## 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:-

We give below winners of the prize and also the citation and the reply from the winner.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.

**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!