mathematical sciences news 2018 contents 2018

Editor-in-Chief Tom Bohman

Contributing Writers Letter from Department 03 Tom Bohman Head, Tom Bohman Joyce DeFrancesco Jocelyn Duffy Janko Gravner Faculty Notes 04 Ben Panko Emily Payne Ann Lyon Ritchie Tomasz Tkocz

Graphic Design and Photography 14 Carnegie Mellon University Marketing & Communications In Memoriam: Jim Greenberg

Mellon College of Science Communications 18 Dives Carnegie Mellon University into Startup World Department of Mathematical Sciences Wean Hall 6113 Pittsburgh, PA 15213 20 Alumna Shafi Goldwasser Receives cmu.edu/math Honorary Degree at Commencement

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©2018 Carnegie Mellon University, All rights reserved. No 28 part of this publication may be reproduced in any form Class of 2018 without written permission from Carnegie Mellon University’s Department of Mathematical Sciences.

1 faculty updates

Letter from Mathematics Department Head, Tom Bohman Math Helps Make Congressional Districts in PA More Fair The Department of Mathematical Sciences pursues a strategy Until recently, the lines drawn on the of developing and maintaining research groups in vibrant congressional map of Pennsylvania looked areas of mathematics that are enriched by natural connections irregular. So irregular, in fact, that the outline with other research domains within mathematics and beyond. of the seventh district looked like a drawing Our department has a tradition of being outward looking. of cartoon characters and was often referred We have research groups in applied analysis, combinatorics, to as “Goofy kicking Donald Duck.” logic and mathematical finance and probability, with a particular interest in research that has connections with other These wildly drawn maps are likely the result areas of strength at Carnegie Mellon. All of these research of gerrymandering, the more than 200-year- groups have been quite successful in recent years and the old practice of redrawing congressional stories in this newsletter are reflections of that success. and legislative lines to benefit a particular political party. , including The quality of our groups was affirmed by the 2018 U.S. News Mathematical Sciences Associate Professor & World Report ranking of graduate programs in mathematics. Wesley Pegden, have been enlisted to the Carnegie Mellon placed 16th in applied mathematics, 7th in forefront of the fight against gerrymandering. discrete mathematics and combinatorics and 6th in logic, all improvements from the previous edition of these rankings. In early 2017, Pegden, with University While rankings of academic programs have their flaws, I feel Cover Image Professor of Mathematical Sciences Alan that these results reflect genuine improvements in the quality Greenberg-Hastings model Frieze and the University of Pittsburgh’s of the research the department has produced over the last Maria Chikina, published a paper in the The cover image is generated by few years. Our rise in the rankings also means that our peers Proceedings of the National Academy of the cellular automaton known as the are taking notice; the survey results are based only on input Sciences that used Markov Chains to redraw Greenberg-Hastings model. This from mathematics departments that grant Ph.D.s in the area Pennsylvania’s congressional maps step by cover image is based on work of in question. step and show that there was little chance David Griffeath of the University of that the existing districts had been drawn at Wisconsin. Griffeath has conducted In this issue of the newsletter, we will introduce you to three random or without bias. The paper caught extensive studies of cellular automata. tenure-track faculty members who joined the department in the eye of lawyers who were challenging 2018. These new faculty represent our continued investment the Pennsylvania congressional map in front For more on the cover image in research (see page 8). of the state Supreme and Greenberg-Hastings itself see Court. Pegden testified pages 16-17 of this newsletter. The cover image for this newsletter was produced by the in the case. The court cellular automata known as the Greenberg-Hastings model found the maps to be (see page 16 for some mathematical details regarding unconstitutional and I hope that the this model). Jim Greenberg, who introduced this model in called for the maps to be many alumni of joint work with Stuart Hastings, passed away earlier this redrawn before the state’s the Department year. Greenberg was head of the Mathematical Sciences May primary elections. of Mathematical Department from 1995 to 2002. For more on Greenberg, see Sciences have the story on page 14. The newly drawn map a chance to appears to be more fair. reconnect with The department’s outward-looking research orientation also In November’s midterm the department has benefited our students. We have constructed a curriculum elections, Pennsylvania’s by visiting that gives excellent support to educational activities across congressional delegation cmu.edu/math the university while preparing mathematics majors for success went from primarily in a wide range of careers both in academia and beyond. You Republican to evenly split, Let us know can read about the success of our students in the stories on reflecting the popular vote “Goofy kicking Donald Duck.” Pennsylvania's 7th district before what's new pages 22-31 of this newsletter to learn more about our current in the state. the newly drawn map. United States Department of Interior. with you! and recent students.

2 3 faculty notes

In his thesis, the prize committee noted, “Vasey undertook a deep and sustained study of classification theory for abstract elementary classes.”

“Among the many theorems he proved, his eventual categoricity theorem for universal classes is recognized as a landmark achievement toward Shelah's conjecture generalizing Morley's theorem Mathematics Alumnus Sebastien on uncountable categoricity to abstract Vasey Receives Prestigious Sacks elementary classes,” the committee Prize in Logic continued. “A second remarkable result is his classification of the stability spectrum for Sebastien Vasey, who earned his doctorate tame AECs, which may well pave the way from Carnegie Mellon University’s Department for connections with, and applications to, of Mathematical Sciences in May 2017, other areas of mathematics.” has been awarded the Sacks Prize by the Association for Symbolic Logic for his The Sacks Prize was founded in 1994 to dissertation. The annual international prize honor noted logic researcher Gerald Sacks recognizes “the most outstanding doctoral of Harvard and MIT. Sacks was known dissertation in mathematical logic.” for serving as an advisor to many Ph.D. students. Vasey shared this year's prize with Vasey conducts research in model theory, Matthew Harrison-Trainor, a postdoctoral one of the main branches of mathematical fellow in mathematics at Victoria University logic. He won the award for his dissertation of Wellington. Last year, Vasey was a titled “Superstability and Categoricity in recipient of the Guy C. Berry Graduate Abstract Elementary Classes,” which he Student Research Award. He is currently completed under the direction of Professor of a Benjamin Peirce Fellow in the Harvard Mathematical Sciences Rami Grossberg. University Department of Mathematics. Hayden Schaeffer Receives observed data, in a sense, reverse engineering NSF CAREER Award models from data. These extracted models can be used to gain insights on the data and Assistant Professor of Mathematical make data-enabled decisions. Kavčić-Moura University vision. In the article, Fonseca potential, specifically through Sciences Hayden Schaeffer received Professor of Mathematical offers advice to young women, his social enterprise startup, a Faculty Early Career Development CAREER Awards are among the most Sciences Irene Fonseca encouraging them to network, Expii. The success of Loh’s (CAREER) award from the National Science prestigious awards for young faculty. They was featured in a special be confident in their skills and teaching methods is evident in Foundation. The award supports his work recognize and support those who exemplify Women’s History Month have fun. his students’ triumphs. As the on optimization and machine learning for the role of teacher-scholars through issue of the Notices of the coach of the U.S. Mathematical understanding features and structures in outstanding research, excellent education American Mathematical Olympiad Team, he led the Associate Professor of time-series data. and the integration of education and research. Society. In the profile the U.S. team to a first place of Fonseca, it notes her Mathematical Sciences finish in the 2018 International Schaeffer, whose research is also supported contributions to the variational Po-Shen Loh was named Mathematical Olympiad ­— the by an Air Force Young Investigator Program Image: Given a set of data (top image), study of ferro-electric one of Pittsburgh Magazine’s United States’ third win since (YIP) grant, develops mathematical tools Schaeffer computes the velocities (second and magnetic materials, “40 Under 40” honorees for Loh took over as coach four for extracting information from data, a task image), uses an approximation to the composites, thin structures, 2017. The program honors years ago. Under his direction, that is important to machine learning, data trajectory to initialize the learning process (third phase transitions and in the people who are making Carnegie Mellon had the mining, image processing and automated image) and learns the best fit trajectory via mathematical analysis of Pittsburgh a better place. second most students finishing analysis. Under the grant, Schaeffer plans an optimization problem (bottom image) to image segmentation, denoising, Loh was recognized for his in the top 500 of all students to construct methods for learning the attempt to find the generating equations that detexturizing, registration and work that challenges people taking the Putnam exam for the underlying process that generates some govern the dynamics. recolorization in computer to learn math and reach their past five years.

4 5 faculty notes

Decoration). In 2014, she was named a University Professor, the highest distinction that can be bestowed on a professor at CMU.

The Kavčić-Moura professorships honor inventors José M.F. Moura and Aleksandar Kavčić, whose scientific research and technological innovations have had a transformative impact on the computing industry for more than a Fonseca Brings Abel to Pittsburgh decade and a half. The Department of Mathematical Sciences “We are delighted to honor José will host some of the world’s preeminent and Alek for their groundbreaking mathematicians for the Abel in Pittsburgh work and passionate commitment to conference. The one-day conference, held advancing research and education on January 11, 2019, is the ninth edition of at Carnegie Mellon,” said President the “Abel in …” series that aims to increase Farnam Jahanian. “The Kavčić- public awareness of mathematics and the Moura Professorships will allow us . to attract and retain outstanding scholars across a broad spectrum The Abel Prize, awarded by the Norwegian of disciplines, providing the funds Academy of Science and Letters for a for brilliant minds to make innovative researcher’s contributions to mathematics Irene Fonseca Named Kavčić-Moura advances in their research.” over the course of a career, is considered to University Professor of Mathematics be the most important prize in the field for a Moura, the Philip L. and Marsha lifetime achievement. Professor of Mathematical Sciences Irene Fonseca was Dowd University Professor of one of four Carnegie Mellon University faculty members Electrical and Computer Engineering, The conference comes to Pittsburgh appointed to new Kavčić-Moura Professorships, and Kavčić, a former doctoral through Irene Fonseca’s position on the Abel designed to provide sustained, long-term support for student of Moura who is an adjunct Committee. Fonseca is co-organizing the scholars across the university whose breakthroughs and faculty member at CMU, developed conference with Professor of Mathematical discoveries have the potential to impact the world where and patented systems and methods Sciences Dejan Slepčev. Scheduled to human life and technology meet. that fundamentally increased the speak are Abel Laureate Yakov Sinai and accuracy with which hard disk drive Abel Prize Committee member Sun-Yung Fonseca has been a faculty member in the Mellon circuits read data from high speed Alice Chang from , Abel College of Science since 1987 and is the director of magnetic disks. Prize Committee member Gil Kalai from the the university's Center for Nonlinear Analysis. One of Hebrew University of Jerusalem and Scott the world's leading researchers in the field of applied The Kavčić-Moura Professorships Sheffield from the Massachusetts Institute mathematics, Fonseca̓s research lies at the interface of are funded by the university’s of Technology. applied analysis with materials and imaging sciences. In proceeds from the 2016 settlement 2017, she was appointed to the Abel Prize Committee, of the patent infringement lawsuit The meeting also will serve as an opportunity which is responsible for selecting the winner of the top against Marvell Technology Group for the Abel Committee, which consists of prize recognizing lifetime contributions to mathematics by Ltd. and Marvell Semiconductor Inc. Fonseca, Chang, Kalai, François Labourie the Norwegian Academy of Science and Letters. In 2012, As outlined in 2016, the majority from the Université de Nice and Hans she became the second woman to be elected president of those proceeds were put into Munthe-Kaas from the University of Bergen, of the Society for Industrial and Applied Mathematics endowment, for the perpetual to meet. The group will recommend a (SIAM). She is a fellow of the American Mathematical support of undergraduate financial candidate for the Society and SIAM. She is a Grand Officer of the Military aid, graduate student fellowships, 2019 Abel Prize on March 19, 2019, and Order of Saint James of the Sword (Grande Oficial da endowed faculty chairs and cross- the prize will be bestowed at an awards Ordem Militar de Santiago da Espada, Portuguese campus research initiatives. ceremony in Oslo on May 20.

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2017 on the intersections of finite sets as part of the Department of Mathematical Sciences' colloquia program.

Outside of his research, Frick said he enjoys attending music concerts, running and Florian Frick watching Netflix. “I don't know Pittsburgh yet, Assistant Professor Florian Frick joined the but I'm hoping it turns out to be exciting.” Department of Mathematical Sciences at Carnegie Mellon in July.

A native of Germany, Frick came to Carnegie Tomasz Tkocz Mellon from Cornell University, where he Assistant Professor Tomasz Tkocz joined was an assistant professor for three years. the Department of Mathematical Sciences at During that time, he also spent a semester Carnegie Mellon in January. as a postdoctoral fellow at the Mathematical Sciences Research Institute in California. He A native of Poland, Tkocz came to earned his Ph.D. in 2015 from Technische Carnegie Mellon after time spent working Universität Berlin, from which he also obtained as a Berlekamp Postdoctoral Fellow at the his bachelor's and master's degrees. Mathematical Sciences Research Institute in California and as a postdoctoral research “I work at the triple-point of combinatorics, associate at Princeton University. He obtained topology and geometry,” Frick said of his his Ph.D. from the Mathematics Institute at the research. He said Carnegie Mellon's strong in the United Kingdom group of faculty who work in combinatorics after studying at the . attracted him to come here. “My research revolves around probability,” An example of Frick̓s research is his work Tkocz said, and the number of Carnegie to calculate whether it's possible to find four Mellon faculty working in this field incentivized Math points that form a square within a curved him to come here. figure on a plane. Another area of mathematics mixtures of Gaussians is important for at CMU he's interested in is one that is relatable to “I'm interested in problems related to machine learning, his work on entropy many people at some point in their lives: fair probabilistic questions that don't have inequalities is important to information rent division. too much structure,” Tkocz said. “I'm also theory, his work on random matrix theory Welcomes interested in applying such theory to related is important to quantum computing and “It's always possible,” he said. fields like convex geometry.” his work on chaining is important to coding theory and compressed sensing.” Three In a paper that will appear in American Math Tkocz said he is "particularly fond" of studying Monthly, Frick and his collaborators found high-dimensional phenomena, the topic on Before joining the faculty, Tkocz New an algorithm that could divide roommates which he based his Ph.D. thesis. An example previously presented at Carnegie so they would be content with their different Tkocz offered of his work in this field is Mellon in December 2016 on Gaussian Faculty rents and rooms, even if one doesn̓t know the studying what one could say about the typical mixtures as part of the Department preferences of one of the people who will be volume in a high-dimensional setting. (For an of Mathematical Sciences' colloquia living in the apartment. introduction to this class of problems, see program. Members pages 12-13.) “I am impressed by his work on the disproof Outside of his research, Tkocz describes of topological Tverberg conjecture,” said Boris “The mathematical areas he's worked in himself as a nature lover, and he enjoys Bukh, associate professor of mathematics. are all essential for theoretical computer hiking and climbing in the various parks science,” Professor of Computer Science around the Pittsburgh area. He also Before joining the faculty, Frick previously Ryan O’Donnell said of Tkocz’s research “His enjoys experimenting with new dishes presented at Carnegie Mellon in December work is important to many fields. His work on and styles in cooking.

8 9 Franziska Weber

Assistant Professor Franziska Weber joined the Department of Mathematical Sciences at Carnegie Mellon this fall.

Weber came to Carnegie Mellon from the University of Maryland, College Park, where she worked as a postdoctoral fellow. She obtained her Ph.D. from the University of Oslo in Norway after receiving bachelor̓s and master's degrees from the Swiss Federal Institute of Technology in Zürich, Switzerland, where she also held a postdoctoral research position.

“There are a lot of really good people in my extended field here,” Weber said of why she came to Carnegie Mellon.

Weber̓s research focuses on partial differential equations, which provide mathematical descriptions of physical phenomena. As these equations cannot be solved explicitly, instead, one must construct an algorithm that can then be used to solve the equation.

“My job is to construct algorithms for certain Before joining the faculty, Weber kinds of equations like those governing fluids previously presented at Carnegie Mellon and waves,” Weber said of her research. For in December 2017 on the structure example, Weber creates algorithms to study preserving numerical methods for the Rosensweig instability, a sequence of nonlinear partial differential equations peaks and valleys that form on the surface of modeling complex fluids as part of the a magnetic fluid in the presence of a strong Department of Mathematical Sciences̓ magnetic field. colloquia program, and in April at the Women and Mathematics at Carnegie “Fluids, such as ferrofluids and liquid crystals, Mellon University conference. show up in daily applications and have special properties,” said Irene Fonseca, Outside of work, Weber enjoys outdoor Kavčić-Moura University Professor. For activities such as running, hiking, skiing example, Weber creates algorithms to study and snowboarding. the Rosensweig instability, a sequence of peaks and valleys that form on the surface of Image: A simulation of the “quarter a magnetic fluid in the presence of a strong five-spot problem” that models magnetic field. water injected in an oil reservoir. This simulation was generated by a finite difference scheme for a model of two- phase flow in porous media and is the joint work of Weber, Nils Henrik Risebro, Sid Mishra and Giuseppe Coclite.

10 11 So, where is the volume concentrated? By the of Dvoretzky's theorem, which says that Convex Sets in HighThe Dimensions Work of John Nash, continued 3σ 3σ-rule, f (x)dx> 0.99 , so more than 99 n-dimensional symmetric convex sets admit by Tomasz Tkocz ∫–3σ percent volume of B is within a symmetric slab sections with roughly log n-dimensional n { x ∈ , |x1| ≤ 3σ } of width 6σ . In high subspaces which are almost Euclidean balls. This note briefly surveys several results, classical dimension n, this width of constant order Interestingly, such Euclidean subspaces are not by now, which shaped high-dimensional convex becomes tiny in comparison with the radius easy to find constructively, however, Milman's geometry and describes its major open problems. n We start with a closer look at Euclidean balls and r of B, which is of the order √ . To challenge probabilistic proof of Dvoretzky's theorem shows then try to make a point that “in high-dimensions, our high dimensional intuitions more, remark that most subspaces yield Euclidean sections. all convex sets, despite their rich structure, behave that the choice of the axis 0x1 was arbitrary Of course, many problems remain unsolved. a bit like Euclidean balls.” and by symmetry the above remains true for A central one, the slicing problem posed by n any other direction, so 99 percent volume of B A set of points A in is called convex if along Bourgain, asks whether there is a universal is within any symmetric slab of width 6σ . with every two points a1, a2 in A, the whole constant c such that for every dimension n, segment [a1, a2] joining a1 and a2 is contained Let us have a look at how much volume there is every n-dimensional convex set with volume in the set A. For example, balls and ellipsoids, or near the surface of B, say in an annulus of the 1 admits a codimension 1 section through its cubes and parallelopipeds are convex, whereas Figure 1: Volume distribution is Gaussian. inner radius r — 1/√ n and the outer radius r. barycentre of volume at least c. (Currently the annuli are not. How is volume distributed in a Plainly, this volume equals, best dimension dependent bound on is of the convex set? Say we have a convex set of volume c n n n order –1/4). The slicing problem is related to r–1/√ n r√ n √2πe n 1. Where is most of its volume concentrated? 1– = 1– 1– 1 ~ 1– e ~1– e ~ 0.98 ( r ) ( r–√ n ) many other natural questions in convex geometry Consider the example of a centered Euclidean ball This means that on one hand, almost all the and beyond. For example, the Busemann-Petty n 2 2 2 B = { x ∈ , x 1 + … + x n ≤ r }. Choose its volume of B is within any symmetric slab of problem asked the following: if symmetric –1/n n radius r such that it has unit volume, r = vn , constant width (hence also in the intersection of convex sets K and L in satisfy vol(K ∩H) where vn is the volume of the n-dimensional ball many of those), and on the other hand, almost ≤ vol(L∩ H) for every codimension 1 subspace n n 1/n with radius 1. Thus, = 1 Γ 1 , which is all the volume of is concentrated near its r π ( + 2 ) B H, does it follow that vol(K ) ≤ vol(L)? After a n √ approximately for large n. For x ∈ , let surface (in an annulus of width 1/√ n ). bit of dramatic history, the problem was solved √2πe f (x ) be the (n – 1)-dimensional volume of the completely and the answer is affirmative if and Asymptotic convex geometry is concerned with cross-section of B with a hyperplane perpendicular only if n ≤ 4 (for n ≥ 10, a cube and a ball give a to the axis passing through ( , , … , ). quantifying such high dimensional phenomena 0x1 x 0 0 counter-example). Proving that vol(K ) ≤ C vol(L) for arbitrary convex sets. One of the major Plainly, for a universal constant C would also solve ∞ breakthroughs was Klartag's central limit the slicing problem in the affirmative. Another f (x)dx= vol(B) = 1. theorem, which guarantees that the same ∫–∞ example concerns information theory: the phenomenon that we observed above for the The function is the probability density of the slicing problem is equivalent to showing that the f ball holds for an arbitrary convex set (properly volume distribution of the ball B in the direction of relative entropy per coordinate of every uniform normalised): the distribution of volume (function distribution on a convex set does not exceed the vector (1, 0, … , 0). What is this distribution Figure 2: Most of volume is in any slab and near the surface. f ) is Gaussian along most of directions. It was like? That cross-section is again a ball, whose a universal constant. The links with information explained by Antilla, Ball and Peresinaki that theory were also fruitful to establish unexpected radius is 2 2 (Pythagoras' theorem). √r –x such a central limit behavior comes from the Therefore, connections between the slicing problem and volume being concentrated in a thin annulus, n–1 other major open problems, such as the Kannan- 2 n–1 x 2 n– as readily observed above for the Euclidean r 2–x2 v = – — r 1 v Lovász-Simonovits conjecture (which in turn f (x)= √ n –1 (1 2 ) n –1 r ball. Klartag established estimates of this type, has its roots in devising efficient algorithms for

We check that r n– 1 v ~ for large and called thin-shell bounds, for an arbitrary convex n –1 ~√e n computing volumes of convex sets). when x is much smaller than r, we can write set K in n, showing that there are sequences 2 – 2 1 – (x/r) ~ e (x/r) and thus conclude that, δn ,εn such that ―> 0 as n ―> ∞ such that, approximately, |x| 1 n Further reading – ex2 vol ∈ 1– | εn ≤ 1– ≤1+ ε ≥1– δ f (x) ~ √ee π vol(K) x , √ n n n { } Giannopoulos, A., Milman, V., Asymptotic This means that f , the volume distribution in provided that K is properly normalised (its convex geometry: short overview. Figure 3: In a convex set in high dimensions the ball B is roughly Gaussian with variance σ of inertia matrix being the identity). Historically, Different Faces of Geometry, 87{162, Int. Math. 1 volume decays exponentially in almost every constant order ( σ = )! Ser. (N. Y.), 3, 2004. √2πe direction and concentrates in a thin shell. asymptotic convex geometry grew out

12 13 feature

well, did not have a large ego, deferred Hrusa added: “He is missed already. The the spotlight to others and avoided direct department is not the same without him.” disputations. He used his brash personality to charm both the administration and the faculty Greenberg's broad range of research to obtain the best for the department. In the interests included hyperbolic conservation naturally slow-motion university setting, Jim laws, fluid and solid mechanics, continuum shocked results into being by his insistence limits of particle systems, cellular automata on instant decision and subsequent action,” (CA) and metastable systems. said William Williams, professor emeritus Among his best-known work is the of mathematical sciences and a friend of Greenberg-Hastings CA, a two-dimensional Greenberg’s for more than 50 years. cellular automaton proposed as a model of Carnegie Mellon colleagues described excitable media named after his work with Greenberg as a strong leader, loyal and Stuart Hastings, who is now on faculty with colorful. David Kinderlehrer, alumni professor the University of Pittsburgh. The CA created of mathematical sciences, remembers the striking image that appears on the cover Greenberg talking with faculty members at La of this newsletter. Prima Espresso in Wean Hall. Hastings first met Greenberg at Case In Memoriam: “Everyone in the department had a personal Western in 1968 and became well relationship with Jim. He knew us, was acquainted at SUNY Buffalo. interested in us and was accessible,” “Jim Greenberg was outrageous — in Jim Greenberg Kinderlehrer said. the positive senses of that word: bold, unusual, startling,” Hastings said. “He James (Jim) M. Greenberg, former head of the Department of In September 2002, Greenberg stepped was outrageous in his generosity. To this Mathematical Sciences, died on August 3, 2018. He was 78. down as department head to accept the outsider at least, he appeared outrageous Greenberg was born July 30,1940, in Chicago, Illinois. role of associate director for mathematical sciences for the U.S. Office of Naval in his ability to get what he wanted for the Greenberg was at Carnegie Mellon at its start. As a fellow at the Research, Global and European Office of mathematics departments he chaired (two Mellon Institute of Industrial Research, he became an assistant Aerospace Research and Development. of them). He was outrageous in his professor of mathematics in 1967 when the institute merged He remained on the Carnegie Mellon openness and his non-compliance with with the Carnegie Institute of Technology to form Carnegie mathematical sciences faculty. He academic conventions.” Mellon University. He left in 1969 to pursue opportunities at continued to participate in CMU seminars Hastings also remembered Greenberg for other universities, including Case Western Reserve University, weekly in his retirement. his genuine interest in others. the State University of New York at Buffalo, the Ohio State University and the University of Maryland, Baltimore County, In addition to his administrative impact, “He could relate to all ages,” Hastings said. where he served as department head. Greenberg was widely known for his research “Some years ago, the young daughter of contributions. a colleague was overjoyed when Jim took He returned to Carnegie Mellon in 1995 when, under the over for her at a recalcitrant CMU Carnival leadership of Robert Mehrabian as university president, Bill Hrusa, professor of mathematical game and won her a prize. And during his Greenberg was selected from a field of competitive candidates sciences, first met Greenberg in final illness, around half a dozen grad school to head the mathematics department. the early 1980s. friends from over 50 years ago made the Greenberg emphasized the importance of faculty research “Jim was very interested in mathematical effort to visit or contact him.” and worked to reduce the teaching loads of research active models and worked in many different areas, Hastings added: “As my daughter, who faculty. He deployed a strategy for targeted faculty hiring that but one common theme was a strong knew Jim from her childhood put it, 'he developed the areas of research focus that are present in the relevance to real-world phenomena. He was bucked the stereotype of ‘mathematician’ department today. very good at striking a balance between finding models that were rich enough to be with consistent irreverence and humor.' He “Jim was very successful in his research but was an even better relevant and yet simple enough to obtain was smart, an innovative researcher, and fun academic administrator and program administrator. He listened results,” Hrusa said. to work with. I will miss him.”

14 15 i = 0 i = 3 i = 6 i = 9 i = 12 i = 15

The Greenberg-Hastings The state space for the model is range three box (i.e., the neighborhood of a point Cellular Automaton { 0, 1, … , k + 1} and the neighborhood of a x is the set of 49 points of l∞ distance at most site x ∈ 2 is x + N. State 0 is the resting 3 from x), the excitation threshold is t = 5, and by Tom Bohman and Janko Gravner state, state 1 is the excited state, and states there are 9 states (so there are k = 7 refractory 2, … , k + 1 are the refractory states. We states). This sequence begins with a random The image on the cover was generated begin with an initial configuration, configuration and shows how local stationary by the Greenberg-Hastings model. This periodic configurations spontaneously emerge and cellular automaton was introduced by Jim 2 f : ―> { 0, 1, … , k + 1} generate waves in the dynamics. Greenberg and Stuart Hastings as a simple 0 discrete model that generates the complex Given the configuration f , the configuration spatial patterns observed in excitable media i f is determined for all points in the integer such as nerve tissue and the Belousov i +1 The correspondence between the states and lattice simultaneously and in parallel. If x is Zhabotinsky chemical reaction. These colors in these image are indicated in this key. i = excited or is in one of the first – refractory 20 k 1 patterns are generally spontaneous, self- states then x proceeds to the next refractory black state 0 resting state exciting, spatially homogeneous oscillations. orange state 1 excited state state; to be precise, if f (x) ∈ { 1, … , k } then The fact that this simple model generates i yellow state 2 refractory state References f (x) = f (x) + 1. If x is in the last refractory such striking inspired interest i +1 i light green state 3 refractory state Robert Fisch, Janko Gravner, and in cellular automata as models of complex state at time i then it is in the resting state aqua state 4 refractory state David Grieath, Metastability in the at time + ; that is, if ( ) = + then light blue state 5 refractory state Greenberg-Hastings model. Ann. systems. We placed this image on the cover i 1 fi x k 1 ( ) = . A position that is resting lavender state 6 refractory state Appl. Probab. 3 (1993), 935-967. in honor of Jim Greenberg. fi +1 x 0 x becomes excited if a sufficient number of its dark blue state 7 refractory state Janko Gravner, Hanbaek Lyu and purple state 8 refractory state The classical Greenberg-Hastings model neighbors are excited. David Sivako, Limiting behavior of 3-color excitable media on arbitrary evolves in discrete time on the 2-dimensional (We also use colors to indicate the states in the graphs, Ann. Appl. Probab. 28 integer lattice. The model has three 1 ∈ : image on the cover, but in a slightly different if | { y x + N fi (y) = 1} | ≥ t (2018), 3324-3357. 2 f (x)=0 f (x)= parameters: a finite setN ⊂ , which i i +1 0 otherwise way. Black again indicates the resting state. The { James Greenberg and Stuart defines the neighborhood of the origin, a orange squares are again the excited states — Hastings, Spatial patterns for positive integer , which is the refractory k The sequence of images depicted above are these are raised on the cover. The refractory states discrete models of diffusion in period for the model, and a positive generated by Greenberg-Hastings using the are in successively darker shades as the refractory excitable media. SIAM J. Appl. integer t, which is the excitation threshold. following parameters. The neighborhood is a states progress toward the resting state.) Math. 34 (1978) 515-523.

16 17 alumni notes

Inspired by his family’s favorite television Consumer Electronics Show (CES) in Las show, Shark Tank, Kovscek pitched his Vegas and garnered widespread attention. children an idea — a device that could Conservation Labs was one of six companies attach to your water pipe to prevent leaks to receive the Innovation Award, which is and tell you about your water consumption sponsored by the Environmental Protection Mathematician for only $100. Agency and the Computer Technology Association, for their potential to reduce energy Over the course of the next three years, use and carbon emissions. Dives into the Kovscek devoted much of his time to researching sensor technologies to develop The company also caught the eye of a a noninvasive device that could fit over the representative from the Alexa-Techstars Startup World Accelerator, a 12-week immersive program outside of a water pipe while measuring the water flow inside the pipe. To reach the in Seattle, funded by Amazon and managed Alumnus Mark Kovscek has developed price point he wanted, Kovscek invested his by seed accelerator Techstars. Conservation a smart water technology through his energy in using machine learning to translate Labs was one of nine startups selected from company Conservation Labs sound signatures to accurate water flow among hundreds of applications. As part of the estimates, leak alerts and water insights. program, companies receive funding from the Customers can access the data and Alexa Fund and access to business strategy, the Mark Kovscek built his career on solving information through an app to save money investor community and Alexa-enabled devices other people’s problems, using his skills in and have peace of mind. to incorporate in their products and services. mathematics and analytics to address multi- billion-dollar challenges. But when it came to “H2know can tell you how long your showers Conservation Labs has reached a number of a problem in his own home — a leaky heating are, how many times the toilet was flushed, milestones this year, from debuting at CES system — a few drops of lost water became an how many times the washing machine was to conducting field tests and moving from unexpected wave of entrepreneurship. used. We can tell you about the whole house prototypes to beta units. While Kovscek is from that device, and we can detect micro excited for what’s to come, he won’t forget Kovscek is the founder and CEO of leaks down to a drop or two a second.” where everything started — in his basement, Conservation Labs, a company whose working as a team of one and holding company mission is grounded in both economic and The strength of H2know lies in its algorithms meetings at the dinner table before leaving his environmental conservation. He’s gearing up to and analytic capabilities, a strength that job as president of software marketing firm bring the company’s first product, H2know, to comes from Kovscek’s background in Velocidi to follow his dream. market next year. H2know is a low-cost, smart mathematics. He earned his bachelor of water monitor that aims to save consumers science in mathematics from Carnegie “It’s been a healthy dose of fear. The level of money by detecting water leaks in the home Mellon University in 1992, and he says he’s risk is really high for someone with a mortgage and helping them manage their water use. used math every day since. and a family of six to take care of, but it’s the excitement, the adrenaline rush of taking a risk It all started in late summer 2014 when the “A lot of what I do is problem-solving on this, on what I believe in, that is worth it,” Kovscek household began having higher than with mathematics. Whether it’s building Kovscek said. normal water bills. A leak in the family’s heating an algorithm through machine learning or system was causing water to go right down the building financial models, math has stuck H2know is only the beginning for Conservation overflow drain. These kind of every day leaks with me every day of my career in some way Labs. The next step will focus on creating a — a dripping faucet, a bad water valve, a leaky shape or form. I get to use my degree every shut off device that can turn your water off toilet — aren’t uncommon and can easily go day; it was an investment well made for me. automatically in the case of a catastrophic unnoticed at the owner’s expense. situation in the home. “My courses not only taught me the “H2know and Conservation Labs started as mathematics, they taught me how to think “We remain passionate about conservation and an intellectual curiosity — Is this a problem I and problem solve, and so there was always what this combination of sensor technology, can solve?” Kovscek said. When looking for a an added complexity that I really liked at machine learning and analytics can do for solution to prevent future leaks, the products CMU.” conservation,” Kovscek said. “If you think about on the market were incredibly expensive, well those three things and how they come together, over $1,500. “I thought there has to be a better In January 2018, Conservation Labs and the ideas are limitless.” way to do this.” H2know were officially unveiled at the

18 19 alumni notes

problems and combinatorial property testing. She is also a two-time winner Alumna Shafi Goldwasser of the Gödel Prize for outstanding papers in the area of theoretical Receives Honorary Degree computer science. Early in 2018, Goldwasser took over at Commencement as the director of the Simons Institute for the Theory of Computing at the Turing Award winner and Mathematical esteemed honorees embody this tradition, University of California, Berkeley. Sciences alumna Shafi Goldwasser received having received preeminent levels of distinction The institute is the world’s leading an honorary doctor of science and technology in their fields and exhibited a record of venue for collaborative research in at Carnegie Mellon̓s 121st Commencement extraordinary contributions to society.” theoretical computer science and on May 20. Goldwasser earned her brings together leading researchers bachelor's degree in applied mathematics Goldwasser is among the world’s elite and the next generation of young from Carnegie Mellon University in 1979. computer scientists. In 2012, she received the scholars to explore deep unsolved Association for Computing Machinery’s Turing problems about the nature and Goldwasser joined Nobel Laureate Ada Award, the highest honor in computer science limits of computation. She had Yonath, philanthropist and investment icon given to an individual for their contributions “of previously been the RSA Professor David Tepper and award-winning actor Ted lasting and major technical importance to the at the Massachusetts Institute of Danson in receiving honorary degrees. computing field.” Technology.

“It is a CMU tradition to award honorary Her pioneering contributions include the degrees to exemplary leaders, who serve as introduction of probabilistic encryption, Image: Shafi Goldwasser speaking role models for our graduates and the entire interactive zero knowledge protocols, at the Mathematical Sciences Carnegie Mellon community,” said CMU elliptic curve primality testings, hardness diploma ceremony during President Farnam Jahanian. “This year's of approximation proofs for combinatorial commencement in May.

20 21 student news

from scratch. Along with six close friends, he established the Carnegie Mellon University Informatics and Mathematics Competition. The annual one-day competition brings high schoolers from across the country to campus to compete in a number of subject tests, written by Kwon and the competition’s organizing team. “It’s something that I’m very proud of Andrew because it has really grown,” Kwon said. Participation has nearly doubled every year for the past three years, Zucker’s starting with 120 competitors in 2016, 230 in 2017 and 400 in 2018. Love of Math Kwon, who plans to pursue a career in Some of the competition’s former academic research in number theory and participants are now students at Takes Him Junior algebraic geometry, said he is excited Carnegie Mellon. “They’ve told me to have his potential as a scholar and ‘your competition made CMU really to the City Andrew researcher recognized through the stand out to me and it’s why I’m here scholarship. now,” Kwon said. “And I think that’s really incredible to be able to have that of Love Kwon Most of his undergraduate research has kind of impact.” focused on number theory and additive combinatorics, a branch of number theory Throughout the scholarship process, Andrew Zucker can’t remember a time Recognized that studies how sets of integers behave Kwon worked closely with Carnegie when he didn’t love math. Growing up in under addition. Last summer, through the Mellon University’s Fellowships and Greeley, Colorado, pursuing an education as Research Experiences for Undergraduates Scholarships Office. “Andrew is and career in mathematics just seemed program funded by the National Science both extremely talented and also natural to him. Foundation, Kwon conducted research very humble. That is an especially When he began his undergraduate studies Goldwater under Joe Gallian at the University of powerful combination,” said Stephanie at the California Institute of Technology, Minnesota, Duluth, on a newer line of inquiry Wallach, assistant vice provost for Zucker initially thought he wanted to study in additive combinatorics. Kwon examined a undergraduate education. “He has Scholar combinatorics, but quickly found that his recent mathematical paper that showed the used his time at Carnegie Mellon and heart was in logic. existence of minimal additive complements elsewhere to deepen his understanding Mathematical Sciences honors student in special cases. He developed a conjecture of mathematics and to build “When I took my first course in logic, Andrew Kwon received a 2018 Barry that identified a new, relatively general relationships with faculty mentors. We it seemed to use all the parts of Goldwater Scholarship. Given by the Barry property of complementable sets that have enjoyed the chance to work with combinatorics that I liked while putting Goldwater Scholarship and Excellence in provides insight into what distinguishes him and we are very pleased that he everything into a beautiful theoretical Education Foundation, the award supports complementable sets and sets that are received this prestigious scholarship.” framework,” said Zucker. undergraduate students interested in not complementable. Kwon submitted pursuing research careers in the fields of his results for publication in a professional Taking the time to write about his As an undergraduate, Zucker made his science, engineering and math. journal, and, if justified, his conjecture could research and interests, personally and first contributions to the field of topological explain all observed phenomena in this area intellectually, helped Kwon prepare dynamics, publishing in the journal Kwon was one of 211 students to receive of study. for the road ahead in applying for Fundamenta Mathematicae. the scholarship, selected from among 1,280 graduate school and fellowships. “The sophomores and juniors nationwide. The Working with younger students is another process took a lot of self-examination After graduating from CalTech, Zucker scholarship provides up to $7,500 per year passion of Kwon’s. As a first-year student, because it was the first time that I had came to Carnegie Mellon’s highly regarded for tuition, fees, books and room and board he took on the ambitious goal of starting a to think about the math I liked doing logic program to continue his studies. for up to two years. math competition for high school students and why it’s worth doing,” he said. Under the tutelage of Associate Professor

22 23 student news

Math Club Wins Clinton Conley and Professor Top Awards Among James Cummings, he researched the interactions between logic, dynamics and descriptive set Blitz Booths at theory. As a doctoral student, he continued to make impressive Spring Carnival contributions, particularly refining the connections between Ramsey theory and topological dynamics. This year’s Spring Carnival brought to life Myths and Legends throughout history. The Carnegie “Andy’s work around metrizable Mellon Math Club combined both magic and universal minimal flows and the historical legend as it transported Carnival-goers generic point problem are his most back to a time of castles, kings and math- impressive contributions to the field, ematical sorcery with its award-winning booth, but they are not his only ones,” Merlin’s Quest. said Conley. “In general, Andy has a very strong research program The design was based around scenes from the connecting combinatorial regularity lessons that Merlin taught the young King Arthur, principles with their dynamical which took first place among all “blitz” booths — counterparts and had remarkable booths that are contained in an 8’x18’ plot. The success advancing this program group also won the environmental award among while still a graduate student.” blitz booths for their attention to energy efficiency, recycling and reusing of building materials. His work was so impressive, Conley said, that it caught the Upon entering, visitors encountered Arthur’s first attention of international experts meeting with Merlin at his hut in the forest. From in the field. When attending a set there, the surrounding walls laid out four “quests,” theory conference at the Centre as introduced by Merlin’s owl Archimedes, that International de Rencontres Arthur must complete on his quest to become Mathématiques near Marseille, king. Visitors could help by solving a few France, Conley heard Zucker’s mathematical riddles along the way. Tasks included theorems referenced in two determining relationships within a flock of geese as separate talks. friends or strangers, taking on Zeno’s paradox and decrypting a secret message using a scytale. The After completing his Ph.D. program journey ended with the ultimate kingly challenge in the spring, Zucker moved to — pulling a sword out of the stone, or in the Math Paris. His graduate work earned Club’s case, pulling a sword out of a box to open him a prestigious National Science the lid and collect a small prize. Foundation post-doctoral fellowship to study at the Université Paris “Some of our favorite visitors were those that came Diderot, which has a large logic in and thoughtfully did our activities, sometimes group and brought him closer to for as long as an hour,” said Elliot Haney, collaborators in Paris, Lyon and mathematical sciences senior and Math Club Marseille, France. booth chair. “We also enjoyed seeing people’s reactions to our booth. On the back wall, we had “Mathematically, I feel like I’ve been a fire extinguisher and a light switch. As a joke, we more productive than ever, and I get wrote above them ‘some gifts Merlin brought from to explore France and elsewhere in the future.’ To our surprise, that was one of the Europe, too,” said Zucker. most popular additions we made.

24 25 student news Summer 2018 Math Meeting of the Minds SURF Research Projects Poster Competition

First Place — David Simmons Prize for David Altizio, Xinyu Wu and Undergraduate Research in Mathematics Taisuke Yasuda Zachary Singer Asymptotic Stability of the Sparse Polynomial Approximations Faraday Wave Problem with Unknown Random Sampling Advisor: Ian Tice

Second Place Bryan Ding and Zilin Wang Matthew Bowen Constructing Local Volatility Surfaces to Price Options The Sprague-Grundy Function for Advisor: Elnur Emrah Some Selective Compound Games

Wenxin Ding Third Place (Tie) Does Gaussian Noise Increase Entropy? Sidhanth Mohanty (CS major) Advisor: Tomasz Tkocz Sum-of-Squares Refutation Threshold for Regular SORT_4 Instances Shuyang (Serena) Gao and International SURF with Novdano Dede Yusuf Nicholas Sieger Assessment of Discrete and Continuous Cycle Double Covers Time Models for Pricing Options Senior, Alp Müyesser Advisor: David Handron Young Researcher Award Zizhuo Chen, David Xu, Yuzhi Guo “Math is largely about finding patterns,” “If a system you're looking at is large Varun Gudibanda and Anni Huang said Carnegie Mellon University senior enough, orderly substructures will form,” Dynamics of the Inextensible Inverted Utility Optimization in the Agency Problem Necati Alp Müyesser. The patterns of Müyesser said. Flag with Piston-Theoretic Forcing Term this undergraduate̓s life have taken him Advisor: Jason Howell from Turkey to Berlin to Pittsburgh, and One highlight of Müyesser's undergraduate from a completely separate college to the career at Carnegie Mellon was his time as Yong Gun Choe Other Math Major Award Winners: Department of Mathematical Sciences. an International Summer Undergraduate Hidden Markov Model and Its Research Fellow. The program funds Applications to Stock Price Prediction First place in the Advisor: Yu Gu When Müyesser, a native of Istanbul, roughly two months of full-time research Statistics Poster Competition entered Carnegie Mellon, he did so as under a faculty member in a foreign country. Chi Fang Zhiyang He Müyesser's ISURF grant took him to the Free a student in College of Engineering's Predicting Delay in Freight Transportation Department of Electrical and Computer University of Berlin, where he said he got to Hypergraphs with Few Berge Paths of Fixed Length Between Vertices through Socioeconomic Events Engineering. meet many well-known mathematicians. “It Advisor: Michael Tait (Joint with Yudi Jin, Frank Kovacs was a good simulation of what my graduate and Maria Rodriguez de la Cruz) “I realized quickly it wasn't for me.” school experience will be like,” Müyesser said. Ruoyuan Liu and Yuepeng Yang

Instead, through his first-year classes, As a senior, Müyesser is looking forward to Recognizing Mesh Structure in Images Dietrich Humanities Award Advisor: Dejan Slepčev he found himself drawn to mathematics and continuing in academia and mathematical Ruth Scherr particularly a field known as Ramsey Theory. research. "I'm confident that I will want to do (Image on left page) Laborious this for a long time," he said. “The catchphrase for the field could be Jung Joo Suh Honorable Mention in the ʻcomplete chaos is impossible,ʼ” Müyesser Along with rock climbing, one of Müyesser̓s Line Sections and Graph Homomorphisms for Graphs Induced by Statistics Poster Competition described. “It is completely opposed to favorite extracurricular activities is just walking Actions of the Integer Lattice Qiuyu Wang anything I would have thought of as math around the campus and the city, he said. Even Advisor: Clinton Conley Assessing the Performance of the False from my high school background.” then, math is never far from his mind. Discovery Rate Procedure in Detecting Cointegrating Stock Pairs Essentially, Ramsey Theory looks at the “It's a good way to do homework,” Müyesser Emily Zhu (Joint with Emily Chen, Christian Manaog conditions that mean "order" that must exist noted of his time walking, in which he can Multicolor Ramsey Numbers for and Boyan Zhang) in a certain set, whether that be a group of think through equations in his head. “It̓s Small Hypergraphs numbers or even a collection of people. easier not to get distracted.” Advisor: Tom Bohman

26 27 Classclass of of 2016 2018

Joseph Briggs Receives Graduate Student Teaching Award

Joseph Briggs was honored with the “He just loves math and gets so excited Mellon College of Science’s 2018 Hugh about math concepts and teaching others,” Young Graduate Student Teaching Award. Filiere wrote. “He would solve the problem and be jumping all around, pumped up at “Joe Briggs embodies everything one how it is that we just solved this problem! To could possibly want in a [teaching see a TA so passionate about math naturally assistant],” Associate Teaching Professor makes you more excited about it as well.” Deborah Brandon wrote in a letter nominating Briggs for the award. “He is a Briggs graduated in 2018 with a thesis gifted expositor. He is extremely reliable on the subject of online algorithms and and very helpful, and he really cares about extremal structures. He is now a post- his students. In addition, he has that rare doctoral fellow in mathematics at the ability to make the learning experience Technion Israel Institute of Technology. memorable (in a good way).” Senior April Li Receives In his nearly five years at Carnegie Mellon Image: Fugassi and Monteverde Award University, Brandon said that Briggs̓ Rebecca Doerge, the Glen de Vries Dean of the flexible teaching style allowed her to assign Mellon College of Science, presenting the MCS April Li received the 2018 Dr. J. Paul him to teach alongside many different Graduate Student Teaching award to Joseph Briggs. Fugassi and Linda E. Monteverde Award. instructors in a variety of courses. The award recognizes the graduating female senior in the Mellon College of “I have never had a TA that I would Science with the greatest academic trust more than Joe to elegantly explain achievement and professional promise. of Mathematical Sciences hosted an mathematical material to undergraduates,” inaugural “Women and Mathematics at Associate Professor Wesley Pegden said As part of the Department of Mathematical CMU” conference that included a mini- in a letter supporting Briggs̓ nomination. Sciences’ Computational Finance course on connections between partial program, Li was a dedicated student differential equations and mathematical A dozen of Briggs̓ former students wrote leader who enriched the experiences and finance. Li helped organize the conference in to confirm his teaching abilities. opportunities for students in the program and served as a panelist. and across campus through her passion “Before an exam, I would take advantage for computational finance. In 2016, Li In addition to her student organization of his office hours to ask him several co-founded the Quant Club, which has leadership, Li was an accomplished questions about the material,” sophomore been an invaluable resource for students teaching assistant and undergraduate Shruti Murali wrote. “He answered all of interested in quantitative finance. researcher. She conducted extensive them and was always excited when I had undergraduate research on utility more questions because he̓s just that In her role as a leader in the Quant Club, indifference pricing in incomplete financial kind of instructor — always going above Li helped organize mock interviews on models with Professor of Mathematical and beyond to make sure his students campus in fall 2017. She conducted Sciences Bill Hrusa from June 2015 understood the material.” resume workshops for students, where through May 2016 and served as a she used her own experience in finding teaching assistant in two of his courses. Junior Makayla Filiere echoed that internships to help students prepare for sentiment about Briggs̓ energy and recruiting season. “April is one of those extremely rare passion for instruction. individuals who excels at everything she Li was also president of the Sales and does. She is creative, her analytical ability Trading Academy at Carnegie Mellon and is extremely high, she is very skilled at a member of the Mortar Board Society of coding, she has excellent mathematical Scholars. In April 2018, the Department intuition, she understands the interplay

28 29 Classclass of of 2016 2018 Class of 2018 between mathematical models and Destinations real-world phenomena and she is an outstanding communicator,” said Early on, Brakensiek developed an interest Hrusa. “I consider April to be more of a in theoretical computer science, an area colleague than a student.” Firms that hired more than one member of the class: that seamlessly connects his interests in Li’s potential for professional success mathematics and computer science. He was prominent since early in her worked closely with Professor of Computer undergraduate career. Following Science Venkatesan Guruswami during his her sophomore year, Li landed an four years at Carnegie Mellon on several internship with Tudor Investments as projects in coding theory, computational a quantitative analyst in the global complexity theory, producing new results in research department. It is unusual for constraint satisfaction and approximation a sophomore to get their foot in the algorithms. door so early. Internships in the finance His work resulted in several publications, industry, which often lead to full-time job manuscripts and presentations at major offers, are generally available to juniors computer science and mathematics and seniors who can begin working Joshua Brakensiek̓s conferences and symposia, including the within a year of finishing their internships. Undergraduate Success Leads Symposium on Discrete Algorithms, APPROX Li managed a similar feat her junior year to NSF Graduate Fellowship and RANDOM. by applying for an internship at four During his time at Carnegie Mellon University, “I am thankful to Carnegie Mellon, major financial institutions — Goldman mathematical sciences graduate Joshua especially the Mellon College of Science, Sachs, JPMorgan Chase, Bank of Brakensiek kept busy. He was a Goldwater for giving me the freedom and opportunity America and Citigroup. She had offers Scholar, a Putnam Fellow and a member to explore research in such depth as an from all four banks before mid-semester. of the Carnegie Mellon Putnam Team undergraduate,” Brakensiek said. She accepted Goldman Sachs’ offer that placed first in 2016. He conducted as a sales and trading summer analyst astrostatistical research with faculty During his senior year, Brakensiek continued intern in their securities division. working in statistics and cosmology and to receive national awards and accolades theoretical computer science research with for his research and academic performance. Since graduation, Li is continuing at faculty in the School of Computer Science. The Computing Research Association Goldman Sachs as a sales and trading Doctoral Programs: He published papers and presented at awarded Brakensiek one of four Outstanding analyst in interest rates products. She conferences around the world. All while Undergraduate Researcher Awards for his is part of a new breed of traders who working toward joint bachelor’s and master’s outstanding potential in computing research. make decisions based on the use CalTech, Applied Math degrees in mathematical sciences through of sophisticated mathematical and the Department of Mathematical Sciences’ Following a national competition, Brakensiek statistical models and who employ Chicago, Math Honors Program. was one of 2,000 awardees, selected from ideas from data analytics and machine 12,000 applicants, to receive a Graduate learning, together with an understanding Duke, Math “I am exceedingly thankful that I was able to Research Fellowship from the National of financial markets and economics to call Carnegie Mellon my academic home for Science Foundation. The fellowship is helping guide their trading decisions. Illinois, Math (2) four years. The innovative opportunities I had to fund his graduate studies at Stanford access to changed my life,” said Brakensiek, University as he pursues his Ph.D. in “The scientific education that I have Minnesota, Economics who graduated this past May. theoretical computer science. received at CMU will allow me to make more informed trading decisions,” Li Stanford, Computer Science Brakensiek came to Carnegie Mellon “Looking back, I am amazed at how well said. “In particular, the training that I have UCLA, Biostatistics University as part of its Knaster-McWilliams Carnegie Mellon equipped me to pursue received in the BSCF program equips Scholars program, which is one of only a graduate studies in my field. As soon as me to have a solid understanding of and UCSD, Math few scholarship-supported programs in the I entered Stanford’s program, I felt that I to be mindful of the risks associated with United States that pairs an honors program had much to contribute due to the many trading complex financial instruments.” Wisconsin, Math with increased access to faculty and early professors, mentors and friends who had research opportunities. helped me at CMU,” Brakensiek said.

30 31 Mathematics is essential to the future of science. It is the backbone of computer science, machine learning, artificial intelligence, engineering and data science. These fields, when combined with the foundational sciences, will lead to tomorrow’s life- changing and breakthrough scientific discoveries. Invest in us as we prepare the next generation of mathematicians to make an impact. Please consider directing your gift to one of these priority funds: To make a gift, visit cmu.edu/math/giving Summer Undergraduate Research Endowed Fund For more information or to have a confidential Gifts to this fund will enable students to undertake discussion about your in-depth research during the summer months. philanthropy, please contact the Mellon Innovation Fund for College of Science Mathematical Sciences Development staff: Gifts to this endowed fund support activities that enrich the undergraduate experience for mathematics majors and faculty recruiting.

Steven Shreve Endowed Fund for the Mathematics of Finance Gifts to this fund will support research and educational programs focused on the mathematics of finance, including funding for faculty recruitment and LISA GEORGI retention, graduate student and postdoctoral scholar Director of Development support, summer undergraduate research, the Quant 412-268-2019 Club and student travel. [email protected]

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