian is a beautiful and fundamental contribution to our understanding of the combinatorics - and later (with Rietsch), the topology - of this space which has important connections to Lusztig's work on canonical bases in representation theory. Professor Williams is also a leader in the exciting new subject of cluster algebras. She (with Musiker and Schiffler) proved an important special case of the famous Laurent positivity conjecture (now a theorem); their proof is a technical tour de force, which unlike some other approaches, yields a transparent combinatorial rule for the Laurent polynomials in question. Her paper with Ardila and Rincon, in which an old conjecture about realisability of positively oriented matroids is finally established, has been hailed by experts as the "climax of the study of positroids in the past decade." Most recently, her work with Kodama brings her expertise into the entirely new direction of soliton solutions of the KP equation and modelling shallow water waves. Beyond her outstanding scientific achievements, Professor Williams has assumed many leadership roles in the mathematical community and is a dedicated PhD and post-doctoral adviser. We congratulate Professor Williams for her well-deserved the Association for Women in Mathematics Microsoft Research Prize!
Response from Lauren Williams: I am deeply honoured to be receiving this award, and would like to thank the Association for Women in Mathematics and Microsoft for their generosity in establishing it, as well as my mentors and colleagues who nominated me for the award. I am profoundly grateful to have had numerous wonderful mentors, from childhood up until now, but I would like to mention in particular my thesis advisor Richard Stanley and my colleague Bernd Sturmfels, as well as Sara Billey and Sergey Fomin. Mathematics is rarely a solitary endeavour these days, and I am happy to acknowledge my many collaborators (now friends), including Sylvie Corteel, Yuji Kodama, Konstanze Rietsch, Federico Ardila and Felipe Rincon, and Gregg Musiker and Ralf Schiffler. Finally I would like to thank the math department and my colleagues at UC Berkeley, for providing me with such a supportive and welcoming mathematical "home." I don't think that anyone completely understands why women are still a minority among mathematicians. But ever since the Association for Women in Mathematics was established, this organization has played an important role in bringing together the community of women mathematicians, and reminding us all that there are many women mathematicians out there doing excellent work. The various activities, meetings, and lectures that the Association for Women in Mathematics has sponsored have provided a lot of inspiration and support to me personally, as I know they have done for countless others. Thanks again!
2018 Melanie Matchett Wood, Wisconsin-Madison.
Citation: The 2018 AWM Microsoft Research Prize in Algebra and Number Theory is presented to Professor Melanie Matchett Wood, in recognition of her exceptional research achievements in Number Theory and Algebraic Geometry.
Melanie Matchett Wood received her doctorate in 2009 from Princeton University. She is currently a Professor at the University of Wisconsin-Madison, after appointments at the American Institute of Mathematics, Stanford University, and Mathematical Science Research Institute.
Wood has made deep and influential contributions to number theory and algebraic geometry. She excels at drawing connections between different areas of mathematics. Her work is a truly remarkable synthesis of number theory, algebraic geometry, topology, and probability. In arithmetic statistics, Wood, with her coauthors, gave the first heuristic account of the variation of the Mordell-Weil rank in families of elliptic curves, which predicts in particular, contrary to widely held belief among the research community, that elliptic curves over the rationals have absolutely bounded rank. Her joint work with Vakil suggests that the limiting behaviour of many natural families of varieties should stabilise in a motivic sense. These results and conjectures have attracted considerable attention and spawned a substantial amount of follow-up research. More recently, she determined the behaviour of the sandpile group of a random graph, thus proving an important conjecture in tropical geometry.
Beyond her outstanding scientific achievements, Wood has assumed many leadership roles in directing undergraduate research and promoting participation of women and girls in mathematics. She coached the first United States team to participate in the China Girls Math Olympiad, an international competition with a proof-based format. She is considered one of the most visible role models for a whole generation of American young women in mathematics. AWM congratulates Melanie Matchett Wood for her well-deserved AWM Microsoft Research Prize.
Response from Melanie Matchett Wood: I am deeply honoured to receive this award. I would like to thank the AWM and Microsoft for their generosity in establishing this prize. I have been lucky to have many amazing mentors and roles models in mathematics, from a very early age. Moreover, the joy I get from working with my collaborators is a continual inspiration in my research. I would like to thank all my mentors and collaborators, and mention in particular Joseph Gallian, Manjul Bhargava, Ravi Vakil, Lillian Pierce, Jordan Ellenberg, and Nigel Boston. Thank you as well to my mentors and colleagues who nominated me for this award. I would like to especially thank the American Institute of Mathematics and the Packard Foundation for providing me flexible funding early in my career, which allowed me to take risks like looking far afield in mathematics for the answers to my questions in number theory. Finally, I would like to thank the University of Wisconsin-Madison for its flexibility in letting me have a faculty position suited for how I wanted to balance my career and family.
2020 Melody Chan, Brown University.
Citation: The 2020 AWM Microsoft Research Prize in Algebra and Number Theory is presented to Professor Melody Chan, in recognition of her advances at the interface between algebraic geometry and combinatorics.
Professor Chan received her doctorate in 2012 from University of California, Berkeley and held an NSF Postdoctoral Fellowship at Harvard University. She is currently an Assistant Professor at Brown University, a Sloan Research Fellow, and has recently won an NSF CAREER Award.
Chan is known for an exceptional combination of strength in both combinatorics and algebraic geometry, as well as her ability to fearlessly digest difficult techniques from other fields of mathematics. Chan has proved numerous conjectures across tropical geometry, graph theory, and algebraic geometry.
In Chan's recent work with Galatius and Payne, they showed that the cohomology of the moduli space of genus curves grows exponentially in a particular degree, an astounding result which contradicts conjectures of Kontsevich and Church-Farb-Putman that said this cohomology should vanish. This breakthrough comes from a deep study of moduli spaces of tropical curves.
Chan's foundational work on the moduli of metric graphs and tropical curves, both solo and with several co-authors, is central to the field, already having important applications, and is expected to continue to lead to further work far beyond the original papers. Chan's work with López Martín, Pflueger, and Teixidor i Bigas proves beautiful new results on the expected number of turns in a random Young tableau and then applies them to give explicit topological information on Brill-Noether varieties that seemed beyond reach before their work.
Beyond her outstanding scientific achievements, Chan has assumed leadership roles to promote the participation of women in research, co-organising Women@AGNES (Algebraic Geometry Northeastern Series) at Brown and Yale; serving as Faculty Advisor for the Horizons Seminar at Brown, featuring talks and workshops on topics including diversity, community, and career development for young mathematicians; and organising the peer Mentoring Network for women in math at Brown.
Researchers call Chan a "leader" and a "major force" and are impressed by both her insights and her technical prowess. AWM congratulates Melody Chan for her well-deserved AWM Microsoft Research Prize.
Response from Melody Chan: I am happy to receive the 2020 AWM Microsoft Prize in Algebra, and I thank the AWM and Microsoft for their generosity in recognising my work. I have learned so much from my close collaborators: Renzo Cavalieri, Soren Galatius, Sam Payne, Nathan Pflueger, Martin Ulirsch, and Jonathan Wise. I'm also grateful for the support and mentorship of Dan Abramovich, Matt Baker, Lucia Caporaso, Joe Harris, Diane Maclagan, and especially my PhD advisor Bernd Sturmfels; and many other mathematicians, including my many supportive colleagues at Brown.
Getting to do research in mathematics is a privilege. After all, basic research in math and science is a long game: we get to study fundamental questions that may have no applications right now but, in totality and over the course of history, make an outsize impact in ways we couldn't have predicted. And we get to have fun while doing it, too. But that whole calculus is predicated on having time, having breathing room, to think about the long game at all. Right now, I'm less and less sure that we have that room. We have a climate crisis on our hands, crises of civil rights and human rights, crises of democracy and disenfranchisement, entire crises of empathy. Our country is taking children from their parents. I've never been more concerned for my country and my community than I am now.
Being a math professor is, I think, still the best thing I personally know how to do. It's getting harder to carry out basic math research - like research on the combinatorics and geometry of moduli spaces, say - with any moral certainty. But there are many parts of the job that do have an immediate impact. We must train students well, and find ways to support and include more of them in the first place; we must be role models, while interrogating our role as scientists; we must make more noise altogether. Let's work together on prioritising these aspects of the profession, even while we push on our foundational research.
2022 Jennifer Balakrishnan, Boston University.
Citation: The 2022 AWM-Microsoft Research Prize in Algebra and Number Theory is presented to Jennifer Balakrishnan in recognition of outstanding contributions to explicit methods in number theory, particularly her advances in computing rational points on algebraic curves over number fields.
Professor Balakrishnan is internationally recognized as a leader in computational number theory. Her doctoral dissertation presents the first general technique for computing iterated p-adic Coleman integrals on hyperelliptic curves. In the course of her collaboration with Minhyong Kim at Oxford, Balakrishnan helped realize the substantial practical potential of Kim's non-abelian Chabauty method, and with her collaborators, turned it into a powerful tool for identifying integral and rational points on curves that are entirely beyond reach using the traditional Chabauty approach. In an impressive tour de force, Balakrishnan, Dogra, Müller, Tuitman and Vonk used the quadratic Chabauty method for computing the rational points on the split Cartan modular curve of level 13. Facetiously known as the "cursed curve" among number theorists because 13 is the only prime level that had stubbornly resisted all such prior attempts, this work represents a major breakthrough. It not only completes the proof of the split Cartan cases of Serre's uniformity conjecture for Galois images of elliptic curves, but also opens an avenue for tackling nonsplit Cartan modular curves at higher level.
Balakrishnan's research exhibits extraordinary depth as well as breadth. In joint work with Besser, Ciperiani, Dogra, Müller, Stein and others, she has worked extensively on computing p-adic height pairings for hyperelliptic curves. Applications of this research include the formulation, along with numerical evidence, of a p-adic analogue of the celebrated Birch and Swinnerton-Dyer conjecture, some new explicit examples in Iwasawa theory, and more. With Ho, Kaplan, Spicer, Stein and Weigandt, Balakrishnan has assembled the most extensive computational evidence to date on the distribution of ranks and Selmer groups of elliptic curves over the rational numbers, thereby providing the most convincing evidence thus far in support of the widely believed conjecture that the average rank of a rational elliptic curve is .
After receiving her doctorate from the Massachusetts Institute of Technology in 2011, Professor Balakrishnan held appointments as an NSF Postdoctoral Fellow at Harvard University as well as a Junior Research Fellow and a Titchmarsh Fellow at the University of Oxford. She is currently the Clare Boothe Luce Associate Professor of Mathematics at Boston University, a Sloan Research Fellow, and a recipient of an NSF CAREER award. Her research is also supported by the Simons Foundation, through the Simons Collaboration in Arithmetic Geometry, Number Theory, and Computation.
Balakrishnan has delivered an impressive array of invited and plenary lectures in locations across four continents. Beyond her outstanding scientific achievements, she has assumed leadership roles in service to her institution and the community, especially in bringing more women into math, devoting untold hours to mentoring and advocating for junior women in the profession, and striving to create supportive environments for them. In addition to her extensive record of student supervision at all levels, she has co-organised numerous research conferences, thematic programs and summer schools, including many Women in Sage gatherings. She serves on the editorial boards for five top quality journals, the AMS Short Course Subcommittee, the Scientific Advisory Board for the Institute for Computational and Experimental Research in Mathematics, the Board of Directors for the Number Theory Foundation, and the Steering Committee for the Women in Numbers Network.
Jennifer Balakrishnan's work is widely known and recognised across the globe within the number theory community and beyond. AWM congratulates her for her well-deserved AWM-Microsoft Research Prize.
Response from Jennifer Balakrishnan: I am honoured to receive the 2022 AWM-Microsoft Prize in Algebra and Number Theory. I would like to thank the AWM and Microsoft for this recognition of my work, which would not have been possible without the support of my mentors and collaborators.
Over the years, I have been very fortunate to have had the encouragement of several mentors: my PhD advisor, Kiran Kedlaya, as well as Dick Gross, William Stein, Barry Mazur, Minhyong Kim, and Henri Darmon. I am deeply indebted to my collaborators, including Amnon Besser, Mirela Ciperiani, Netan Dogra, Steffen Müller, Jan Tuitman, and Jan Vonk, who have been generously working with me for several years and have taught me so much. Boston University has provided a wonderful research environment, and I am very grateful for the support of my colleagues, including Margaret Beck, Tasso Kaper, David Rohrlich, and Glenn Stevens. The Women in Numbers network and the Simons Collaboration in Arithmetic Geometry, Number Theory, and Computation (with special thanks to Noam Elkies, Brendan Hassett, Bjorn Poonen, Andrew Sutherland, and John Voight) have also provided a warm sense of community and the inspiration to follow various research directions I would not have otherwise pursued.
2024 Yunqing Tang, University of California, Berkeley.
The AWM-Microsoft Research Prize in Algebra and Number Theory was presented to Professor Yunqing Tang, in recognition of her breakthrough work in arithmetic geometry, including results on the Grothendieck-Katz -curvature conjecture, a conjecture of Ogus on algebraicity of cycles, arithmetic intersection theory, and the unbounded denominators conjecture of Atkin and Swinnerton-Dyer. An observer wrote that Tang "has a knack for absorbing difficult ideas with lightning speed, making them her own, and then applying them in creative and unexpected ways."
Citation: The -curvature conjecture lies in the field of arithmetic geometry: it predicts that for a certain vector bundle associated to a variety over a number field, if an invariant called the -curvature vanishes for all but finitely many primes, then an associated "monodromy representation" has finite image. Tang has made progress toward this conjecture by proving for example that the conclusion holds if the -curvature vanishes for all primes, when the variety is the projective line minus three points. In the area of -adic Hodge theory, Tang has proved Ogus' conjecture (which predicts that cycles in de Rham cohomology which are invariant by almost all crystalline Frobenii are Hodge cycles) for a large class of abelian varieties.
With collaborators, Tang has developed a program in arithmetic intersection theory on Shimura varieties that can prove a phenomenon of interest occurs at infinitely many primes. This has had many interesting consequences. As a first example, Ananth Shankar and Tang have proved that an abelian surface with real multiplication over a number field is isogenous to a product of elliptic curves when reduced modulo infinitely many primes. As a second example, with Ananth Shankar, Arul Shankar, and S Tayou, Tang's work proves that a K3 surface over a number field with everywhere good reduction has the property that the Picard rank of the reduction jumps, at infinitely many places.
Recently, in joint work with F Calegari and V Dimitrov, Tang has presented a proof of the 50-year-old "unbounded denominators conjecture," originally posed by Atkin and Swinnerton-Dyer. This conjecture can be framed (roughly speaking) as the statement that a modular form for a finite index subgroup of , expanded as a Fourier series in , has integral coefficients if and only if it is a modular form for some congruence subgroup of .
Yunqing Tang is an assistant professor at University of California, Berkeley. She received a PhD from Harvard University in 2016, and she was awarded the AWM Dissertation Prize. Tang subsequently was a Member at the IAS, an Instructor at Princeton University, a junior researcher (Chargée de recherche) at CNRS/Université Paris-Sud, and an assistant professor at Princeton University. Her work is supported by the NSF, and Tang has recently been awarded a Sloan Research Fellowship and the SASTRA Ramanujan prize. A press release from the Ramanujan Prize committee wrote that Tang's "wide ranging contributions are bound to have impact in the decades ahead."
Response from Yunqing Tang: I am very honoured to receive the 2024 AWM-Microsoft Prize in Algebra and Number Theory. I would like to thank the AWM and Microsoft for their generosity in recognising my work. I have been very lucky to have several amazing mentors: my PhD advisor, Mark Kisin, as well as Peter Sarnak and Shouwu Zhang; they have been supportive over the years and shared with me numerous mathematics insights. I am deeply indebted to my collaborator Ananth Shankar, with whom I have been working since graduate school time; our numerous discussions have shaped part of my research program. I also would like to give a special thank you to my collaborators Wanlin Li and Vesselin Dimitrov for numerous zoom discussion and working sessions to keep me stay productive during the pandemic. I would like to thank all my collaborators: Frank Calegari, Victoria Cantoral Farfán, Elena Mantovan, Davesh Maulik, Rachel Pries, Arul Shankar, Sho Tanimoto, Salim Tayou, and Erik Visse; I am very grateful to have the opportunities to work with them and learn interesting math from them.
I would like to thank the math department and my colleagues, especially the algebraic geometry and number theory group, at UC Berkeley for a supportive working environment. Many of my works have been done during my stay at Princeton, CNRS, Université Paris-Saclay, IAS and Harvard and I am grateful for the excellent working environment at these places. Finally, I would like to thank AWM again for providing the community of women mathematicians and for recognising my work at an early stage through the dissertation prize.
The Association for Women in Mathematics Microsoft Research Prize serves to highlight to the community outstanding contributions by women in the field and to advance the careers of the prize recipients. The award is made possible by a generous contribution from Microsoft Research.We give below winners of the prize and also the citation and the reply from the winner.
Winners of the Microsoft Research Prize in Algebra and Number Theory:
2014 Sophie Morel, Princeton University.
Citation: The 2014 AWM-Microsoft Research Prize in Algebra and Number Theory is presented to Professor Sophie Morel, in recognition of her exceptional research in number theory. Professor Morel received her doctorate in 2005 from l'Université Paris-Sud. After appointments at the Institute for Advanced Studies, the Clay Mathematics Institute and Harvard University, she is currently a Professor of Mathematics at Princeton University. Professor Morel is a powerful arithmetic algebraic geometer who has made fundamental contributions to the Langlands program. Her research has been called "spectacularly original, and technically very demanding." Her research program has been favorably compared to that of several Fields medalists. She accomplished one of the main goals of the Langlands program by calculating the zeta functions of unitary and symplectic Shimura varieties in terms of the L-functions of the appropriate automorphic forms. To achieve this, she introduced an innovative t-structure on derived categories which had been missed by many experts. Her book 'On the cohomology of certain noncompact Shimura varieties' published in the Annals of Mathematics Studies Series is described as a tour-de-force. Professor Morel found another remarkable application of her results on weighted cohomology. She gave a new geometric interpretation and conceptual proof of Brenti's celebrated but mysterious combinatorial formula for Kazhdan-Lusztig polynomials, which are of central importance in representation theory. We would like to congratulate Professor Morel for her substantial achievements.
2016 Lauren Williams, University of California, Berkeley.
Citation: The 2016 AWM-Microsoft Research Prize in Algebra and Number Theory is presented to Professor Lauren Williams, in recognition of her exceptional research in algebraic combinatorics. Professor Williams received her doctorate in 2005 from the Massachusetts Institute of Technology. After appointments at MSRI, Berkeley, and Harvard, she is currently an Associate Professor of Mathematics at the University of California, Berkeley. Professor Williams is a powerful and broad combinatorialist, whose scientific reach extends into representation theory, algebraic geometry and physics. Her early work on the totally nonnegative Grassmannian is a beautiful and fundamental contribution to our understanding of the combinatorics - and later (with Rietsch), the topology - of this space which has important connections to Lusztig's work on canonical bases in representation theory. Professor Williams is also a leader in the exciting new subject of cluster algebras. She (with Musiker and Schiffler) proved an important special case of the famous Laurent positivity conjecture (now a theorem); their proof is a technical tour de force, which unlike some other approaches, yields a transparent combinatorial rule for the Laurent polynomials in question. Her paper with Ardila and Rincon, in which an old conjecture about realisability of positively oriented matroids is finally established, has been hailed by experts as the "climax of the study of positroids in the past decade." Most recently, her work with Kodama brings her expertise into the entirely new direction of soliton solutions of the KP equation and modelling shallow water waves. Beyond her outstanding scientific achievements, Professor Williams has assumed many leadership roles in the mathematical community and is a dedicated PhD and post-doctoral adviser. We congratulate Professor Williams for her well-deserved the Association for Women in Mathematics Microsoft Research Prize!
Response from Lauren Williams: I am deeply honoured to be receiving this award, and would like to thank the Association for Women in Mathematics and Microsoft for their generosity in establishing it, as well as my mentors and colleagues who nominated me for the award. I am profoundly grateful to have had numerous wonderful mentors, from childhood up until now, but I would like to mention in particular my thesis advisor Richard Stanley and my colleague Bernd Sturmfels, as well as Sara Billey and Sergey Fomin. Mathematics is rarely a solitary endeavour these days, and I am happy to acknowledge my many collaborators (now friends), including Sylvie Corteel, Yuji Kodama, Konstanze Rietsch, Federico Ardila and Felipe Rincon, and Gregg Musiker and Ralf Schiffler. Finally I would like to thank the math department and my colleagues at UC Berkeley, for providing me with such a supportive and welcoming mathematical "home." I don't think that anyone completely understands why women are still a minority among mathematicians. But ever since the Association for Women in Mathematics was established, this organization has played an important role in bringing together the community of women mathematicians, and reminding us all that there are many women mathematicians out there doing excellent work. The various activities, meetings, and lectures that the Association for Women in Mathematics has sponsored have provided a lot of inspiration and support to me personally, as I know they have done for countless others. Thanks again!
2018 Melanie Matchett Wood, Wisconsin-Madison.
Citation: The 2018 AWM Microsoft Research Prize in Algebra and Number Theory is presented to Professor Melanie Matchett Wood, in recognition of her exceptional research achievements in Number Theory and Algebraic Geometry.
Melanie Matchett Wood received her doctorate in 2009 from Princeton University. She is currently a Professor at the University of Wisconsin-Madison, after appointments at the American Institute of Mathematics, Stanford University, and Mathematical Science Research Institute.
Wood has made deep and influential contributions to number theory and algebraic geometry. She excels at drawing connections between different areas of mathematics. Her work is a truly remarkable synthesis of number theory, algebraic geometry, topology, and probability. In arithmetic statistics, Wood, with her coauthors, gave the first heuristic account of the variation of the Mordell-Weil rank in families of elliptic curves, which predicts in particular, contrary to widely held belief among the research community, that elliptic curves over the rationals have absolutely bounded rank. Her joint work with Vakil suggests that the limiting behaviour of many natural families of varieties should stabilise in a motivic sense. These results and conjectures have attracted considerable attention and spawned a substantial amount of follow-up research. More recently, she determined the behaviour of the sandpile group of a random graph, thus proving an important conjecture in tropical geometry.
Beyond her outstanding scientific achievements, Wood has assumed many leadership roles in directing undergraduate research and promoting participation of women and girls in mathematics. She coached the first United States team to participate in the China Girls Math Olympiad, an international competition with a proof-based format. She is considered one of the most visible role models for a whole generation of American young women in mathematics. AWM congratulates Melanie Matchett Wood for her well-deserved AWM Microsoft Research Prize.
Response from Melanie Matchett Wood: I am deeply honoured to receive this award. I would like to thank the AWM and Microsoft for their generosity in establishing this prize. I have been lucky to have many amazing mentors and roles models in mathematics, from a very early age. Moreover, the joy I get from working with my collaborators is a continual inspiration in my research. I would like to thank all my mentors and collaborators, and mention in particular Joseph Gallian, Manjul Bhargava, Ravi Vakil, Lillian Pierce, Jordan Ellenberg, and Nigel Boston. Thank you as well to my mentors and colleagues who nominated me for this award. I would like to especially thank the American Institute of Mathematics and the Packard Foundation for providing me flexible funding early in my career, which allowed me to take risks like looking far afield in mathematics for the answers to my questions in number theory. Finally, I would like to thank the University of Wisconsin-Madison for its flexibility in letting me have a faculty position suited for how I wanted to balance my career and family.
2020 Melody Chan, Brown University.
Citation: The 2020 AWM Microsoft Research Prize in Algebra and Number Theory is presented to Professor Melody Chan, in recognition of her advances at the interface between algebraic geometry and combinatorics.
Professor Chan received her doctorate in 2012 from University of California, Berkeley and held an NSF Postdoctoral Fellowship at Harvard University. She is currently an Assistant Professor at Brown University, a Sloan Research Fellow, and has recently won an NSF CAREER Award.
Chan is known for an exceptional combination of strength in both combinatorics and algebraic geometry, as well as her ability to fearlessly digest difficult techniques from other fields of mathematics. Chan has proved numerous conjectures across tropical geometry, graph theory, and algebraic geometry.
In Chan's recent work with Galatius and Payne, they showed that the cohomology of the moduli space of genus curves grows exponentially in a particular degree, an astounding result which contradicts conjectures of Kontsevich and Church-Farb-Putman that said this cohomology should vanish. This breakthrough comes from a deep study of moduli spaces of tropical curves.
Chan's foundational work on the moduli of metric graphs and tropical curves, both solo and with several co-authors, is central to the field, already having important applications, and is expected to continue to lead to further work far beyond the original papers. Chan's work with López Martín, Pflueger, and Teixidor i Bigas proves beautiful new results on the expected number of turns in a random Young tableau and then applies them to give explicit topological information on Brill-Noether varieties that seemed beyond reach before their work.
Beyond her outstanding scientific achievements, Chan has assumed leadership roles to promote the participation of women in research, co-organising Women@AGNES (Algebraic Geometry Northeastern Series) at Brown and Yale; serving as Faculty Advisor for the Horizons Seminar at Brown, featuring talks and workshops on topics including diversity, community, and career development for young mathematicians; and organising the peer Mentoring Network for women in math at Brown.
Researchers call Chan a "leader" and a "major force" and are impressed by both her insights and her technical prowess. AWM congratulates Melody Chan for her well-deserved AWM Microsoft Research Prize.
Response from Melody Chan: I am happy to receive the 2020 AWM Microsoft Prize in Algebra, and I thank the AWM and Microsoft for their generosity in recognising my work. I have learned so much from my close collaborators: Renzo Cavalieri, Soren Galatius, Sam Payne, Nathan Pflueger, Martin Ulirsch, and Jonathan Wise. I'm also grateful for the support and mentorship of Dan Abramovich, Matt Baker, Lucia Caporaso, Joe Harris, Diane Maclagan, and especially my PhD advisor Bernd Sturmfels; and many other mathematicians, including my many supportive colleagues at Brown.
Getting to do research in mathematics is a privilege. After all, basic research in math and science is a long game: we get to study fundamental questions that may have no applications right now but, in totality and over the course of history, make an outsize impact in ways we couldn't have predicted. And we get to have fun while doing it, too. But that whole calculus is predicated on having time, having breathing room, to think about the long game at all. Right now, I'm less and less sure that we have that room. We have a climate crisis on our hands, crises of civil rights and human rights, crises of democracy and disenfranchisement, entire crises of empathy. Our country is taking children from their parents. I've never been more concerned for my country and my community than I am now.
Being a math professor is, I think, still the best thing I personally know how to do. It's getting harder to carry out basic math research - like research on the combinatorics and geometry of moduli spaces, say - with any moral certainty. But there are many parts of the job that do have an immediate impact. We must train students well, and find ways to support and include more of them in the first place; we must be role models, while interrogating our role as scientists; we must make more noise altogether. Let's work together on prioritising these aspects of the profession, even while we push on our foundational research.
2022 Jennifer Balakrishnan, Boston University.
Citation: The 2022 AWM-Microsoft Research Prize in Algebra and Number Theory is presented to Jennifer Balakrishnan in recognition of outstanding contributions to explicit methods in number theory, particularly her advances in computing rational points on algebraic curves over number fields.
Professor Balakrishnan is internationally recognized as a leader in computational number theory. Her doctoral dissertation presents the first general technique for computing iterated p-adic Coleman integrals on hyperelliptic curves. In the course of her collaboration with Minhyong Kim at Oxford, Balakrishnan helped realize the substantial practical potential of Kim's non-abelian Chabauty method, and with her collaborators, turned it into a powerful tool for identifying integral and rational points on curves that are entirely beyond reach using the traditional Chabauty approach. In an impressive tour de force, Balakrishnan, Dogra, Müller, Tuitman and Vonk used the quadratic Chabauty method for computing the rational points on the split Cartan modular curve of level 13. Facetiously known as the "cursed curve" among number theorists because 13 is the only prime level that had stubbornly resisted all such prior attempts, this work represents a major breakthrough. It not only completes the proof of the split Cartan cases of Serre's uniformity conjecture for Galois images of elliptic curves, but also opens an avenue for tackling nonsplit Cartan modular curves at higher level.
Balakrishnan's research exhibits extraordinary depth as well as breadth. In joint work with Besser, Ciperiani, Dogra, Müller, Stein and others, she has worked extensively on computing p-adic height pairings for hyperelliptic curves. Applications of this research include the formulation, along with numerical evidence, of a p-adic analogue of the celebrated Birch and Swinnerton-Dyer conjecture, some new explicit examples in Iwasawa theory, and more. With Ho, Kaplan, Spicer, Stein and Weigandt, Balakrishnan has assembled the most extensive computational evidence to date on the distribution of ranks and Selmer groups of elliptic curves over the rational numbers, thereby providing the most convincing evidence thus far in support of the widely believed conjecture that the average rank of a rational elliptic curve is .
After receiving her doctorate from the Massachusetts Institute of Technology in 2011, Professor Balakrishnan held appointments as an NSF Postdoctoral Fellow at Harvard University as well as a Junior Research Fellow and a Titchmarsh Fellow at the University of Oxford. She is currently the Clare Boothe Luce Associate Professor of Mathematics at Boston University, a Sloan Research Fellow, and a recipient of an NSF CAREER award. Her research is also supported by the Simons Foundation, through the Simons Collaboration in Arithmetic Geometry, Number Theory, and Computation.
Balakrishnan has delivered an impressive array of invited and plenary lectures in locations across four continents. Beyond her outstanding scientific achievements, she has assumed leadership roles in service to her institution and the community, especially in bringing more women into math, devoting untold hours to mentoring and advocating for junior women in the profession, and striving to create supportive environments for them. In addition to her extensive record of student supervision at all levels, she has co-organised numerous research conferences, thematic programs and summer schools, including many Women in Sage gatherings. She serves on the editorial boards for five top quality journals, the AMS Short Course Subcommittee, the Scientific Advisory Board for the Institute for Computational and Experimental Research in Mathematics, the Board of Directors for the Number Theory Foundation, and the Steering Committee for the Women in Numbers Network.
Jennifer Balakrishnan's work is widely known and recognised across the globe within the number theory community and beyond. AWM congratulates her for her well-deserved AWM-Microsoft Research Prize.
Response from Jennifer Balakrishnan: I am honoured to receive the 2022 AWM-Microsoft Prize in Algebra and Number Theory. I would like to thank the AWM and Microsoft for this recognition of my work, which would not have been possible without the support of my mentors and collaborators.
Over the years, I have been very fortunate to have had the encouragement of several mentors: my PhD advisor, Kiran Kedlaya, as well as Dick Gross, William Stein, Barry Mazur, Minhyong Kim, and Henri Darmon. I am deeply indebted to my collaborators, including Amnon Besser, Mirela Ciperiani, Netan Dogra, Steffen Müller, Jan Tuitman, and Jan Vonk, who have been generously working with me for several years and have taught me so much. Boston University has provided a wonderful research environment, and I am very grateful for the support of my colleagues, including Margaret Beck, Tasso Kaper, David Rohrlich, and Glenn Stevens. The Women in Numbers network and the Simons Collaboration in Arithmetic Geometry, Number Theory, and Computation (with special thanks to Noam Elkies, Brendan Hassett, Bjorn Poonen, Andrew Sutherland, and John Voight) have also provided a warm sense of community and the inspiration to follow various research directions I would not have otherwise pursued.
2024 Yunqing Tang, University of California, Berkeley.
The AWM-Microsoft Research Prize in Algebra and Number Theory was presented to Professor Yunqing Tang, in recognition of her breakthrough work in arithmetic geometry, including results on the Grothendieck-Katz -curvature conjecture, a conjecture of Ogus on algebraicity of cycles, arithmetic intersection theory, and the unbounded denominators conjecture of Atkin and Swinnerton-Dyer. An observer wrote that Tang "has a knack for absorbing difficult ideas with lightning speed, making them her own, and then applying them in creative and unexpected ways."
Citation: The -curvature conjecture lies in the field of arithmetic geometry: it predicts that for a certain vector bundle associated to a variety over a number field, if an invariant called the -curvature vanishes for all but finitely many primes, then an associated "monodromy representation" has finite image. Tang has made progress toward this conjecture by proving for example that the conclusion holds if the -curvature vanishes for all primes, when the variety is the projective line minus three points. In the area of -adic Hodge theory, Tang has proved Ogus' conjecture (which predicts that cycles in de Rham cohomology which are invariant by almost all crystalline Frobenii are Hodge cycles) for a large class of abelian varieties.
With collaborators, Tang has developed a program in arithmetic intersection theory on Shimura varieties that can prove a phenomenon of interest occurs at infinitely many primes. This has had many interesting consequences. As a first example, Ananth Shankar and Tang have proved that an abelian surface with real multiplication over a number field is isogenous to a product of elliptic curves when reduced modulo infinitely many primes. As a second example, with Ananth Shankar, Arul Shankar, and S Tayou, Tang's work proves that a K3 surface over a number field with everywhere good reduction has the property that the Picard rank of the reduction jumps, at infinitely many places.
Recently, in joint work with F Calegari and V Dimitrov, Tang has presented a proof of the 50-year-old "unbounded denominators conjecture," originally posed by Atkin and Swinnerton-Dyer. This conjecture can be framed (roughly speaking) as the statement that a modular form for a finite index subgroup of , expanded as a Fourier series in , has integral coefficients if and only if it is a modular form for some congruence subgroup of .
Yunqing Tang is an assistant professor at University of California, Berkeley. She received a PhD from Harvard University in 2016, and she was awarded the AWM Dissertation Prize. Tang subsequently was a Member at the IAS, an Instructor at Princeton University, a junior researcher (Chargée de recherche) at CNRS/Université Paris-Sud, and an assistant professor at Princeton University. Her work is supported by the NSF, and Tang has recently been awarded a Sloan Research Fellowship and the SASTRA Ramanujan prize. A press release from the Ramanujan Prize committee wrote that Tang's "wide ranging contributions are bound to have impact in the decades ahead."
Response from Yunqing Tang: I am very honoured to receive the 2024 AWM-Microsoft Prize in Algebra and Number Theory. I would like to thank the AWM and Microsoft for their generosity in recognising my work. I have been very lucky to have several amazing mentors: my PhD advisor, Mark Kisin, as well as Peter Sarnak and Shouwu Zhang; they have been supportive over the years and shared with me numerous mathematics insights. I am deeply indebted to my collaborator Ananth Shankar, with whom I have been working since graduate school time; our numerous discussions have shaped part of my research program. I also would like to give a special thank you to my collaborators Wanlin Li and Vesselin Dimitrov for numerous zoom discussion and working sessions to keep me stay productive during the pandemic. I would like to thank all my collaborators: Frank Calegari, Victoria Cantoral Farfán, Elena Mantovan, Davesh Maulik, Rachel Pries, Arul Shankar, Sho Tanimoto, Salim Tayou, and Erik Visse; I am very grateful to have the opportunities to work with them and learn interesting math from them.
I would like to thank the math department and my colleagues, especially the algebraic geometry and number theory group, at UC Berkeley for a supportive working environment. Many of my works have been done during my stay at Princeton, CNRS, Université Paris-Saclay, IAS and Harvard and I am grateful for the excellent working environment at these places. Finally, I would like to thank AWM again for providing the community of women mathematicians and for recognising my work at an early stage through the dissertation prize.