MATHEMATICAL SCIENCES NEWS 2020 MATHEMATICAL SCIENCES NEWS CONTENTS
Department of Mathematical Sciences The Mathematical Sciences News is published yearly by the Department of Mathematical LETTER FROM DEPARTMENT HEAD, 03 Sciences at Carnegie Mellon University for its TOM BOHMAN students, alumni and friends to inform them about the department and serve as a channel of communication for our community. Readers with comments or questions are urged to send them to the Mathematical Sciences News. The department is headed by Tom Bohman. Editor-in-Chief Tom Bohman, Department Head FACULTY NOTES 04 Editor Jocelyn Duffy, Associate Dean for Communications Contributing Writers Tom Bohman, Boris Bukh, Dan Carroll, Jocelyn Duffy, Ben Panko, Emily Payne, Colleen Storm Publications Managers Emily Payne, Communications Manager
Rachel Keeney, Publications Manager and Graphic Designer RESEARCH ROUNDUP 10 Photography & Images Jonah Bayer, Digital Producer Courtesy of Carnegie Mellon University, unless otherwise noted Online Magazine Gigi Wiltanger, Web Manager Carnegie Mellon University Department of Mathematical Sciences Wean Hall 6113 5000 Forbes Avenue Pittsburgh, PA 15213 FEATURE 16 1 1 www.cmu.edu/math _3 4
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Carnegie Mellon University does not discriminate in admission, employment or administration of its programs or activities on the basis of race, color, national origin, sex, handicap or disability, age, sexual orientation, gender ALUMNI NEWS 18 identity, religion, creed, ancestry, belief, veteran status or genetic information. Furthermore, Carnegie Mellon University does not discriminate and is required not to discriminate in violation of federal, state or local laws or executive orders. Inquiries concerning the application of and compliance with this statement should be directed to the university ombudsman, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, telephone 412-268-1018. Obtain general information about Carnegie Mellon University by calling 412-268-2000. STUDENT STORIES 20 Visit the online version of the Mathematical Sciences Newsletter for additional content. magazine.mcs.cmu.edu/math/
DEPARTMENT OF MATHEMATICAL SCIENCES 1 LETTER FROM MATHEMATICS DEPARTMENT HEAD, TOM BOHMAN
This has been a challenging year due to the COVID-19 pandemic, but I am happy to report that the department nimbly adapted to the new reality and responded in a number of productive ways. After evacuating campus in March and moving all classes online, faculty developed teaching innovations tailored to the new learning format, and we believe that we are delivering a high-quality educational product. While we hope to gradually move operations back to campus as soon as we are able and we believe that in-person learning is generally superior to its online counterpart, some of lessons learned over the last few months will impact and improve our course delivery even after we are teaching fully in-person again. In addition to the adaptation of our course delivery, math faculty launched research and entrepreneurial efforts in response to COVID-19. Po-Shen Loh developed and deployed the COVID-19 tracking app, NOVID, which uses ultrasound to detect anonymous positive cases within your network of recent contacts. The app can discern your social proximity to other NOVID users and the length of time of your interaction with them. NOVID works through self- reporting but does not collect personal information. For more details see the story on page 6. Wes Pegden and Maria Chikina (Chikina is faculty at the University of Pittsburgh) developed a novel COVID-19 public health response strategy informed by classical mathematical modeling of the pandemic. Two key features of their approach were recognizing that people would not be able to remain quarantined indefinitely, and that the virus would still be present even after the first wave of infections passed. Thus, any strategy has to account for the impact of subsequent waves of infection. Neither one of these notions were prevalent in discussions during the early days of the pandemic when Chikina and Pegden released their potentially life-saving study. You can read more on page 7. Many of our students had their summer plans canceled due to the pandemic. The department responded by increasing our summer course offerings and expanding our traditional summer research program (SURF). We also introduced the Summer Experiences in Mathematical Sciences (SEMS), a new ABOUT THE COVER slate of summer mathematics activities for our majors. The department has been developing a culture of engagement with undergraduate research over The image on the cover depicts a closed curve in three- the last several summers. This put us in an excellent position to develop the dimensional space and a parallelogram that is inscribed SEMS program on very short notice. For details see pages 10-13. by this curve. Despite the COVID-19 upheaval, the department continues to produce In recent work that was published in Forum of impressive research results. This issue of the newsletter features recent work Mathematics, Sigma Assistant Professor Florian Frick in geometry, which is a significant interest of some of the faculty who have and his co-authors proved that every close curve in the joined the department in the last few years. Much of this work is in discrete three-dimensional space inscribes such a parallelogram. and convex geometry, an area of mathematics that features not only a wealth This resolves a nearly 50-year-old conjecture of of beautiful classical problems but also close and deepening ties with coding Hadwiger and is closely related to the square peg theory and data science. problem, which is a notoriously difficult open problem We extend our deepest appreciation and gratitude for your continued that was originally posed by Toeplitz over 100 years ago. support and engagement with the Department of Mathematical Sciences. For more details see the discussion on page 15. Likewise, we would like to hear from you! Please keep the department updated by contacting us directly or joining our LinkedIn group: https://www.linkedin.com/groups/12371860.
2 MATHEMATICAL SCIENCES NEWS LETTER FROM DEPARTMENT HEAD 3 FACULTY NOTES
CENTER FOR NONLINEAR ANALYSIS RESEARCHERS RECEIVE NSF GRANT TO STUDY TOPOLOGICAL DEFECTS
Researchers from the Department of Mathematical Sciences’ Center for Nonlinear Analysis (CNA) have received a grant from the National Science Foundation to better understand the ubiquitous phenomena associated with topological defects. The interdisciplinary team will aim to bring together knowledge from the fields of engineering, physics and mathematics to create a unifying framework for describing and understanding topological defects and the phenomena associated with them. These phenomena occur when two uniform patterns of organized matter — from a jet turbine blade to a galaxy — are offset from one another, rotated differently or otherwise have some form of deformation on the surface at which they meet. The crack or curve created where the joining surface ends and the two patterns separate is called a topological defect, and depending on the situation and material, its effect can range A Familiar Face from interesting to catastrophic. Mathematical Sciences Ph.D. “Our team envisions a future where new graduate Clive Newstead returns knowledge from our research can be to the department as an assistant leveraged to seed advances in technology teaching professor relevant for grand challenges in seismic forecasting, infrastructure renewal and energy-efficient transportation,” the After Clive Newstead received his Ph.D. so it was nice to be able to interact with them researchers wrote in their proposal. in mathematical sciences in 2018 from and show them that I'm a real human being Carnegie Mellon University, he joined the after all,” said Newstead. Through mathematically defining how Department of Mathematics at Northwestern topological defects agree and differ within This fall, Newstead officially joined the University as a lecturer, teaching courses these different materials, scientists and department as an assistant teaching in multivariable calculus, linear algebra and engineers in civil engineering, advanced professor. He’s teaching two courses abstract algebra. manufacturing, astronomy, seismology and remotely, Concepts of Mathematics — the countless more fundamental fields will have Shortly after, he returned to Carnegie Mellon very course he wrote the textbook for — and a universal tool for defining one of the most — this time as a faculty candidate. His visit to Multivariable Calculus and Vector Calculus ubiquitous phenomena in our physical world. campus certainly did not go unnoticed. for Engineers. The team of researchers is led by Professor Newstead was a bit of a legend among Newstead’s research is in category theory of Civil and Environmental Engineering students. Not only had he been a highly and its connections with logic. “I like to think Amit Acharya and includes CNA Director respected teaching assistant, winning about category theory as the ‘mathematics and Kavčić-Moura University Professor both the Mellon College of Science’s and of mathematics,’” Newstead said of his work. of Mathematics Irene Fonseca; Assistant university’s Graduate Student Teaching “Whereas mathematics tries to identify patterns Professor of Mathematical Sciences Franziska Award, but he also wrote a free textbook, “An and structures in the real world and make Weber; Professor of Physics Ira Rothstein; Infinite Descent into Pure Mathematics,” that sense of them abstractly, category theory does Emeritus Professor of Physics Bob Suter; became the main text for two courses in the the same thing with mathematics itself.” department. Hamerschlag University Professor Emeritus A native of Yorkshire, England, Newstead of Civil and Environmental Engineering During his interview, many students attended likes cooking hearty British dishes. Outside of Jacobo Bielak; Assistant Professor of Civil and Disclinations and interfaces in a solid (Top to Bottom): his job talk and one student even asked him research and teaching, he enjoys biking and Penta-twin, Finite strain stress field of a penta-twin, Stress- Environmental Engineering Jerry Wang; and to sign a copy of the front cover. relaxing with a good dose of reality TV and is in shielding of penta-twin by dislocations. researchers from the University of Arizona. the midst of planning his upcoming wedding. “A lot of students only know me as the Dan Carroll mysterious figure who wrote their textbook, Emily Payne
4 MATHEMATICAL SCIENCES NEWS FACULTY NOTES 5 Visit the online magazine to watch a walkthrough of the NOVID app
to another and anonymously records the reflecting off ceilings without significantly distance between users. It later notifies a user affecting distance measurement. if they have come in contact with a person Since launching, the app has been featured who just reported that they tested positive or a in Wired, Forbes, Fast Company and the person who has been in contact with someone New Yorker. NOVID’s user-friendly interface who reported that they tested positive. STOP allows users to follow their social interactions Another way NOVID is different from many and see how well they are playing their part THE SPREAD other apps is that it doesn’t rely solely on to stop the spread. Bluetooth to measure distances between With its early warning network, you can smartphones. Instead, NOVID augments PO-SHEN LOH DEVELOPS NOVID APP see how many users in your network you’ve Bluetooth with ultrasound — soundwaves TO TRACK CORONAVIRUS WITHIN met through direct and distant contact and with frequencies just outside the range of YOUR USER NETWORK receive a notification if any user you’ve been human hearing, which modern mobile phones in contact with has self-reported a positive can emit and record. By accurately measuring COVID-19 diagnosis. NOVID has also shown As the world continues to focus on slowing the time sound takes to travel, ultrasound it avoids false exposure alerts with 99.6% the spread of COVID-19 and determining can more accurately measure the distance accuracy — to date, NOVID is the only COVID how to safely resume everyday life, a team between devices. app in the world that displays the estimated of designers, mathematicians, computer distance from other devices. scientists and physicists, led by Mathematical “I had an idea of how to help Emily Payne Sciences Professor Po-Shen Loh, developed NOVID — an innovative new app designed to the world return to normalcy.” help track the spread of COVID-19. It is widely known that Bluetooth is unable MATHEMATICAL SCIENCES PROFESSOR Charts: Infection and ICU usage rates under various NOVID presents a new paradigm in exposure to accurately calculate distances and can PUBLISHES COVID-19 MODELING modeled scenarios for age-targeted COVID-19 mitigation. notification: it gives users a heads-up when incorrectly register interactions between someone in their user network, with up to 12 people in different rooms when signals pass Research from Carnegie Mellon University degrees of separation, has tested positive for through walls or ceilings. Ultrasonic signals and the University of Pittsburgh published the coronavirus. improve accuracy because they can also in the journal PLOS One modeled how be detected through bags “I had an idea of how to help the world return age-targeted strategies to mitigate the and pockets and can go to normalcy,” said Loh. “And the Carnegie transmission of COVID-19 could lead to around obstacles by Mellon and broader Pittsburgh community better final outcomes in mortality and had everything I needed to start making that hospitalization from the novel coronavirus. idea into reality.” “COVID-19 has a striking age-dependent Using social media and crowdsourcing, mortality rate,” said Maria Chikina, an Loh brought together a team, assistant professor at the University of including many CMU students and Pittsburgh Medical School. “An infected alumni working in mathematics, 75-year-old is 1,000 times more likely to die human-computer interaction, than an infected 15-year-old, and an infected app development and information 55-year-old is 20 times more likely to die security — all areas where CMU excels than an infected 25-year-old.” — to develop the free app. Unlike many COVID-19 apps in development or in Chikina co-authored this study with Associate use, NOVID is completely anonymous. Professor of Mathematical Sciences Wesley The notification system doesn’t Pegden to dig into this “unusual characteristic” ask for personal information. of the disease. Using a standard “Susceptible, Instead, each user is assigned Infected, Removed” epidemiological model, a random number, which the pair sought to model the dynamics of is used when their phone how often people in various age groups have contact with each other, how that contact senses and communicates strategies were more effective compared to affects the transmission of COVID-19 and with other smartphones. homogeneous strategies targeting all age how adjusting those contacts would affect the The app also doesn’t store groups equally, in situations where population impact of the pandemic. GPS locations. immunity plays a role in controlling the NOVID allows people to The researchers concluded that heterogenous epidemic. In particular, their modeling shows anonymously self-report if they have strategies focusing on reducing transmission that age-targeted mitigations can allow tested positive for COVID-19. The app of COVID-19 to and among the highest-risk immunity to contribute to epidemic control periodically senses if one user comes close age groups are most effective at reducing even while significantly reducing the burden of mortality and intensive care unit usage. These infection among at-risk groups.
6 MATHEMATICAL SCIENCES NEWS FACULTY NOTES 7 Zheng Family Postdoctoral Fellowship
he postdoctoral program in the Department of Mathematical Sciences offers fertile ground for budding researchers to flex their scholarly muscles. We are Texcited to announce a new endowed position to our growing department: the Zheng Family Postdoctoral Fellowship. The Zheng Family Postdoctoral Fellowship is a generous gift made by Fang Zheng, a longtime admirer of mathematics and a parent of a Carnegie Mellon mathematics student. Zheng is the managing director and chief investment officer of Keywise Capital, an asset management company based in Hong Kong. He regularly hosts Carnegie Mellon faculty, alumni and events in Hong Kong. An engineer by training, Zheng understands the need for strong mathematics research and its applicability to all aspects of human life. Without a solid mathematical foundation, many of society’s innovations would not be possible. Providing such a strong foundation requires extensive preparation and experience. An important element of curating mathematics excellence is through robust postdoctoral training. Once a scholar has completed their dissertation, a postdoctoral position affords them the space and time to invest in their research and gain valuable experience teaching and collaborating with a large network of colleagues. The Zheng Family Postdoctoral Fellowship adds prestige to the department by joining the Zeev Nehari Visiting Assistant Professor position. Together, these named positions offered by the Department of Mathematical Sciences add stature and a competitive edge to the department. The department offers several avenues for postdoctoral recruitment and training. These opportunities range from funding through individual faculty members’ grants, positions supported by income from the Masters of Science in Computational Finance program, general postdoctoral positions funded by the department itself and postdocs affiliated with the Center for Nonlinear Analysis (CNA). The CNA has a long history of garnering research funding that William Chan, Zheng Family Postdoctoral Fellowship recipient Fang Zheng, Managing Director & Chief Investment Officer, Keywise Capital supports postdocs, who bring new expertise, perspectives and energy to the department. As new researchers to the field, our postdocs regularly engage in interdisciplinary and multidisciplinary projects at Carnegie Mellon and with other institutions. This strengthens the reputation of the department nationally and internationally and is conducive to building relationships across institutions, which is necessary to continue working on important and applicable research. William Chan is the first recipient of this fellowship. Graduating with a Ph.D. in mathematics from the California Institute of Technology in 2017, Chan accepted a National Science Foundation Postdoctoral Fellowship and Research Assistant Professor position at the University of North Texas from 2017-2020. His research, while far-reaching, is centered in descriptive set theory. Topics of interest range from continuity and partition properties at small uncountable cardinals to analysis of Borel equivalence relations. Throughout his research, you can find traces of the application of determinacy in the context of games. The Fang Zheng Family Postdoctoral Fellowship will deepen the impact of our postdoctoral program for many years to come.
Colleen Storm
8 MATHEMATICAL SCIENCES NEWS FACULTY NOTES 9 RESEARCH ROUNDUP Image from the SURF project of Minsung Cho. expecting to have internships help offset the cancellations has been critical in figuring in areas like math finance in other programs due to out my career path, as the and software engineering COVID-19. Obviously, we had research I’ve done with him the opportunity to stay to forgo the usual in-person always involved practical engaged in mathematics over group research meetings and applications of math and the summer. Many of the my students were spread bits of machine learning. It’s participants described it to be across the country, but I largely due to my research a meaningful and enjoyable was very impressed at how with Professor Schaeffer that I way to spend their time. well they managed to work feel like I know what it means together as a group by using to be an applied math major,” “I applied to the research online collaboration tools,” said student Hannah Milano. program because I really said Associate Professor Ian wanted to see how math Participants also found SEMS Tice. “It’s always a pleasure to research is conducted, and to lay insightful groundwork watch my students grow as what questions people for more rigorous research. researchers over the course of think are interesting,” said the summer program. Getting “I'm glad I did the program mathematical sciences major to continue this tradition because I used to be really Xinjie He. He continued, “I during the chaos of the intimidated by math research, was also introduced to a new pandemic was a much needed but now it seems more area and frontier questions morale booster for me.” approachable, and I feel like in math, and some advanced I know a little bit more about techniques that I may not be In line with departmental what I can expect I will be able to learn in classes.” goals, the research doing when I do research for A Proliferation projects were intellectually my master's thesis,” junior diverse and included both “It's always a Carolyn Lee said. of Summer applications of mathematics pleasure to watch and pure mathematics Though the summer of 2020 my students grow research. While some brought about many changes Undergraduate as researchers over students were concerned in procedure for Carnegie with gaining research Mellon, the Department of ummertime in The SEMS program is an the course of the experience for potential Mathematical Sciences found Research Pittsburgh traditionally exciting new addition to summer program. careers in academia, others a way to use it to provide Sushers in a flurry the Carnegie Mellon menu Getting to continue were greatly invested in more opportunity for our of out-of-class activities of undergraduate summer industry-related topics. undergraduates. We are for students at Carnegie programs in which math this tradition looking forward to resuming “I’ve had the good fortune to Mellon University. Many students can participate. on-campus research and during the chaos of work with Professor [Hayden} of them disperse, traveling These include the Summer study when conditions allow. Schaeffer, whose guidance home, studying abroad, Undergraduate Research the pandemic was a or for some mathematics Fellowship (SURF), Summer much needed morale Colleen Storm undergraduates, interning Undergraduate Applied booster for me.” in the finance industry on Mathematics Institute Wall Street. But, this year, (SUAMI) and Mathematical - Ian Tice as the global coronavirus Finance Summer Implementing the SEMS pandemic brought the world Undergraduate Research to a screeching halt, many Program (MFSURP). Similar to program was seamless. In students found that their SUAMI and MFSURP, SEMS the last several summers, summer programs had is a math-only experience, the department has made a been canceled. However, administered by the concentrated effort to invest in true Tartan spirit, they Department of Mathematical more time in collaborating on compelling research were eager to have the Sciences. Even more, SEMS is productive summer they had not strictly a research-based and study projects with undergraduates. Thus, the been planning. Answering program. It also includes this call, the Department in-depth reading groups department was naturally well positioned to execute of Mathematical Sciences and weekly seminars, which curated a brand new students can participate such a big project with little time to prepare. summer program: Summer in alongside their projects. Image from the SURF project of Venessa Jiang and Ben Yuan. Image from the SURF project of Experience in Mathematical The variety of experiences “I took on twice as many Shanjiawen Zhao, Kevin Zhou and Vinod Krishnamoorthy. Sciences (SEMS). afforded students who were students as usual this year to
10 MATHEMATICAL SCIENCES NEWS RESEARCH ROUNDUP 11 SUMMER 2020 UNDERGRADUATE RESEARCH PROJECTS
SURF PROJECTS SEMS PROJECTS
Neil Chen, Daria Mashanova, Noah Stevenson Neil Chen, Daria Mashanova, Hannah Milano and Mukund Traveling Wave Solutions to Hannah Milano and Mukund Subramaniam the Multilayer Free Boundary Subramaniam Neural Network Based Incompressible Navier-Stokes Neural Network Based Model Image from the SURF project of Jonathan Jenkins, Carolyn Model Approximation for Equations Approximation for Discovering Lee, Yuxuan Liu, Ethan Lu and Desmond Reed. Discovering Dynamical Advisor: Ian Tice Dynamical Systems Systems Advisor: Hayden Schaeffer Image from the SURF project of Chloe Ireland. Advisor: Hayden Schaeffer Zimu Xiang Upper Bound of the Yun Cheng, Albert Xu and Zhijie Chen Expectation of the Longest Yixue Liu Numerical Scheme for Common Sequence for a Large Combinatorial Structures with Allen-Cahn Equation and its Alphabet Gaussian Weights Energy Stability Advisor: Boris Bukh Advisor: Tomasz Tkocz Advisor: Franziska Weber Zhuqing Yang Ari Fiorino, Eric Li, Justin Sun Minsung Cho Particle Filtering Method and Xiao Liu Weakly Cop Win Infinite and its Application in Heat Detecting Sumsets Graphs Diffusion Problem Advisor: Kaave Hosseini Image from the SURF project of Yidan Hu. Image from the SURF project of Eugene Lee. Advisor: David Offner Advisor: Yu Gu Vanessa Jiang and Ben Yuan Yidan Hu Shanjiawen Zhao, Kevin Zhou Sparse Approximation of Self-Avoiding Property of and Vinod Krishnamoorthy High-Dimensional Functions Multi-Dimensional Gaussian On Log-Rank Conjecture for and its Application to Learning Processes AND Functions Governing Equations ANDREEV MEMORIAL Advisor: Jiawei Li Advisor: Kaave Hosseini Advisor: Hayden Schaeffer ENDOWED SURF Chloe Ireland Alexander Zheng Tudor-Dimitrie Popescu, Xinjie Elizabeth and Konstantin Andreev are lifelong “In 2016, we were looking to make a more Improving the Ramsey Entropy-Regularized Time- He and Ling Hu math lovers. In fact, as a high school student meaningful gift to an organization, and Bounds on Berge-K3 Discrete Optimal Transport Structure of Complement of Konstantin attended a math-focused school Carnegie Mellon was a top contender from Hypergraphs Paths with Multimarginal Sum-Free Sets in Sofia, Bulgaria, with the full support of his the beginning, due to our engagement with Advisor: Tom Bohman Constraints Advisor: Kaave Hosseini father. Later, in 2002, Elizabeth and Konstantin the department and our ability to be involved Advisor: Robert Pego met at Carnegie Mellon while enrolled in in how the contribution would be used,” Jonathan Jenkins, Carolyn Shanjiawen Zhao, Kevin Zhou mathematics degree programs. Elizabeth was Elizabeth said. Lee, Yuxuan Liu, Ethan Lu Hongyi Zhou and Vinod Krishnamoorthy an undergraduate in the department, and The SURF program offered up a lot of and Desmond Reed Coarsening Regularity of On Log-Rank Conjecture for Konstantin was a Ph.D. student in Algorithms, potential for this contribution. As previous Surfactant Dynamics from Integer Partitions AND Functions Combinatorics and Optimization. students, the Andreevs understand the the Arnold Perspective Advisor: Boris Bukh Advisor: Kaave Hosseini Their commitment to mathematics has benefit of such a unique, research-based Advisor: Ian Tice served as a tether to the academic world, program for budding mathematicians. Eugene Lee which they departed from after graduation. “Konstantin in particular, having done a Ph.D., An Invisible Search Game MEETING OF THE MINDS POSTER COMPETITION Always interested in the goings-on of the feels it’s very valuable for undergrads to have Department of Mathematical Sciences, the Advisor: Anton Bernshteyn a testing ground for how research works,” Andreevs make a point to stay informed Grand Prize Winner Runners Up Elizabeth said. Alexander Lum-Havrilla about the stellar accomplishments of the Khinchin-Type Inequalities faculty and students. Specifically, they Every summer, the Andreev Memorial Fei Peng Max Aires find student performance in renowned SURF endowment will provide a stipend Advisor: Tomasz Tkocz What Can You Draw? On the Number of Edges in of $3,500 for a student pursuing summer a Maximally Linkless Graph mathematics competitions highly impressive. Anrey Peng Our Summer Undergraduate Research research in mathematics. Eugene Lee was Nonlinear Cantilever Limit Philip Lamkin Fellowship (SURF) is no exception to the inaugural recipient of the award; he Cycle Oscillations Log-Convexity of Moments their attention. Year after year, the SURF completed a research project entitled, “The Advisor: Jason Howell of Averages program churns out serious and competitive Invisible Search Game” under the direction undergraduate research in mathematics, of Anton Bernshteyn. The Andreevs made Fei Peng Noah Stevenson this endowed gift in memory of Konstantin’s which is exciting for both the Andreevs and Lonely Runners and Coprime Characterizations of father, Andrei, who was a researcher at the the department. Mappings Screened Sobolev Spaces Bulgarian Academy of Sciences. Advisor: Tom Bohman
12 MATHEMATICAL SCIENCES NEWS RESEARCH ROUNDUP 13 Excellence in Discrete & Convex Geometry
The faculty of the Department of Mathematical Sciences have continued to grow their expertise and influence in the field of discrete and convex geometry. Four recent results in particular highlight that success and offer promise for future innovations in mathematics and related fields.
KELLER'S CONJECTURE HADWIGER'S COVERING CONJECTURE Teaching Professor John Mackey was More than 60 years ago, famed Swiss first introduced to Keller's Conjecture mathematician Hugo Hadwiger published as a graduate student in Hawaii a series of intriguing, unresolved roughly 30 years ago. problems, including what came to be called his “covering conjecture.” “The conjecture is pretty natural: if you tile a plane with identical square The question asks “can every tiles, then some pair will have to share n-dimensional convex body be covered RESOLVING A KNOTTY PROBLEM REDUCING DISCREPANCY an entire side,” explained Mackey. “If by 2n smaller copies of itself?” Hadwiger you tile three-dimensional space believed it was possible, and since then In 1911, the German mathematician Ongoing work from Associate with identical cubes, then some pair mathematicians, including Assistant Otto Toeplitz posited that any closed Professor Boris Bukh aims to make will have to share an entire square Professor Tomasz Tkocz, have worked to loop on a plane inscribes, or contains, integration in higher-dimensions more face. [Eduard Ott-Heinrich] Keller prove that conjecture true or false. four points that can form a square. feasible and accurate. conjectured that this pattern continues While breakthroughs have been made “Besides aiming at understanding “Computing integrals exactly is almost in all higher dimensions.” to show this “square peg problem” geometric properties of convex sets never possible as that would require to be true in certain situations, this Since his introduction to it, Mackey has (which play an important role in many infinitely many additions,” Bukh said. deceptively simple conjecture has been regularly returned to the tessellation areas of mathematics), Hadwiger’s “For that reason, our only choice is to resistant to being fully solved. “Even problem. The conjecture had been conjecture touches upon the idea of a approximate integrals.” now, more than a century later, this is proved true up through six dimensions covering, one of the simplest and most still an open problem,” said Assistant While it is relatively easy to in the 1940s, and other researchers fundamental in mathematics,” Tkocz said. Professor Florian Frick. approximate integrals of functions had been able to harness powerful For decades, the best upper bound defined on low-dimensional domains, computers to disprove it in ten Hadwiger proposed a variant of the on the conjecture was provided by higher-dimensional integrals have dimensions and higher. In 2002, Mackey problem in 1971, asking whether any English mathematician Claude Ambrose proved difficult for mathematicians. was able to create a counterexample closed loop in three-dimensional space Rogers, who proved that for an to disprove Keller’s conjecture in eight inscribes a parallelogram. “You could One can improve their approximation arbitrary n-dimensional convex body, and nine dimensions, leaving only the think of a piece of rope that might be with a low “discrepancy” set of points to approximately 4n√n smaller copies seventh dimension unresolved. knotted in complicated ways such that examine in a domain; however, it is not of the body are sufficient to cover it. the two ends of the rope are fused yet known which sets of points have the Last year, in collaboration with However, in new research published in together to make a loop,” Frick explained. smallest discrepancy. In the technical alumnus Joshua Brakensiek, now collaboration with mathematicians from An illustration of a closed curve with an feature of this publication, Bukh a Ph.D. candidate at Stanford the University of Michigan, the Weizmann inscribed parallelogram is given on the relates this issue of high-dimensional University, and Associate Professor Institute of Science in Israel and the cover of this newsletter. integration with another area of of Computer Science Marijn Heule, University of Alberta, Tkocz was able to mathematics — convex geometry. Mackey was able to finally show that use tools from information theory and In collaboration with researchers no counterexample exists for Keller’s asymptotic convex geometry to improve from North Carolina State University, “By doing so, not only we can leverage conjecture in seven dimensions. on Rogers' upper bound. Brandeis University, Cornell University decades of research in numerical and the Université du Québec á integration to attack problems in Montréal, Frick was able to prove convex geometry but also bring ideas Top Right Image: It is possible to cover the a closed n-dimensional cube with 2n copies of the interior of the cube. This is achieved by assigning a copy of the interior to each of the 2n vertices of the cube and shifting each Hadwiger's conjecture true. from the latter area,” Bukh said. copy of the interior a small distance in the direction of its assigned vertex. The image above depicts such a covering of the square (i.e., the two-dimensional cube). The covering conjecture of Hadwiger assets that every See the next page for more details on convex body in n-dimensional Euclidean space can be covered by 2n copies of its interior. Bukh's work. Ben Panko
14 MATHEMATICAL SCIENCES NEWS RESEARCH ROUNDUP 15 Discrepancy 1 _3 & Holes 4 _1 4
ow to spread n points uniformly in the maximum is over all axis-parallel boxes. For 0 _1 _3 d 0 1 unit cube [0, 1] ? A natural impulse example, discrepancy of the regular grid is 2 4 is to arrange them into a regular grid. (n1−1/s). Koksma and Hlawka showed that H Figure 2: Halton-Hammersley set of size 16. Figure 3: A 5-hole in a planar set. For many applications, this is a poor way. Θthe approximation error for integration of The number of points in every not-too-small One such application is numerical functions of bounded variation is behaves dyadic rectangle is exactly proportional to its area. 1 integration: given a function f, the aim is to like –n discP in the worst case. Intuitively,
compute the definite integral ∫[0,1]d f. If f has integration with respect to a high-discrepancy a nice antiderivative, we may compute the set assigns undue weight to some subregions integral using the Fundamental Theorem of of [0, 1]d at expense of the others. Calculus. Sadly, most functions have no nice It is a fascinating open problem to find sets antiderivatives and so one must resort to of smallest discrepancy in [0, 1]d. The best numerical approximations. existent constructions have discrepancy of d−1 Consider the approximation of ∫[0,1]d f by a asymptotic order log n in dimension d. One 1 finite sum–n p P f(p), where P is an n-points such construction is the Halton–Hammersley set. What happens∑ ∈ if P is the grid? The good sets. It is easy to define these sets ford =2: news is that by the very definition of the given a natural number m whose base-2
Riemann integral, the sum converges to the expansion is at · · · a1a0, its reversal is the integral as ∞. The bad news is that the binary number ( )=0. · · · . k n r m a0a1 at Planar Halton–Hammersley sets with n=2 showed that there exist arbitrary large sets in convergence →is quite slow even for very well- Halton–Hammersley set is {(m/n, r(m)) : points have a peculiar property: they have zero d without f(d)-holes for some function f that behaved functions. m=0, 1, . . . , n − 1}. The higher-dimensional discrepancy on every axis-parallel rectangle of ℝis slightly larger than the factorial. Recently, Halton–Hammersley sets are defined similarly s s t t in a collaboration with Professor Holzman For example, for the step function the form [a1 /2 , b1 /2 ) × [a2 /2 , b2 /2 ) with using digit reversals of m when written in f(x , . . . , x )=1 the approximation error is s + t ≤ k. In discrepancy theory, a set with from Haifa, graduate student Ting-Wei 1 d x1≤ several prime bases. proportional to nα−1/d for the most values this property is called a net. Chao and the author showed that the sets of Horton and Valtr are Halton–Hammersley of n. This is impractically large even for In addition to their uses in numerical sets in disguise. d = 5. One might be tempted to blame this on algorithms, nets are also useful in the discontinuity in f. However even for the combinatorial geometry. One such To be precise, Horton’s set is obtained from 2 −2/d function x 1 the error is proportional to n . application is to the problem of convex holes. the planar Halton–Hammersley by stretching 1 d It turns out that the grid points are a poor A convex hole in a set P is a subset y-coordinates; the y-coordinate 0.a0a1 · · · (in
choice for our integration method because H P that are vertices ⊂of ℝa convex polytope; binary) becomes a0 Y0 + a1 Y1 + · · · where
some axis-parallel boxes contain too few it is⊂ called -hole if |H|= . Erdős asked if, for Y0 Y1 · · · is a quickly-decaying sequence points, whereas others contain too many. every fixedℓ , all sufficientlyℓ large planar sets of real≫ numbers.≫ A similar transformation Here, by axis-parallel box we mean a set in general positionℓ contain an -hole. Horton turns any net into a set without large holes. answered Erdős’s question in negative:ℓ he Using this relation, new sets in d without cd- of the form B=[a1,b1) × · · · × [ad,bd), and by too few and too many we mean that the constructed arbitrarily large planar sets holes were found. It is conjecturedℝ that the number of points of P that are in B deviates without 7-holes. Valtr generalized Horton’s exponential bound is tight but the best lower significantly fromn vol(B), which is the 0 construction to higher dimensions; he bound for d ≥ 3 is mere 2d + 1. number of points that we expect to fall into B 0 1 had it been completely uniformly distributed. Boris Bukh Discrepancy of a set B is the difference Figure 1: Discrepancy of a box is the difference between the expected and the disc P(B)=|P B|−n vol(B). Discrepancy actual number of points inside.
of P is discP∩=max discP(B), where the
16 MATHEMATICAL SCIENCES NEWS FEATURE 17 ALUMNI NEWS
companies. At US Bank, Andreev became the league ballpark, and the next day, it's something At the Top of Her Game first municipal bond salesperson to receive the totally different, like financing an airport.” US Bank Summit Award for performance and As an underwriter, Andreev works between the leadership in 2014. worlds of buyers and sellers for these municipal Elizabeth Andreev is making her mark in municipal finance After nearly eight years, Andreev took issuers. Her job is to structure and price the another leap to become an underwriter at bond deal to help her issuer clients meet all UBS, which was expanding its small municipal their needs and objectives while also creating lizabeth Andreev has sat in many seats in bond ETFs and worked on a collaboration bonds group after deciding to reenter the bonds that the sales force can market to Ethe finance industry — she’s been a buyer, with investment management firm VanEck realm of public finance. investors. Andreev has impressed not only her a seller, a market maker and an underwriter. to structure and seed the first High Yield clients and investors with her market insights Andreev finds municipal bond finance She’s brought her sharp quantitative skills to Municipal Bond ETF. and leadership but also the industry at large. these roles at hedge funds, large commercial fascinating. Investment banks, like UBS, help But Andreev soon realized that she wanted to banks and multinational investment banks. public entities — governments, nonprofits, Last year, the Bond Buyer, one of the most try different things, an outlook she attributes etc. — structure and sell municipal bonds to respected and widely circulated daily She credits this expanse of experience to the lingering effects of the financial crisis. investors to help fund public projects. newspapers for the municipal bond industry, to beginning her career just as the most “I think for people of my generation, compared named Andreev as one of its 2019 Rising Stars, “It’s different than any other market because we tumultuous financial crisis in modern history to those before who might have held a job an honor recognizing the best and brightest are not just looking at and crunching numbers hit the United States and the world. for 20 years at the same firm, there's a little municipal finance professionals under age 40. bit more mobility or interest in branching all day. We're working with people who are “It was overwhelming and hard to process all out and trying new things because we all trying to get money to do things that often “I was honored to be nominated and selected. that was going on. Not only because I was know that situations can change very quickly. tangibly affect your everyday life,” she said. It was definitely a highlight of my career,” said so new but because it was kind of new to Developing breadth and new skill sets feels Andreev, who was also promoted to executive everybody. Things that were breaking down Andreev sees projects come across her desk very important,” Andreev said. director at UBS earlier this year. weren't supposed to be able to break down,” from municipal issuers like a hospital trying Emily Payne Andreev recalls. to finance a new wing to a toll authority trying to build a new road. Andreev had just started her professional career “It’s different than any other at Susquehanna International Group, a hedge Every day is something new, Andreev says. A fund headquartered in the Philadelphia area. market because we are not just facet that is both challenging — to know looking at and crunching numbers about all these different municipal When the mathematics major was issuers and their needs — and graduating from Carnegie Mellon University, all day. We’re working with people exciting. “One day, we will be she never pictured a career in finance, but working on financing a minor she did know that she wanted to find a who are trying to get money to do career where she could use math to solve things that often tangibly affect problems in everyday life. your everyday life.” She joined Susquehanna in 2006 as a trader in equity options but because of her quantitative skills, the company quickly In 2010, she transitioned from being a buyer moved her to the commercial mortgage- to a seller on the institutional sales and backed securities trading desk. Shortly trading team at US Bank, transacting in thereafter, the subprime mortgage crisis bonds for a wide range of customers completely obliterated that market. such as hedge funds, mutual funds and insurance Her position was effectively eliminated, but the group offered her a position as a trader and market-maker for municipal bonds and municipal bond exchange traded funds (ETF). While at Susquehanna, Andreev developed a proprietary intraday pricing model for municipal
18 MATHEMATICAL SCIENCES NEWS ALUMNI NEWS 19 STUDENT STORIES
“The Market Making Game is not just based on problem- solving but teamwork, too.”
QUANT CLUB AND CITADEL HOST The Quant Club has been hosting the Market Making Game for a few years, but this was their first university-wide event. Former MARKET MARKING GAME Quant Club executive Max Chen brought the game to the club after participating in a trading competition at another university. Chen, a computational finance graduate, went on to earn his master’s in machine learning from CMU before joining Citadel as a trader in August this year. He wrote all the In February 2020, 150 students gathered “They can assume we will make the choice that questions for the event and, along with Wang, to test their financial prowess at a Market maximizes our profit or minimizes our loss,” helped organize the partnership with Citadel. Making Game. The event was hosted by said Quant Club President Tony Wang, who Citadel representatives gave advice to teams Carnegie Mellon University’s Quant Club and oversaw one of the stations. Essentially, if the during the game and chatted with students at sponsored by Citadel Securities, one of the liquidity station buys a contract, the team a pre-event lunch. world’s biggest market makers. It brought then knows they undervalued the contract. If together undergraduate and graduate the station sells the team a contract, the team The event was undoubtedly a success for the students from all across the university. knows it overvalued the contract. Quant Club. “I think the Market Making Game is my favorite Quant Club event,” said Lily The game was a thought-provoking mixture It’s a lot of uncertainty, but that’s what makes Qiao, a computational finance student and of estimation, optimization and negotiation. the game all the more exciting. club executive. “It is a lot of fun because it’s Teams of participants took on the role of “The most challenging part is the uncertainty so interactive. It’s not just based on problem- market makers as they tried to maximize their that comes from not knowing the exact solving but teamwork, too.” capital by trading with other teams. value of the contract and having to trade Emily Payne Twenty-five teams started the game with it anyways if you want any chance to contracts and some (fake) cash. Throughout make money,” said Dallas Foster, recent five rounds of play, teams had to use their computational finance graduate. cash to buy contracts or sell contracts to Fellow 2020 computational finance graduate make more cash. However, there was a Jason Ren agreed. His team managed that catch. They didn’t actually know the value of uncertainty by splitting up into different the contracts. Instead, each round rotated roles based on their strengths. Shengming through questions about Citadel securities, Luo, a Ph.D. candidate in statistics, and the economy, mathematical puzzles and Justin Cao, another computational finance general trivia that teams had to solve to recent graduate, solved the problems to reveal the price of the contracts. find out the value of the contracts; Ren and “These were tricky problems,” said Bill Hrusa, mathematical sciences student Wynn Huang mathematical sciences professor and faculty drove hard bargains as the team’s traders; and advisor to the Quant Club. “For example, computational finance major Cindy Ding also one probability-focused question asked the traded with other teams and managed the number of times you need to roll a six-sided contracts and communication for the team. die in order that the product is going to be The team took home first prize, ending the bigger than 20200201, the date of the event.” day with a 10% return on their contracts. Hrusa was not surprised to see teams using a Ren’s favorite part about the event was variety of strategies throughout the game. interacting with other teams. “With a room of One of those strategies is to visit the liquidity 150 people, it can get pretty intense,” he said. station. The liquidity station knows the hidden “Multiple groups are trying to compete at the value of all the contracts. Teams could trade same time. It almost felt like a trading floor with the station once per round to gain where people are shouting on the floor and information on the true value of their contracts. the phone.”
20 MATHEMATICAL SCIENCES NEWS STUDENT STORIES 21 SENIOR NOAH STEVENSON SELECTED AS
For example, one of Stevenson’s projects GOLDWATER focused on the rigorous analysis of micropolar fluid equations, which SCHOLAR describe the dynamics of viscous fluids with microstructure such as blood, Mathematical Sciences student Noah colloidal suspensions or liquid crystals. Stevenson has received a 2020 Barry Mathematicians contribute to the Goldwater Scholarship. Given by the Barry understanding of fluid flow by studying the Goldwater Scholarship and Excellence in nonlinear partial differential equations that Education Foundation, the prestigious award govern their motion. To that end, a major supports students who intend to pursue focus of modern mathematical analysis research careers in the natural sciences, is determining when solutions to fluid engineering and math. equations exist and studying if and how Stephanie Wallach, assistant vice provost singularities can form. for undergraduate education, said that CMU Stevenson is pursuing the Mathematical can nominate four students annually for the Sciences Honors Degree Program. He takes “Working in this area is challenging even award. Brittany Allison, assistant director both undergraduate and graduate level for graduate students, but Noah's strong in the Office of Undergraduate Research classes and will graduate from Carnegie background and honors program preparation and National Scholarships, oversees the Mellon University next year with his B.S. and made it possible for him to do so as a rising university’s Goldwater nomination process, M.S. in mathematical sciences. junior,” said Tice. which includes a committee of faculty At CMU, Stevenson has molded his interests Through his analysis of fluid equations, “Receiving the Goldwater Scholarship members who discuss the nominations, and in both teaching and research. He has served Stevenson discovered and analyzed a is reassuring to me that I am traveling also includes working closely with each as a teaching assistant for the mathematical particular stability in micropolar fluids. His along the right academic and career nominee before the final submission. sciences’ courses Math Studies: Analysis work has been submitted as a manuscript to path,” said Stevenson. He adds that the “Dr. Allison did a double take when she I and II, and he has conducted several be published. scholarship process helped him learn how to saw that all four of our nominees were research projects studying partial differential communicate mathematical ideas to non- These experiences have shaped Stevenson’s selected,” Wallach said. “This is an award equations in fluid mechanics and function experts, a skill that will come in handy when career plans to earn his Ph.D. and become a that recognizes our CMU strengths in the spaces, advised by Associate Professor of applying to graduate school. researcher and professor of mathematics. He fields of engineering, natural sciences and Mathematical Sciences Ian Tice. believes the Goldwater Scholarship will help Stevenson is one of 396 students selected for mathematics and the exceptional ability His projects are influenced by his interest in him achieve this goal not only for its financial the scholarship from an applicant pool of over of our faculty to involve our immensely applying abstract mathematical concepts to support but because of what it means to be 5,000 sophomores and juniors nationwide. He talented and highly motivated students in solve problems in the physical world. chosen as a Goldwater Scholar. is also one of four Carnegie Mellon recipients. undergraduate research. Emily Payne
22 MATHEMATICAL SCIENCES NEWS STUDENT STORIES 23 CHRISTOPHER COX RECEIVES GRADUATE ANTOINE REMOND-TIEDREZ STUDENT RECEIVES GRADUATE RESEARCH STUDENT TEACHING AWARD AWARD
Mathematical Sciences Ph.D. graduate lectures, quizzes, homework assignments, Mathematical Sciences Ph.D. graduate Specifically, Cox has studied problems Antoine Remond-Tiedrez was awarded midterms and exams to provide repeated Christopher Cox received the 2020 Guy C. relating to information theory and coding the 2020 Hugh Young Graduate Student opportunities for students to develop their Berry Graduate Research Award. theory, fields which focus on solutions to Teaching Award. mathematical skills. efficiently communicating information across “He is one of two or three best graduate noisy channels. “When I assigned TAs to various courses, I “The course is difficult to teach because it students in Discrete Mathematics that I have knew that I could assign Antoine anywhere, is the first proof-based course for most of seen in the department in my nearly eight “While passing through a noisy environment, knowing that he would be an asset to any our students,” wrote Irene Gheorghiciuc, years here,” wrote Boris Bukh, associate a message may be corrupted: some parts of instructor,” wrote Deborah Brandon, former associate teaching professor. professor of mathematical sciences and Cox's the message may be erased and other parts associate teaching professor, in supporting major professor, in nominating him for the may be completely changed,” Cox explained. However, students had nothing but praise for his nomination for the award. award. Cox is now a postdoctoral fellow at “The main way to combat this noise is to add Remond-Tiedrez. Iowa State University. redundancy to the message.” As testament to this fact, Remond-Tiedrez “Antoine did well in breaking down complicated was a teaching assistant for seven semesters Describing him as driven and independent, Through his work related to the Shannon- problems and directing us to find an approach for nine different courses. He was committed Bukh highlighted the fact that Cox has Hartley theorem, Holzman noted, Cox and Bukh to each step,” wrote student Joanna Yao about to becoming a better educator with each written five papers as a graduate student, achieved a breakthrough on a “notoriously her experience in Concepts of Mathematics. course he taught and he poured countless compared to the average of one to three that difficult problem” of calculating the upper “His teaching style gave me more chances to hours of preparation into doing so. a mathematical sciences student produces on bound of the Shannon capacity of a fixed graph. improve my mathematical thinking.” their way to getting a Ph.D. “I like how the students’ eyes light up In response to winning the Berry award, Cox If you ask his students for someone who is when they have an ‘aha moment.’ I like “In a nutshell, Chris is one of the most chose to express his feelings about it in the dedicated and effective in teaching, they say to hear students voice their sense of brilliant young combinatorialists that I have form of puns. to look no further than at Remond-Tiedrez. accomplishment after completing a difficult encountered over my thirty years of activity in “I'm Berry pleased to have won this award. yet rewarding class; and I like the challenge Brandon agrees: “Antoine puts a lot of this area,” wrote Ron Holzman, a professor of I'd like to thank Boris Bukh, my adviser, and of finding the right teaching strategies for thought and effort into his teaching, in mathematics at the Technion-Israel Institute Dejan Slepčev, denominator, for their non- a particular cohort in a particular class,” addition to having a natural talent for it. He is of Technology, to support the nomination. Cox trivial support,” Cox said. “Being a discrete Remond-Tiedrez said. a wonderful (and versatile) teacher.” worked with Holzman in 2018 after receiving mathematician, I'm glad I could count on a grant allowing him to travel to Israel and When he became the instructor for the them. Hopefully my research doesn't atrophy collaborate on research there. course Concepts of Mathematics in the after winning a trophy.” summer of 2017, he designed the course “My research focuses on the interplay between himself using only the syllabus and the algebraic, geometric and probabilistic tools in textbook, taking great care with planning extremal combinatorics,” Cox said.
24 MATHEMATICAL SCIENCES NEWS STUDENT STORIES 25 CLASS OF 2020 Director of Undergraduate Jersey, he’s starting his CLASS OF 2020 Studies Jason Howell. “He graduate studies virtually. He SPOTLIGHT: accomplished a great deal spends his time in classes, DESTINATIONS during his four years here, seminars, reading groups TRAJAN and I know that the future for and in finding his doctoral Trajan is very bright.” research path. Currently, much of his research consists HAMMONDS Number theory, which Firms that hired more than of trying to understand uses a variety of tools from one member of the class: what has been done in the other subjects like algebraic Trajan Hammonds spent his field before. A lot of number geometry and harmonic undergraduate years cultivating theory research consists of analysis to answer basic experiences that would trying to figure out the right questions about whole give him as much exposure way to think about things, numbers, is Hammonds as possible to research in Hammonds notes. And he’s overarching area of study. And mathematics, a journey that looking for the central story what led him to continue his has prepared him for his next that links everything together. steps as a Ph.D. candidate at studies at Princeton. “I want to identify key tools and Princeton University. “I chose Princeton primarily components in the landscape for its abundance of faculty in Each summer, Hammonds that would be useful for number theory, including two participated in a Research problems I may try to solve recent Fields medal winners, Experience for Undergraduates in the future. Moreover, it's who together cover a wide (REU) program, always hoping important to really understand range of topics in number to find a new problem to how things work in a classical theory,” Hammonds said. tackle. He spent his first setting in order to try to summer at the Mathematical While Hammonds has frame things in a more general Sciences Research Institute physically moved to New setting,” he said. Undergraduate Program, the next at Williams College and his final summer at the University of Minnesota, Duluth. “All three REU programs were instrumental in my decision to pursue a research career in Doctoral Programs: mathematics,” said Hammonds. His passion for research University of Chicago, culminated in an impressive five papers during his time as Statistics an undergraduate, three of which have been published: Penn State University, one in the Houston Journal Mathematics of Mathematics, one in the Journal of Number Theory and another in the International Princeton University, Journal of Number Theory. Mathematics “Trajan has been a vibrant and valuable member of the mathematics community at University of California, CMU,” said Hammond’s advisor, Los Angeles, Mathematics (x2)
University of North Carolina at Chapel Hill, Bio Statistics
University of Wisconsin, Mathematics
26 MATHEMATICAL SCIENCES NEWS STUDENT STORIES 27 www.cmu.edu/math/giving For more information on the Mellon College of Science priorities, please contact Nancy Felix at [email protected] or 412-268-6442.
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