“If someone writes a paper with the positive Grassmannian in it, she always looks at it,” said the physicist in question, Yuji Kodama. G Musiker, L Williams. “I enjoyed being given a sequence of numbers and having to find the pattern and predict the next number.” The overture shocked Williams for two reasons: No one else on the Harvard faculty did her kind of math, and no one else looked quite like her. To understand how the Grassmannian breaks into pieces, recall that each point on it encodes the properties of a line or multidimensional plane passing through the origin. at. 43: 2016: Matrix formulae and skein relations for cluster algebras from surfaces. This is just one example of a Grassmannian. It’s referred to as the “positive” part of the Grassmannian because, roughly, all the lines that cut through it have a non-negative slope. That perspective captivated Williams, and over the last two decades she’s established many of its most far-reaching implications. Main 404. lwilliams@mercyhurst.edu. “If you understand the Grassmannian well, you can go in many different directions. It is a 24-dimensional shape. Williams, who grew up in Los Angeles, is the second woman to be tenured in Harvard’s Math Department and the Seaver Professor at the Radcliffe Institute. But that’s not what motivates Williams. So she invited him to Berkeley to explain it to her in person. Emboldened by the matching pictures, Kodama and Williams looked for deeper connections between the positive Grassmannian and shallow water waves. (Notices of the American Mathematical Society 65 (2018) 1025-1026.) The most famous example of the phenomenon is a tsunami. Due to the nature of quantum interactions, such collisions are not strictly deterministic. I wasn’t sure what the atmosphere would be like.” But Williams had loved her four years at Harvard as an undergraduate — and three subsequent years as a postdoctoral researcher — and that made her more inclined to consider the university’s offer. lwilliams@mercyhurst.edu 814-824-2226 Dr. Williams joined the department in 2013, after earning her PhD in mathematics from the University of Wisconsin Milwaukee. Only it’s not one exact geometric object, but rather a family of them. LAUREN WILLIAMS CURRICULUM VITAE 3 14. Cluster algebras: an introduction, Bulletin of the American Mathematical Society, 51 (2014), 1–26. “As you move [from line to line] you’re moving from one point to another on the Grassmannian,” said Ardila. Information and translations of lauren williams in the most comprehensive dictionary definitions resource on the web. Prior to joining FCW, Williams was the tech reporter for … Her research interests lie at the intersection of algebra and combinatorics, and she has published articles in the Transactions of the AMS and The Journal of Combinatorics. International Mathematics Research … Dwight Parker Robinson Professor of Mathematics at Harvard and Sally Starling Seaver Professor at the Radcliffe Institute Science Center Room 510, Cambridge MA 02138 e-mail: williams @ math. As a professor at Harvard University — where she became only the second tenured woman mathematician in the university’s history — she has uncovered correspondences far more bewildering than anything she learned in grade school. Lauren Williams is a professor in the Mathematics department at University of California Berkeley - see what their students are saying about them or leave a rating yourself. Over the last three years, Williams has expanded this correspondence. Lauren K. Williams - Papers Compact formulas for Macdonald polynomials and quasisymmetric Macdonald polynomials (with Sylvie Corteel, Jim Haglund, Olya Mandelshtam, and Sarah Mason). Moderators are staffed during regular business hours (New York time) and can only accept comments written in English. cancel. “It was useful for me that fairly early in my career I had these collaborations with slightly older women who were making it work.”. The STANDS4 Network ... Lauren Kiyomi Williams is an American mathematician known for her work on cluster algebras and tropical geometry. Pamela Harris: The Mathematical Rise and Social Contribution of a Dreamer. in Secondary Education with a concentration in Mathematics. Lauren K. Williams. Lauren Williams On the cover: Crisscrossing Zigzags II by C. Kenneth Fan. https://www.radcliffe.harvard.edu/news/in-news/new-faculty-lauren-williams The experience was reassuring, but Williams was firmly situated in California by that point. The amplituhedron is a simpler way of calculating amplitudes. An earlier version of this article incorrectly stated that the Grassmannian Gr(4,10) would be a 10-dimensional shape. Positivity for cluster algebras from surfaces. It involved a geometric object called the amplituhedron that had been proposed as the answer to one of the knottiest problems in physics. Williams. Starting in graduate school, Williams proved a number of different features of the way points from the positive Grassmannian sort into cells. When she was 16, she spent a summer at a math research program for high school students hosted by MIT. “You have some very complicated object that is hard to understand,” said Melissa Sherman-Bennett, a graduate student at Berkeley who works with Williams. Yet in the end, she felt something of an obligation to take the position, as a way of encouraging other women to pursue careers in mathematics. Harris and three students, all math majors at Williams. At night, she’d stay up past her bedtime with a lamp pulled under her comforter (she eventually burned a hole through her sheets). The somewhat clunky prevailing method for calculating amplitudes is something called a Feynman diagram, named for its inventor, Richard Feynman. It was meant to help physicists predict what happens when fundamental particles collide. New York model Brooke Banker previously claimed to have the longest limbs at 47 inches in 2013. Dr. Williams joined the department in 2013, after earning her PhD in mathematics from the University of Wisconsin Milwaukee. She is a professor of mathematics at Harvard University. Transactions of the American Mathematical Society 368 (1), 337-363, 2016. Professor Williams received her doctorate in 2005 from the Massachusetts Institute of Technology. My first question is, what excites you about mathematics? In some sense you have to be a warrior, but Lauren has handled things with such grace,” said Ardila. “She’s not afraid to build bridges where they didn’t seem to exist.”. (Stanley would eventually become Williams’ doctoral adviser.). Ken: Hi Lauren, thank you for agreeing to do this interview! As a result, information about the relatively well-studied positive Grassmannian transfers to the relatively unexplored amplituhedron. Lauren Williams has charted an adventurous mathematical career out of the pieces of a fundamental object called the positive Grassmannian. Department of Mathematics, Harvard. The geometric object that weaves through Williams’ work is called the positive Grassmannian. Williams’ work demonstrates how the pieces of the positive Grassmannian can be translated into other areas of mathematics and physics. Lauren Williams is a professor in the Mathematics department at University of California Berkeley - see what their students are saying about them or leave a rating yourself. “I think she’s one of the masters of capturing the combinatorial nature of objects that don’t seem combinatorial,” Ardila said. He is a summa cum laude graduate of the University of Maryland, with a B.S. 3, 699--709. math.CO/0206077 MR2113918 PDF Dissertation: Invariant Polynomials on Tensors Under the Action of a Product of Orthogonal Groups. Arkani-Hamed and his collaborators had defined the amplituhedron in relation to the positive Grassmannian. “I recognized that coming to Harvard would give me a chance to make a positive impact on a department I cared a great deal about,” Williams said. She traveled to Cambridge and had dinner with her prospective colleagues. You can think of these seven cells as the seven puzzle pieces that make up the positive Grassmannian. This process of sorting points by sign patterns to reveal the shapes of puzzle pieces works especially well for the positive part of the Grassmannian. In the 1970s and 1980s, Gian-Carlo Rota and his student Richard Stanley came up with a new way of thinking about sophisticated mathematical shapes. Lauren Williams joined the department in 2018. https://www.radcliffe.harvard.edu/news/in-news/new-faculty-lauren-williams Andrea Patiño Contreras for Quanta Magazine; Courtesy of Lauren Williams. “Ever since I was a kid, I’ve always loved patterns,” said Williams. There is an announcement of the recent three appointments in the Harvard Gazette. “Ever since I was a kid, I’ve always loved patterns,” said Williams. They become apparent when you sort points by their sign patterns — all the points with a given sign pattern fill out the shape of a single cell, or puzzle piece. Transactions of the American Mathematical Society 368 (1), 337-363, 2016. She is a professor of mathematics at Harvard University. All three incoming professors will bring a focus on theoretical research and a knack for mentoring students when they start their new appointments. Williams, a former math concentrator, won a tenured associate professorship at UC Berkeley in 2013—and a full tenured professor position there … (Notices of the American Mathematical Society (63) 2016, 1164-1170.) They also have “subdeterminants,” which are calculated based on a subset of the values in the matrix; a 1 × 3 matrix has three subdeterminants. The American Mathematical Society is generously offering a 25% discount on the two book set To get a sense for a single Grassmannian, let’s start with just two numbers, 1 and 3. And I She worried about uprooting her husband and her young kids, and she also recognized that making such a high-profile move within the math world might increase public scrutiny of her and her work. When Williams arrived at MIT, she read foundational works from the 1990s by George Lusztig and his student Konstanze Rietsch that had introduced the positive Grassmannian, and more recent papers by Alexander Postnikov that applied a combinatorial perspective to the shape. Advisor 1: Lauren Kiyomi Williams No students known. Model better known as The Classic Giraffe who had the longest legs in the U.S. at 49 inches as of 2014. There she started studying a mathematical object with many guises: the Grassmannian. “I enjoyed being given a sequence of numbers and having to find the pattern and predict the next number.” Abusive, profane, self-promotional, misleading, incoherent or off-topic comments will be rejected. But to really understand its place in mathematics, mathematicians first had to learn how to take the Grassmannian apart. “The combinatorics is extremely rich,” said Rietsch. “It was just this delightful surprise.”. The result foreshadowed a lot of the innovative work later in her career. Lauren Williams . Williams tried reading Kodama’s work, eager to see how the Grassmannian might fit in, but it was too far from her own research for her to understand it. J. Gregg Musiker and Ralf Schiffler and Lauren Williams (2011). Postnikov was at MIT, and Williams spent a lot of time talking with him. Given a set of particles on a collision course, you can use their properties to construct a geometric object — the amplituhedron. Lauren Kiyomi Williams is an American mathematician known for her work on cluster algebras and tropical geometry. Ph.D. University of Wisconsin-Milwaukee 2013. The outline of Lauren Williams ’s mathematical career was present very early on in her life. Williams grew up in the Los Angeles suburbs, the oldest of four sisters. “Ever since I was a kid, I’ve always loved patterns,” said Williams. Adv. Login Sign Up. “I’d rather not be working on what everybody else is working on,” she said. Via Quanta Magazine: The outline of Lauren Williams’s mathematical career was present very early on in her life. The outline of Lauren Williams ’ mathematical career was present very early on in her life. She demonstrated that in some cases the combinatorial properties of the positive Grassmannian — the way its points sort into cells — carries over to the amplituhedron via this transformation process. Williams’ research took another surprising turn in 2009, soon after she joined the faculty at Berkeley. Sylvie Corteel and Lauren Williams (2011). The Grassmannian gets its name from Hermann Grassmann, who first formalized it in the mid-1800s. In often dramatic fashion, Williams has also proved that the pieces of the positive Grassmannian can be reassembled in a form that explains everything from the movement of tsunami waves to particle collisions at the frontier of quantum physics. “I sort of don’t like feeling I’m competing against other people toward the same goal.”, Her collaborators have also noticed this unusual trait. Lauren Williams. Now picture a line that runs through the origin. Overall Quality Based on 8 ratings. She has modeled for global brands including most notably Nike. ... an African-American mathematician in … The 3 indicates that we’re in three-dimensional space. “There are so many hidden obstacles for female math professors at top research institutions. Harvard University Center of Mathematical Sciences and Applications. Famous Birthdays #1: Today's Birthdays & Celebrity Deaths. It was like we were speaking different languages,” said Williams. Williams’ preference for less-trafficked problems found an outlet right after graduate school, when she wrote a series of papers with the mathematician Sylvie Corteel that explored an unexpected link between the combinatorics of the positive Grassmannian and statistical physics. Shannon O'Brien. Jump To Ratings. There are now three women who are tenured, full professors in the department. She’s also laid out a precise conjecture that predicts how many tiles are needed to triangulate any amplituhedron. “She’s always been fearless,” said Federico Ardila of San Francisco State University, who was in graduate school with Williams. She and her co-founders want to showcase the contributions of mathematicians who look like her. Professors. Lauren Williams (ice hockey) (born 1996), Canadian ice hockey player Lauren Williams (footballer) (born 1994), American-born professional footballer for Saint Kitts and Nevis Lauren Williams (journalist), American journalist and former Lauren Williams (mathematician), American mathematician at … It’s a shape that keeps track of other shapes. As a child, Williams learned the violin and dreamed of becoming a writer. as space-time expands. “It’s almost like a remote control.”. Points sort into different buckets, or “cells,” based on their sign pattern. Cluster algebras: an introduction, Bulletin of the American Mathematical Society, 51 (2014), 1–26. 15. That means the cells of the positive Grassmannian can serve as the tiles needed to triangulate the amplituhedron. Lauren Williams has the longest-known legs in the US, with her eye-catching limbs measuring an incredible 49 inches from hip to heel. As a child Williams enjoyed playing the violin and writing poetry, but she loved reading most of all. There are lots of ways to chart a career in math. “There were no women and no combinatorialists,” she said. These diagrams involve vast, tedious computations that are hard to carry out with precision as particle collisions grow more complex. Many matrices have a measurement called a determinant, which is a single value calculated using the numbers in the matrix. Lauren Kiyomi Williams is an American mathematician known for her work on cluster algebras, tropical geometry, algebraic combinatorics, amplituhedra, and the positive Grassmannian. AWM-Microsoft Research Prize in Algebra and Number Theory 2016 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. Get highlights of the most important news delivered to your email inbox. G Musiker, L Williams. The Mathematical Sciences Research Institute (MSRI), founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the National Science Foundation, foundations, corporations, and more than 90 universities and institutions. Chair, Department of Mathematics, Associate Professor. A Mathematician’s Unanticipated Journey Through the Physical World Lauren Williams has charted an adventurous mathematical career out of the pieces of a fundamental object … When she went to college, at Harvard, she was excited to join a wider intellectual world. When she arrived at Harvard as a member of the Class of 2000, Lauren Williams knew she liked math, but she wasn’t certain about making a career of it. Physicists model them using the Kadomtsev-Petviashvili, or KP, equation: Feed the equations the position of a wave and the equations output its height at any time in the future. Next, imagine positioning a sphere so that it’s centered around the origin. “It weighed on me very heavily when I was trying to make up my mind. Harvard has hired three new professors in the Mathematics Department: Laura DeMarco, Mihnea Popa, and Melanie Matchett Wood.. Their appointments began in July. reveal patterns in the waves FOUNDER AND PRESIDENT C. Kenneth Fan BOARD OF ADVISORS Connie Chow Yaim Cooper Julia Elisenda Grigsby Kay Kirkpatrick Grace Lyo Lauren McGough Mia Minnes Bjorn Poonen Beth O’Sullivan Elissa Ozanne Katherine Paur Liz Simon Gigliola Staffilani Bianca Viray Karen Willcox Lauren Williams On the cover: Honk! It embodies the particle interaction in a precise way: By calculating its volume, in effect you’re calculating the amplitude for the given collision. Eventually she did — following one more surprising pattern, with echoes of the numerical sequences that had fascinated her as a kid. at. If we’d started with the numbers 4 and 10, we’d instead be thinking about four-dimensional planes passing through the origin in 10-dimensional space — and the Grassmannian, Gr(4,10), would be the shape in which each point represents one of those four-dimensional planes. This means that if you know the coordinates of a point on the northern hemisphere, you know everything about the one-dimensional line through the origin that passes through that point. Lauren K. Williams - Papers Compact formulas for Macdonald polynomials and quasisymmetric Macdonald polynomials (with Sylvie Corteel, Jim Haglund, Olya Mandelshtam, and Sarah Mason). Quanta Magazine moderates comments to facilitate an informed, substantive, civil conversation. While looking for new results on the positive Grassmannian, she noticed that a physicist at Ohio State University had cited her work in his research on shallow water waves. 43: 2016: Matrix formulae and skein relations for cluster algebras from surfaces. One of their papers also included a haiku Kodama and Williams wrote, partly in recognition of their shared Japanese heritage: Arrangements of stones Math. Lauren Williams is a professor in the Mathematics department at Mercyhurst University - see what their students are saying about them or leave a rating yourself. MathSciNet. “To a first approximation, [physicists] are interested in the volume of the amplituhedron, and one way to compute volume is to break it into smaller pieces. It’s central in math,” said Bernd Sturmfels of the University of California, Berkeley, who is also the director of the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany. Login . If you have additional information or corrections regarding this mathematician, please use the update form.To submit students of this mathematician, please use the new data form, noting this mathematician's MGP ID of 92849 for the advisor ID. That's part of why she founded Lathisms, a project that features Latinx mathematicians during Hispanic Heritage Month. In the fall of 2019 Williams and Arkani-Hamed were two of the co-organizers of a semester-long program at Harvard that brought together mathematicians and physicists to explore the link between the positive Grassmannian and the amplituhedron. Expository papers for mathematicians. In 1975, Robinson became the first woman mathematician to be elected to the National Academy of Sciences, and, in 1983, she became the first woman president of the American Mathematical Society. International Mathematics Research … During summer days, “I spent hours sitting in the branches of an apricot tree in our backyard, reading a book and eating apricots,” she said. LAUREN WILLIAMS CURRICULUM VITAE 3 14. Beginning in the early 1990s, many mathematicians started to focus on a particular part of the Grassmannian called the positive Grassmannian. As they talked, Kodama sketched simple schematics to illustrate patterns of wave interaction: Two lines representing two waves converge at a point, and then a single line representing a new wave emerges. https://commons.wikimedia.org/wiki/Category:Lauren_Kiyomi_Williams The numbers were immediately familiar to Williams: She’d encountered them as a graduate student working on a different (and unrelated) problem on a version of the positive Grassmannian 16 years earlier. The Positive Grassmannian (from a mathematician’s perspective) Lauren K. Williams, UC Berkeley Lauren K.Williams (UCBerkeley) ThePositiveGrassmannians March2014 1/40 Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials. She quickly recognized that they mirrored pictorial representations called planar bicolored graphs that mathematicians use to describe points on the positive Grassmannian. Sports … For Gr(1,3) — one-dimensional lines in three-dimensional space — each line through the origin is specified by a 1 × 3 matrix, such as: The numbers in the matrix serve as the coordinates for the point in the Grassmannian that encodes the line. She is originally from Houston, Texas. “He’s a physicist and an older Japanese man, and we had a lot of trouble understanding each other that first day. She played sports competitively growing up and got a scholarship for volleyball at the University of Southern California. “We were groping around in the dark for quite a while, but [her work] on the positive Grassmannian was a shining light throughout this process,” said Arkani-Hamed. Her father was an engineer, and her mother, a third-generation Japanese American, taught English. In our example, Gr(1,3), it is one-quarter of the northern hemisphere. The letter’s author was Lauren Williams ’00, would herself grow up to become a Harvard professor. “Being a writer or a poet was one of my childhood dreams, and I thought, I’ve got tenure now, I can be a little crazy,” Williams said. Instead of thinking of it as a single geometric object, they were trying to understand it in terms of the pieces that make it up. . One is to devote yourself to developing a new theory or attacking a prominent open problem. According to our current on-line database, Lauren Williams has 7 students and 7 descendants. Math rose to her attention in fourth grade when she participated in an elementary school math competition. Around the time Williams started graduate school at the Massachusetts Institute of Technology in 2001, mathematicians were developing a new way of thinking about the positive Grassmannian. (The first, Sophie Morel, spent three years at Harvard before leaving in 2012; this fall, Harvard hired two more tenured women mathematicians: Laura DeMarco and Melanie Matchett Wood.). She is Dwight Parker Robinson Professor of Mathematics at Harvard University. “There was no end of classes and activities I wanted to try, and my peers were just as excited as me about learning and doing everything.” She majored in mathematics at Harvard, where she graduated in 2000, and went on to graduate school at MIT. You can construct infinitely many different Grassmannians by starting with distinct pairs of whole numbers. It’s as if the Grassmannian, uncovered a century ago to formalize the mathematics of lines and planes through the origin, also happens to index phenomena in the physical world — a bizarre correspondence that Williams still can’t fully explain. Kodama’s work focused on a particular type of wave called a soliton, or solitary wave. But by looking at it from an entirely new perspective, Williams, 42, has proved that it’s the key to decoding the secrets behind a wide range of seemingly unrelated phenomena in math — and in nature. She is a professor of mathematics at Harvard University. Williams, whose work focuses on algebraic combinatorics, joined Harvard in 2018 . 15. (with Yuji Kodama) A Deodhar decomposition of the Grassmannian and the regularity of KP The leftover northern hemisphere is the Grassmannian formed by one-dimensional lines in three-dimensional space. View Lauren Williams, M.Ed.’s profile on LinkedIn, the world’s largest professional community. They demonstrated that it’s possible to change a positive Grassmannian into the amplituhedron by multiplying it by a type of matrix, effectively providing a mathematical recipe for moving points on the positive Grassmannian over to points on the amplituhedron. The Grassmannian itself contains infinitely many points, which can’t be counted in a discrete, finite way. Instead, they’re described by an amplitude, which is like a probability that the collision plays out in a given way. The 1 means we’re going to think about one-dimensional lines within that space. “When I was much younger, I worried about whether it was possible to be a successful academic and also have a family,” she said. In addition to their mathematical results, Williams gained something else by working with Corteel, who had a baby during their first collaboration. Lauren. Sports-loving Lauren Williams has the longest-known legs in the US The 26-year-old model from Texas says her family are all well over 6ft Models everything from sports and … She has also worked as a science educator at a children's museum in Houston. She was scouted as a model numerous times from a young age. Williams and the physicist Nima Arkani-Hamed were co-organizers of a recent program at Harvard that explored connections between the positive Grassmannian and particle physics. Get Quanta Magazine delivered to your inbox. The size of a matrix depends on the Grassmannian. Quantum Bruhat graph and Schubert polynomials, A. Postnikov: Proceedings of the American Mathematical Society 133 (2005), no. in Mathematics and a B.S. It can be hard for people to imagine a career for themselves when they don’t see people like them with that career.”, Williams started at Harvard in the fall of 2018 and became the second woman ever to hold a tenured position in the university’s math department. So the question is how to calculate the volume. “Whenever I had a free moment, I’d find myself wandering back to [those numbers] and wondering how to make that connection,” she said. These days, her mind is made up. Or, as mathematicians write it, Gr(1,3). the level of mathematical sophistication. Williams, whose work focuses on algebraic combinatorics, joined Harvard in 2018. At the same time Williams was moving across the country, she was deep in a new Grassmannian project. The letter’s author was Lauren Williams ’00, would herself grow up to become a Harvard professor. The drawings looked familiar to Williams. Ph.D. “She is open-minded and adventurous.”. He holds a Ph.D. in Computer Science from MIT. “I found it to be a very beautiful confluence of ideas,” Williams said. “The large-scale behavior of a wave formation is entirely determined by which cell your point in the positive Grassmannian lies in,” Williams said. That was her first serious introduction to combinatorics — an area of math concerned with breaking complicated objects into pieces and then classifying and counting them. And for technical reasons, mathematicians don’t need to consider one of them, (000), which leaves just seven categories for those infinite points to be divided into. “But you can break it into pieces that give you more insight into this big complicated thing.”. “I unexpectedly won the contest, and the two teachers who organized it sort of took me under their wing,” she said. This turns something infinite and uncountable into something discrete and possible to sort: While there are infinitely many different 1 × 3 matrices, their three subdeterminants can have only eight different sign patterns: (000), (00+), (0++), and so on. 814-824-2226. During the event, Williams was talking with two physicists who mentioned a sequence of numbers related to triangulations of the amplituhedron. Her other honors include Noether Lecturer 1982, MacArthur Foundation Fellowship 1983, and election to the American Academy of Arts and Sciences. 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To help physicists predict what happens when fundamental particles collide according to our current on-line database, Lauren,! Arrays of numbers related to triangulations of the pieces of a recent program at Harvard that explored between. At the same time Williams was firmly situated in California by that point 368 1... Such collisions are not strictly deterministic different languages, ” said Williams on! She invited him to Berkeley to explain it to be a very beautiful confluence of ideas ”... Violin and writing poetry, but Lauren has handled things with such grace, ” Williams said, taught.. Information about the relatively unexplored amplituhedron counted in a 2013 paper by Nima Arkani-Hamed were of... Teaching '' award all three incoming professors will bring a focus on theoretical research a... Famous Birthdays # 1: Today 's Birthdays & Celebrity Deaths more widespread parts of mathematics at Harvard.! Patterns, ” she said defined by vectors that can be described in a theory.
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