mathematical sciences news 2019 contents 2019
Editor-in-Chief Tom Bohman
Contributing Writers Letter from Department 03 Tom Bohman Head, Tom Bohman Jocelyn Duffy Bruce Gerson Florian Frick 04 Ben Panko Emily Payne Faculty Notes Ann Lyon Ritchie Franziska Weber
Graphic Design and Photography 10 Carnegie Mellon University Expanding the Boundaries on Marketing & Communications Summer Undergraduate Research
Mellon College of Science Communications 14
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Produced for the Department of Mathematical Sciences by Marketing & Communications, November, 2019, 20-137.
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1 Letter from Mathematics Department Head, Tom Bohman
Undergraduate research has become a hallmark of a Carnegie Mellon University education, with students citing that it has helped to prepare them for their futures in academia and the workforce. Mathematics hasn't traditionally been seen as a field where students do meaningful research; over the course of the last few years, the department has made a concerted effort to change that.
In this issue of the Department of Mathematical Sciences Newsletter, we highlight undergraduate research in the department and the programs we have developed to provide research experiences for our students. Our main interest is in summer programs that allow students to focus on a project without the distractions of coursework. With the institution of these programs, the number of students engaged in summer research has grown over the last four years from a handful to more than 60 working on high-quality, open-ended mathematics projects. This is an extraordinary program that very few, if any, other mathematics departments can match.
In addition to exposing students to research, the summer Cover Image programs, which include the Summer Undergraduate Research What Can You Draw? Fellowships (SURF) and the Mathematical Finance Summer Undergraduate Research Program (MFSURP), give students For his research project during the the opportunity to be mentored by our renowned faculty. The summer of 2019, Fei Peng raised the mathematical finance program, which has grown dramatically following deceptively simple question: in the last few years, is administered by Bill Hrusa. The SURF What can you draw? program involves a large fraction of the faculty, including Florian Frick, who recently joined us as an assistant professor and was Your canvas is the plane 2 — colored awarded a Sloan Research Fellowship this year (see the articles white to begin with — and you are given on pages 14 and 16 for more on Professor Frick and his research). a pencil, which produces a black unit disk wherever it meets the canvas, and The summer research programs have been made possible an eraser, which produces a white unit I hope that the through the support of our alumni, including David Simmons, disk. There are no further restrictions on many alumni of Larry Jennings, David and Jacqui Martin, and Elizabeth and your artistic freedom. You may raise the tool the Department Konstatin Andreev. To learn more about the research our off the canvas, that is, there is no continuity of Mathematical students engage in, see page 28 for details on the 2019 SURF requirement for the centers of disks you draw Sciences have projects and page 10 for an article on the evolution of the and you can switch tools as many times as a chance to summer programs. Additionally, the image on the cover of desired. Peng and his research mentor Florian reconnect with this issue is drawn from the SURF project of Fei Peng, a junior Frick showed the main result is that drawability the department mentored by Professor Frick. cannot be characterized by local obstructions. by visiting A bounded set can be locally drawable, while cmu.edu/math Of course, there have been many other developments in the not being drawable. The figure presented here department. In this issue you can also read about our new is an example of such a set; the boundary Let us know faculty, faculty awards (including Po-Shen Loh's receipt of the has curvature less than one, but the set is not what's new Presidential Early Career Award in Science in Engineering) drawable in the large. with you! and student awards.
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Ian Tice Wins Julius Ashkin Award Giovanni Leoni Wins Richard Moore Award Assistant Professor Ian Tice was awarded Additionally, Tice is a strong supporter of the Mellon College of Science's 2019 Julius undergraduate math research. His experience Professor Giovanni Leoni was honored Another focus of Leoni's has been to equip Ashkin Teaching Award for his devotion and as a research advisor inspired him to apply with the Mellon College of Science's postdoctoral students with the tools to effectiveness in teaching. for a National Science Foundation (NSF) 2019 Richard Moore Award. This award is effectively teach upper-level mathematics CAREER grant. NSF awarded him $38,500 presented every year to a faculty member in courses. In the summer of 2017, he Tice is an inspiring educator, both to fund summer research for 11 students the college who is making substantial and implemented an orientation program for challenging and guiding his students to over the next five years. One of Tice's favorite sustained contributions to the educational all incoming postdocs to instruct them become astute mathematicians equipped activities is organizing and supervising the mission of the college. on how to write syllabi in line with Faculty with inquisitive minds. Tice has crafted Department of Mathematical Sciences' Senate guidelines, how to conduct student hundreds of homework problems and written poster competition at Carnegie Mellon's Leoni created the department's assessments, how to work with teaching over 1,300 pages of notes for his courses. annual undergraduate research symposium, Undergraduate Curriculum Committee, assistants and how to use Canvas, the Meeting of the Minds. which has completely transformed and While his homework problems are known university's learning management system. strengthened the undergraduate program Leoni also provided constructive feedback for taking hours to complete, by the end, Beyond teaching, Tice has been a driving over the course of the last decade. Since for postdocs through "mock" classes. students understand step-by-step how to force in developing and modernizing parts its inception, Leoni has served as chair of assemble a proof for a full result; this, of the undergraduate and graduate the committee with the goal of providing In addition to his focus on the broader Tice says, helps students synthesize curriculum in the department. excellent mathematical education to all educational and administrative aspects mathematical arguments and trains them Carnegie Mellon University students, of the department, Leoni cares deeply to think like mathematicians. He completely revamped the undergraduate computational and applied mathematics regardless of major or college. The about his students and dedicates his time Tice knows he demands a lot of his students, concentration, helped reform the graduate committee's main achievements include a to making sure they succeed. He holds but his door is always open. Students will curriculum in analysis and partial differential complete redesign of the introductory course recitations for courses that do not have a spend hours in his office, be it to get help on equations (PDE) and revised courses in the sequence for math and computer science recitation and conducts research projects a problem, to revisit a concept from lecture math honors program. He has developed four students and a redesign of the honors with students over the summer. that they didn't understand or to talk more new courses, restructured the department's course sequence for mathematics majors. "Giovanni Leoni regularly steps in and about their interests in math. mathematical modeling and graduate PDE Leoni has worked with faculty across takes charge of administrative details that courses and took on teaching a number of "Professor Tice is someone who very clearly the university to tailor the mathematical have a very high impact on the education these courses, both new and old. loves teaching," wrote former student Jose sciences program to suit students from all of students in mathematics courses," Arrillaga. Arrillaga spoke with Tice about "Ian is an enthusiastic and devoted teacher majors. His efforts have also helped grow wrote Tom Bohman, Alexander M. Knaster his difficulty understanding some results in who has demonstrated the ability to create and strengthen the mathematics major at Professor and head of the Department of convergent series, "He made sure to put an excellent courses, inspire students and train Carnegie Mellon; each cohort has nearly Mathematical Sciences, in his nomination additional exercise on that week's problem some of the best undergraduates," wrote 100 math majors, which is three times more letter for Leoni. "He is a significant part of set, which helped immensely in clarifying Professor Dejan Slepčev in supporting than the number of students before the new the great success of our undergraduate my uncertainties," he said. Tice's nomination. curriculum was implemented. program and mathematics."
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Po-Shen Loh Wins University's Top Teaching Award
Associate Professor Po-Shen Loh came Bohman also noted in his nomination letter to Carnegie Mellon in 2010 and quickly the course Discrete Mathematics, which became known for his enthusiasm, has earned a "sterling reputation" under effectiveness and overall excellence Loh's teaching. as an instructor. For these reasons, he received Carnegie Mellon University's 2019 "As a department head, I wish that I could William H. and Frances S. Ryan Award for assign him to more courses," Bohman said. Meritorious Teaching. Loh also is the lead coach for the U.S. Tom Bohman, Alexander M. Knaster International Mathematical Olympiad team Professor and head of the Department of of high school students. In 2015, he brought Mathematical Sciences, said Loh's most the team's training program to CMU, where impressive contribution to teaching at CMU the team prepares for the competition. The has been his restructuring and expansion U.S. has won four world championships in of the Putnam Seminar, which prepares the last five years. undergraduate students to participate in the CMU junior Deborah Blank commended premier Putnam Mathematical Competition. Loh for his collaborative teaching approach. After Loh began teaching the class, enrollment more than tripled to about 200 "Professor Loh would ask the students students each year. for ideas or intuitions on how to solve the problem...which fostered an inclusive "He teaches it in the style of an classroom environment," she said. "He 'inverted' or 'flipped' classroom, with made me feel that I mattered as an student discussion of the problems and individual and a teammate, and instilled in us presentation of solutions taking up a a hunger to do our best and to be excited part of each class," Bohman wrote in his about competition math." nominating letter. "Rather than viewing this class as preparation for a competition, Senior Nina Edwards said she has never Po re-envisioned it as a problem-guided seen a teaching style quite like Loh's. She tour of mathematics, which develops praised his dedication to students, creativity core problem-solving skills that enhance in designing problems and his "remarkably Po-Shen Loh Honored with Loh's research focuses on the intersection of the ability to learn and use higher-level engaging and dynamic style of lecturing." Presidential Early Career Award discrete systems, probability and computer mathematics." science. He recently attended the World Junior Theodore Li said Loh was a "math Po-Shen Loh has received the prestigious Economic Forum's Annual Meeting of New Loh holds seminar sessions each day evangelist" who makes math fun. Presidential Early Career Award for Scientists Champions in Dalian, China, as a member of and tailors each class to the students' and Engineers (PECASE). The PECASE is the Forum's Young Scientists community. "presumed knowledge," Bohman said. "Dr. Loh shares his enjoyment and passion the highest honor bestowed by the United for math and can rekindle any jaded States government to outstanding scientists Established in 1996, the PECASE Loh's efforts have paid dividends as student's love for mathematics, which he and engineers who are beginning their acknowledges the contributions scientists and CMU placed first in the competition in would consider more of a success for his independent research careers and who engineers have made to the advancement 2016 and for the last seven consecutive class than our constant top placings," show exceptional promise for leadership in of science, technology, engineering and years has had the second-most students Li said. science and technology. mathematics education, and to community from schools in the U.S. and Canada place service as demonstrated by scientific among the top 500 individuals. In 2016, A global ambassador for mathematics, leadership, public education and community three students earned the Putnam Fellow Loh travels the world giving talks to diverse outreach. The awards are conferred annually at designation by finishing among the top audiences, from fellow mathematicians to the White House following recommendations five individuals. high school students. from participating departments and agencies.
6 7 faculty notes
"I am interested in the structural properties of graphs," Offner said about his research in the field of combinatorics. Graphs can be thought of as informal "networks" with nodes and connections, he notes, and his research looks at questions such as how the large version of these networks can be decomposed into copies of smaller sub-networks, or how many of those connections in a network must be severed to remove all the copies of a sub-network within it.
"Much of my work is related to questions in theoretical computer science and Martin Larsson optimization that are also beautiful and challenging pure mathematics problems Martin Larsson joined the Department of in their own right," Offner said. "There are Mathematical Sciences as an associate a number of easy to state conjectures in professor in fall 2019. A native of Sweden, this field that nonetheless have proven hard Larsson received his Ph.D. from Cornell to understand and will require new insights University and was most recently an assistant to solve." professor at the Swiss Federal Institute of Technology in Zürich. Several years ago, Offner organized a collaborators developed a theory for a class research collaboration to study the topic "My research is in mathematical finance," of models that have become increasingly Math of polychromatic colorings of graphs. Larsson said. "It's about developing the important for describing volatility, thus In a paper published last year in the mathematics needed to model phenomena making them more practical to work with. at CMU Journal of Combinatorics, Offner and his that we see in financial markets and economic collaborators specifically looked at these systems more broadly." "These models are tricky to analyze colorings on hypercubes, geometric because they lack various nice features that Welcomes figures in four or more dimensions that are Larsson said he found himself attracted to people are used to working with," Larsson analogous to regular cubes in the three- this area of mathematics because it blends said. "We show that despite this, concrete Two New dimensional world. theory with applications. "I get to talk to all computations are still possible in these kinds of people, from pure mathematicians to more challenging models." Offner has an impressive record as a finance academics to bankers and regulators," Faculty mentor of undergraduate research projects. Larsson said, noting that his research touches "Carnegie Mellon exerts a strong pull on He has advised over 30 students who on everything from interest rates to energy anyone doing research in mathematical Members completed projects in the last few years, prices to models for volatility. finance," Larsson said of what drew him to and many of these students went on to join the faculty here. Besides its research, present their work at professional meetings "Lots of interesting things pop up when Larsson also described Carnegie Mellon's David Offner or publish them in mathematics journals. you start to think about individuals that bachelor's and master's programs in interact, often in a highly strategic manner, computational finance as "trailblazers" in David Offner joined the Department of Outside of his research, Offner enjoys in an environment with lots of noise and his field. "Being part of that is very exciting." Mathematical Sciences as an associate spending time with his wife and two randomness," Larsson said. "And for teaching professor in fall 2019. He received children and coaching and watching the record: it's not about predicting the Outside of his work, Larsson enjoys his Ph.D. from Carnegie Mellon and was sports of all kinds. stock market!" being outdoors with his wife and is most recently an associate professor at looking forward to exploring the hiking Westminster College in New Wilmington, "I am excited to work with the talented and An example of Larsson's work is a opportunities around Pittsburgh, as well as Pennsylvania. Offner also previously served hard-working students at Carnegie Mellon study published in October 2019 in The its breweries, coffee shops and restaurants. as a Shelly Visiting Associate Professor at and to interact and collaborate with brilliant Annals of Applied Probability titled "affine "So far we've only begun to scratch the Carnegie Mellon. colleagues," Offner said. Volterra processes." In it, Larsson and his surface!" he said.
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Since 2015 there has been a dramatic increase in the number of undergraduates pursuing mathematics research on campus in the summer months.
A traditional view among mathematicians mathematics SURFs in the summer of is that mathematics is not well suited to 2019 (see page 29 for details on the SURF undergraduate research. Students generally projects from 2019). need several years of course work before they can understand the questions that Furthermore, the quality of these projects are most interesting to mathematicians. has improved as the faculty has become Expanding the However, there is growing demand for more engaged with students' research. such experiences from undergraduates Almost half of the permanent faculty have who are considering moving on to graduate advised at least one SURF project in one Boundaries study. Furthermore, research experiences of the last three summers. "This is special can be key in helping students secure jobs in itself," said Department Head Tom on Summer and internships. Bohman, who created an incentive program for faculty who mentor summer research. In response to this growing demand and The department also introduced the David Undergraduate the fact that a successful undergraduate Simmons competition for undergraduate research project can be a transformative research in mathematics and appointed Irina Research experience, the department is developing Gheorghiciuc to the position of director of programs that make these experiences more undergraduate research in mathematics. In available. This began with an effort to expand this role she helps match students who have the number of mathematics students who an interest in research with faculty mentors. receive Summer Undergraduate Research Fellowships (SURF), a program under the SURF projects rely on one-on-one advising university's Undergraduate Research Office and being able to provide close faculty that awards students stipends to pursue contact has had a direct effect on students' full-time summer research academic growth. One such student, senior on campus. Varun Gudibanda, worked with Assistant Professor Franziska Weber last summer Recognizing the value of this on numerical analysis of liquid crystals to program, the department started produce numerical schemes for the system to directly fund math students and simulate the system in MATLAB. "My pursuing SURFs, leveraging summer research helped me find confidence gifts from alumni, including Larry in my plans to attend graduate school Jennings and David Simmons, in and beyond," said Gudibanda. He also 2016. The size of this program grew from worked on a numerical analysis project with a previous norm of one or two mathematics Associate Teaching Professor Jason Howell SURFs each summer to a total of 14 in the summer of 2018 and now plans to
Images. Summer Undergraduate Research Fellowship work of the last few years.
10 11 obtain a Ph.D. in numerical analysis, with Following her first-year, current junior Lily the goal of eventually providing numerical Qiao and her partner Laura Li conducted frameworks for exciting real-world research on portfolio optimization with problems such as Formula 1 race car order-size impact. They won the Young aerodynamics. Researcher Award at the 2019 Simmons Competition. As the SURF program grew and improved, it became clear that there was a need for "This is one of the most important topics a separate program focused exclusively in finance right now," said Qiao. on the mathematics of finance. Professor Bill Hrusa launched the Mathematical Current financial theory assumes that the Finance Summer Undergraduate price of a security doesn't depend on how Research Program (MFSURP) in the many shares an investor buys. Qiao and summer of 2017. Li developed simple models for order size impact and used them to explore the effect For sophomore Minty Xia, she Most recently, the program added a Twelve students participated in the first on the structure of optimal portfolios. appreciates that MFSURP gives students weekly market watch seminar series, MFSURP. This year, that number increased a chance to work on the same problems where students discuss what's going on to a staggering 38 students. Qiao says that internship recruiters that people in the industry are currently in current financial markets, which also were impressed by her research, and grappling with. gave Xia a leg up. MFSURP is an early research experience she credits MFSURP with giving her for first-years and sophomores to work on the confidence to connect with people This summer, Xia's group studied utility "Before going to my final interviews, I had to mathematical problems that are relevant currently working in finance. indifference pricing. In certain situations, prepare for market knowledge. Having to finance. Normally, students with an a given security may have more than been through that experience interest in finance look to secure an "I'm really grateful that Dr. Hrusa offers one price that is consistent with during MFSURP was really internship by their junior year. Having done this undergraduate research experience," the basic theoretical framework. helpful," she said. research on a topic in finance has made added Qiao, who secured an internship The idea of utility indifference a big difference for those students come with Bank of America for next summer pricing is to single out a price As it's grown, MFSURP internship recruiting season. over a year in advance. that is consistent with has attracted corporate investors' risk preferences. sponsorship from Principal Xia's group analyzed the limits Financial and Goldman of utility pricing to see how Sachs and also garners current models would support from the CMU behave when order Provost's Office. sizes get large. These last four years "We were doing things that have shown that not we knew would be meaningful only are students highly in a real-world context. That motivated to pursue research was the most meaningful but that it has a strong part to me," Xia said. impact on their futures. The department continues to raise She also believes the program endowed funds to support has practical outcomes. the growing number of students Xia will intern with the Bank of undertaking summer research. America next summer, an impressive The Martin Fund for Undergraduate feat for a sophomore. "Most of my Research supports mathematics SURF interviewers asked about my research students and the Steven Shreve Fund for because they would say it's really rare Mathematics supports the MFSURP. for a first-year student to have a research experience," she said. Visit math.cmu.edu/giving for details.
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area of mathematics he's interested in is one that has been relatable to many people at some point in their lives: fair rent division.
In a paper published this year in The American Mathematical Monthly, Frick and his collaborators introduced an algorithm based on Sperner's lemma proving that one can achieve "rental harmony" even while only taking into consideration the preferences of two out of three of the roommates sharing an apartment.
Another study of Frick published this year in the Society for Industrial and Applied Mathematics Journal on Applied Algebra and Geometry found new obstructions for intersection patterns to arise from arrangements of convex open sets in Euclidean space. Frick's These obstructions completely classify intersection patterns Fellowship for a more flexible topological relaxation of arrangements of convex Figure. Sperner PL Map Assistant Professor Florian open sets. Frick was among 126 This question has applications to recipients of 2019 Sloan neuroscience, particularly for studying cofiring patterns of so-called "place Research Fellowships, which For Frick, who has collaborators scattered cells" in the brain that fire when an around the United States and Europe, the honor early career scholars. organism is in a certain place in its fellowship will allow him to invite them to environment. "Sloan Research Fellows are the best young Carnegie Mellon to develop new ideas and scientists working today," said Adam F. Falk, further their work on common projects. A native of Germany, Frick came to president of the Alfred P. Sloan Foundation Carnegie Mellon in 2018 from Cornell in announcing the fellowships in February. "Over the next two years, I hope to expand University, where he was a H.C. Wang "Sloan fellows stand out for their creativity, the reach of topological methods and find assistant professor. During that time, he for their hard work, for the importance of numerous new problem areas that can also spent a semester as a postdoctoral the issues they tackle and the energy and benefit from a topological viewpoint," fellow at the Mathematical Sciences innovation with which they tackle them. To Frick said of his plans for the fellowship. Research Institute in California. be a Sloan fellow is to be in the vanguard of He noted that these geometric and He earned his Ph.D. in 2015 from 21st century science." topological methods can yield fruitful results Technische Universität Berlin, from in other branches of mathematics and their which he also obtained his bachelor's The Sloan Fellowships are open to applications. scholars in eight scientific and technical and master's degrees. fields — chemistry, computer science, Frick describes his research as working "at economics, mathematics, computational the triple-point of combinatorics, topology and evolutionary molecular biology, and geometry." An example of Frick's neuroscience, ocean sciences and physics. research is his work to determine whether Winners receive a two-year, $70,000 it's possible to find four points that form a fellowship to further their research. square within a curve in a plane. Another
14 15 2 1 Two years after the publication of Birch's points in either one point is contained in Avoiding Coincidences d + 1 of the points of X , as it must contain paper, Norwegian graduate student Helge the triangle spanned by the other three or by Florian Frick at least one vertex per simplex. For d=1 this Tverberg proved the same result, unaware the four points come in two pairs such that recovers the notion of median. Birch already of the earlier work by Birch. After learning the segments connecting the pairs intersect. The complete graph on five vertices is recognized that one can be more economical about Birch's theorem for the planar case Another way of saying this is that for any non-planar that is, for any drawing in the about the number of points. He conjectured and the corresponding conjecture for straight-line drawing of a tetrahedron in the plane, two edges share a point outside of a that any (n– 1)(d + 1) + 1 points in d higher dimensions, Tverberg visited Birch in plane two vertex-disjoint faces must intersect. common vertex. The real projective plane may be partitioned into n sets X , … , X 1 n Manchester in 1964. Tverberg would later It is natural to ask whether the drawing needs P 2 does not embed into 3, that is, whose convex hulls contain a common point describe the weather in Manchester during to consist of only straight lines, or whether the 2 3 and confirmed this conjecture ford = 2. It is any continuous map f : P ―> has his visit as "bitterly cold," and unable to sleep overlap of vertex-disjoint faces is a topological a double point. In fact, any immersion of easily checked that for a sufficiently generic d for the last night of his visit, since the heater in feature that holds more generally for any 2 3 set of (n– 1)(d + 1) points in even the P into has a triple point. This article the hotel had gone off, the solution to Birch's continuous drawing. Thus, in 1976, Imre affine hulls of any partitionX , … , X into n discusses some fragments of the story of 1 n conjecture finally dawned on Tverberg. Since Bárány conjectured that for any continuous parts have empty intersection, and thus this developing a theory of non-embeddability then Birch's conjecture has been known as map from the (n– 1)(d + 1) – dimensional result would be optimal. and extensions to higher multiplicity Tverberg's theorem and is now considered a simplex to d there are n pairwise disjoint coincidences (such as the existence of triple central result in discrete geometry. faces whose images have a common point of points, quadruple points, etc.) from the intersection. The case of an affine map exactly viewpoint of discrete geometry. corresponds to Tverberg's theorem, since the image of a face under an affine map is the For real numbers … x 0 ≤ x 1 ≤ ≤ x2n convex hull of its vertices. the set n [ x ,x ] consists ∩ i = 0 i 2n – i only of the point xn— the median of the xi, which leaves precisely n of the xi to either side of it. Bryan Birch, of Birch and Swinnerton-Dyer fame, showed in a 1959 paper that one may observe a similar phenomenon for tuples of real numbers: Any 3n points in the plane 2 can be Figure 2. Ten points in the plane and a split into n triples of points such that the color-coded partition into four parts with corresponding triangles all have a point n intersecting convex hulls. in common.
For higher dimensions, Birch asked whether d any (d + 1) n points X in determine n The n=2 case of Tverberg's theorem, that vertex-disjoint simplices of dimension d that any d+2 points in d can be split into all contain a common point of intersection. two sets, whose convex hulls intersect, had Any such point of intersection c would be a Figure 1. Any nine points in the plane can already been established decades earlier center point for X : Any half-space containing be split into three triangles that surround a by Johann Radon in 1921. The planar case Figure 3. A linear and a continuous drawing c also contains at least a fraction of common point — the red cross. of Radon's result states that for any four of a tetrahedron in the plane.
16 17 Bárány's conjecture became known as the transform vanishing obstructions for topological Tverberg conjecture and was the existence of equivariant maps from soon after — in 1981— shown to hold for the configuration space ofX into maps n a prime. Özaydin later showed the case X ―> d that avoid n –fold points — at least of n a power of a prime. This dependence as long as X has codimension d in d . on divisibility conditions of the intersection n multiplicity n should come as a surprise. The latter can be used to lift maps with no It stems from the following construction: n -fold points in positive codimension to the case, where may have dimension much If f : X ―> d avoids n -fold points of X larger than — the case of interest for the coincidence, then it induces a map from d topological Tverberg conjecture. n -tuples of pairwise distinct points in X , Gifts from friends and alumni of the The David and Jacqueline Martin Endowed the configuration space of X , to ( d )n that The constraint method proves results about Department of Mathematical Sciences Fund for Undergraduate Research in avoids the diagonal, that is, the subspace the existence of n -fold points in a generalized Mathematics, which has the exclusive purpose d n of ( ) , where all n factors are equal. fashion, encapsulating graph planarity, the have been crucial in the development of of supporting Summer Undergraduate d Moreover, this map respects the symmetries embeddability of manifolds into , and their our summer research programs. Endowed Research Fellowships in mathematics. of permuting coordinates. Obstructions higher multiplicity generalizations. Perhaps funds have been established to ensure The Steven Shreve Endowed Fund for the for the existence of such equivariant maps surprisingly, it also exposes chromatic that students now and in the future have Mathematics of Finance, which supports depend on whether n is a prime power or numbers of uniform hypergraphs as a more research and educational programs in the not, since the symmetries have a different rigid version of the same theory. access to these valuable programs. Department of Mathematical Sciences structure depending on the number of Many seemingly simple questions remain focused on computational finance. The distinct prime divisors of n . More precisely, If you would like to help support unsolved. For instance, given three red, three department is currently using this endowed an elementary abelian group of symmetries green, three blue and one yellow point in the undergraduate research in the fund to support the Mathematical Finance acts transitively on coordinates if and only if plane, can they always be split into four sets Summer Undergraduate Research Program. is a prime power. department, consider giving to: n without repeated colors whose convex hulls That the topological Tverberg conjecture all have a point in common? could only be settled in the case of n a prime power was believed to be an artifact of the proof method, which relies on lower bounds for the equivariant topology of the associated configuration space. Recent (2015) counterexamples to the topological Tverberg conjecture for every n with at least two distinct prime divisors thus were a surprise to many. The main ingredients for these counterexamples are an extension of the Whitney trick, which removes pairs of double points to construct an embedding, to higher multiplicity intersections due to Isaac Mabillard and Uli Wagner, and the "constraint Figure 4. Three red, three green, three blue and method" of Pavle Blagojevic, Günter Ziegler, one yellow point in the plane and a colorful and the author. The former provides a way to 4-fold Tverberg partition. For details on giving visit cmu.edu/math/giving
18 19 in memorium
Union exposed economic conditions much worse than depicted in the Patsy J. state press. In a 2016 interview, Balas described the difficulty of his transition after spending decades "P.J." trying to make economic sense of Marxism and socialism. But he also McCarthy said this disillusionment helped spur his turn to mathematics. In 1959, Longtime office manager and at the age of 37, Balas immersed himself in the then-emerging field administrative assistant to the of linear programming, gaining heads of the Department of recognition with a novel solution to a Mathematical Sciences, died timber-harvesting problem. He called on Sept. 27, 2019. She was 68. his solution the Additive Algorithm, similar to what is known as implicit numeration or constraint propagation For the last 34 years, which her family today. He later earned Ph.D. degrees described as "very good years," Patsy (P.J.) in economics (University of Brussels, McCarthy could be found working at 1967) and mathematics (University of Carnegie Mellon, first in the College of Paris, 1968). Humanities & Social Sciences, then in the math department's Wean Hall offices. She He was recruited to Carnegie Mellon came to CMU from Westinghouse Research, in 1967, where he continued to where she worked until the company's develop new methods in the field of Harmar, Pennsylvania offices closed. Egon Balas integer programming, in particular disjunctive programming. At CMU, Having come from a staid corporate University Professor of Balas was born into a Hungarian-Jewish family he co-founded the Algorithms, environment, McCarthy had to adapt to the Combinatorics and Optimization different world of academic management. Industrial Administration and in Cluj, Romania, in 1922. After high school, he wanted to continue his studies in physics but program, a joint Ph.D. program She quickly made the transition and enjoyed Applied Mathematics and was blocked by anti-Semitic laws. Determined between the Mellon College of the setting. Her adaptability was further Thomas Lord Professor of to fight Nazism, he joined the underground Science, Tepper School of Business demonstrated by her success in working Operations Research, died on Hungarian Communist Party and was arrested and School of Computer Science. with four different department heads, William by Fascist Hungarian authorities in 1944. He Williams, Jim Greenberg, Nic Nicolaides March 18, 2019. He was 96. In 1995, he earned the John von was sentenced to 14 years of hard labor but and Tom Bohman, all of whom had greatly Neumann Prize from the Institute escaped during transport to Germany. On his differing personalities and styles. for Operations Research and the "A beloved member of the CMU faculty return home to Cluj, he found that all of his Management Sciences, which "Part of P.J.'s job was to execute directives for more than half a century, Egon Balas immediate family had been killed along with is considered the Nobel Prize of by the department head, but on occasion was a preeminent and legendary scholar most Jews who lived in the town. operations research. In 2001, he she did not do that right away because she who was enormously influential in the won the EURO Gold Medal, and in knew the department head should reconsider fields of operations research and applied Still in the Communist Party, Balas taught 2006, he was elected to the National what he was doing. By consciously waiting, mathematics," said Farnam Jahanian, himself economics and changed his birth Academy of Engineering and the she gave him a chance to do this," said Orion president of Carnegie Mellon University. name, Blatt, a common Jewish surname, IFORS Operational Research Hall Hoch University Professor Steven. E. Shreve. "Throughout his long and distinguished career to Balas in order to serve in the Romanian of Fame. "I don't think the heads ever knew that she as a researcher and teacher, he applied bold, government as economics director in the Ministry of Foreign Affairs. During a power did this intentionally. They were just relieved focused and independent thinking to solve Balas is survived by his wife of 70 struggle in 1952, he was arrested by party when they changed their mind to find out that complex problems and also demonstrated years, Edith Balas, an emeritus leaders and put in solitary confinement for she had not yet acted. An assistant like this is a profound sense of humility, character and professor of Art History at Carnegie more than two years. the most valuable kind." good humor. His extraordinary life and legacy Mellon; two children, Anna Balas will continue to serve as an inspiration to the Released from prison in 1954, Balas became and Vera Balas Koutsoyannis; All agreed that she was indispensable to the entire CMU community." disenchanted with communism, especially three grandchildren; and four department. During the transitions between after a trip with his wife, Edith, to the Soviet great-grandchildren. department heads, McCarthy was credited
20 21 in memorium
with holding the department Moore was seen as an educational leader together. by the department, college and university. He planned most of the developments in "The secret of her success the department's undergraduate curriculum lay in a cheerful cynicism that happened during his tenure and was and great good humor," said involved in curriculum planning at the Professor Emeritus Williams. university level.
McCarthy was an efficient and He received the university's William dedicated administrator who H. and Frances S. Ryan Award for kept the department running Meritorious Teaching. In his nomination smoothly for many years. She Richard A. for this award, he was characterized as a managed the department's "student's professor," referring to the extra administrative staff. She Moore commitment he gave to his students. was known for running a supportive workplace, Professor of mathematics and Moore also made significant contributions with most staff under her to graduate education at Carnegie Mellon, supervision staying with the former department head, died playing a key role in the development of the department for decades. on Oct. 20, 2018. He was 94. Doctor of Arts in Mathematics program, which was designed to prepare students to Her work contained numerous become teachers at the collegiate level. responsibilities, including Born in Ohio in 1924, Moore grew up in managing correspondence, Cleveland, New York City and St. Louis. As an administrator, he was known to securing visas for new faculty, After graduating from high school in 1942, efficiently handle the thankless details acting as a liaison to facilities he attended Yale University for one semester that kept the department running, such management, coordinating before joining the infantry to serve in World as scheduling classes, assigning instructors, documentation for the War II from 1943-45. After completing his advising students, supervising teaching department's advisory board service, he returned to Washington University assistants and coordinating with the rest report and much, much more. where he earned his bachelor's degree in of the university. 1948, master's degree in 1950 and doctorate In addition to her work with in 1953. He then served as an instructor at His commitment to education reached the department, McCarthy the University of Nebraska and at Yale. beyond Carnegie Mellon. He consulted was a history buff and with local school districts and area colleges enjoyed being a pen pal Moore came to Carnegie Mellon, then called and represented the university to groups through Pocket Pals. the Carnegie Institute of Technology, in such as the Mathematical Association of September 1956 as an assistant professor America, the National Council of Teachers McCarthy is survived by of mathematics and remained with the of Mathematics, the Pennsylvania Council of her husband of 42 years, department for the next 30 years until his Teachers of Mathematics and Mathematics Barry McCarthy. retirement as a professor. He served two Council of Western Pennsylvania. terms as associate department head, and was department head from December 1971 In 1995, the Mellon College of Science to June 1975. established the Richard Moore Education Award to recognize faculty for their "The word that best described Dick substantial and sustained contributions to was 'affable,'" said Professor Emeritus the educational mission of the college. William Williams. "He was always low key, unflappable and pragmatic in dealing with Moore is survived by his wife of 69 years, both faculty and students. An ideal associate Ruth Neuhoff Moore; three children, Peter A. department head, he always found the way Moore, Susan Moore McJunkin and Sarah to implement policy and procedure smoothly M. Moore; four granddaughters; and six with minimal irritation to all parties." great grandchildren.
22 23 alumni news
of the program, selected startups spend three months in Silicon Valley developing their company and preparing their pitch to investors.
With seed funding from Y Combinator and an investment fund of $50,000, from names such In marketplace lending, investors back as Ron Conway and Max Levchin, Edmundson personal consumer loans — such as debt and Agrawal hit the ground running to build consolidation, business or home improvement their company. loans. Replacing the traditional role of a bank, companies like Prosper and Lending Club help They started by hiring the best quantitative connect investors directly to consumers to talent they could find. "We are a tech start-up fund these peer-to-peer loans. more than anything else," said Agrawal. "Most of our team is engineers and researchers, In this process, investors generally see a and we are essentially using algorithms and handful of simple metrics such as FICO scores, machine learning to optimize the data and income and the interest rate of the loan to make our decisions." determine buying a loan. On the other hand, the lenders — i.e., Prosper and Lending Club What Theorem essentially does is fact-check — look at upwards of hundreds of different marketplace lending loans through their own factors when pricing a loan. underwriting models to evaluate the pricing of loans and quality of investment for clients. "There's a huge information asymmetry between the originator of the loan and the Their loan scoring technology analyzes more person who is actually bearing the risk and than 1,000 data points to identify the best buying the loan," said Edmundson. This loans, reduce default rates and build strong, asymmetry could lead to a problem if loan diversified portfolios for their clients. lenders over-originate loans or originate lower To prove their models work, Edmundson quality loans. says, the company had to play high stakes In coming up with the idea for Theorem, show-and-tell by buying a bunch of loans Edmundson saw an opportunity to move from and letting them ride for 18-24 months. understanding risk at the portfolio level (what One of the hardest challenges was building investors see) to the loan level (what lenders that institutional credibility. But, in just under see), but there was one missing piece — an two years, the company secured their At 22 years old, Hugh Edmundson took a ability to consider the multitude of factors that first institutional client, a large university risk he wasn't sure would pay off. He left go into originating loans. endowment, and within three years reached Lending his first post-graduate job as a trader at their first $100 million in assets. Morgan Stanley, moved to San Francisco Enter Abeer Agrawal, Theorem's chief As their clients and assets grew, so did the and convinced his fellow computational technology officer. Agrawal, who studied Data company's talent. In the last 18 months, finance classmate Abeer Agrawal to start computational finance and electric and Theorem has grown from a team of six to a their own company. computer engineering at Carnegie Mellon, to Loans began his professional career at Google and team of over 30. Five years later that company, Theorem, then as a founding employee at an early "We are the least smart people at the manages over $1 billion in assets for a stage startup, MobileSpan, which helped Alumni fuse technology and company," said Agrawal. He credits the quality variety of clients. It's safe to say that the businesses access and secure their content of their team and the culture their team has finance to make investing risk is paying off. on mobile devices. Coincidentally, around the built with Theorem's continued success. smarter and safer time the company was acquired by Dropbox, Theorem is an investment management Edmundson and Agrawal had reconnected. "We're still hiring!" added Edmundson. and technology company rolled into one — Edmundson and Agrawal encourage anyone the company uses data science, machine In 2014, the pair joined Y Combinator, a seed interested in what Theorem does to take a learning and software engineering to invest accelerator that has helped launch companies look at their careers page. in marketplace lending loans. such as Dropbox, Airbnb and Reddit. As part
24 25 student news
Under Associate Professor Clinton Conley, Suh has been focusing especially on clopen sets — a set that is both open and closed in Giovanni a specific topological space. Suh said he enjoys doing research, and that its challenges make discovering new ideas Gravina rewarding. "When you're stuck on a problem, Receives you can't ask for help," Suh noted. "Jung Joo is a wonderful representative of Graduate Carnegie Mellon's robust undergraduate mathematics program, where he has built deep foundations in several subfields and Student taken on ambitious research under the mentorship of Professor Clinton Conley," Teaching said Richelle Bernazzoli, assistant director Jung of the Undergraduate Research Office. Award "This is the sort of intellectual boldness that Giovanni Gravina was honored with the with a tone of camaraderie, as if Giovanni Joo Suh the Goldwater Scholarship seeks out and Mellon College of Science's 2019 Hugh was someone they felt very comfortable nurtures. We are delighted to see Jung Joo's Young Graduate Student Teaching Award. with," Associate Teaching Professor Dana Recognized work recognized with this prestigious national A 2019 graduate, Gravina is now a postdoc Mihai wrote about her experience teaching award and we look forward to following his at St. Charles University in Prague. with Gravina in 2015. "Throughout the class, career as a mathematician." I came to rely on Giovanni with confidence as Goldwater Besides research, Suh has also spent time "I was very grateful for the recognition that that everything will be done well." I have received for my work and I was Scholar working as a peer tutor, grader and teaching assistant. He's excited about his Goldwater reminded of the enthusiasm needed to grow Several of Gravina's former students Scholarship, and he said applying for it as a teacher," Gravina said of the honor. also wrote letters attesting to Gravina's Undergraduate student Jung Joo Suh has helped him organize his research into teaching prowess. "His thoughtfulness, was named a 2019 Goldwater Scholar by "Giovanni takes his teaching extremely a coherent narrative. Overall, Suh believes respect for students and genuine interest in the Barry Goldwater Scholarship and seriously and our students really appreciate being recognized as a Goldwater scholar will helping us learn still stand out to me," wrote Excellence in Education Foundation. him," wrote Professor William Hrusa in aid his goal of attending graduate school. "It Meredith Wong about Gravina's teaching nominating Gravina. "He has been an Suh, a native of South Korea who came shows that you're ready to be a researcher," of her 2017 Differential Equations course. outstanding citizen around the department." to Carnegie Mellon from New Jersey, plans Suh said. "I felt like he genuinely cared about my and to enter a Ph.D. program in mathematics Hrusa noted that Gravina substituted for other students' learning because he was so He is one of 496 recipients of the scholarship, after graduation, with a focus on research or assisted with a wide variety of expressive in his teaching and was always which recognizes second- and third-year in descriptive set theory and analysis. undergraduate courses in his time at willing to put in the extra time to make sure college students intending to pursue research Carnegie Mellon, and earned the respect we understood things." "I've been interested in math for a long careers in mathematics, engineering and the of the department's faculty. In 2016, for time," Suh noted, especially set theory, natural sciences. A total of 1,223 students "He was always exceptionally prepared, example, the department entrusted Gravina which studies well-behaved subsets were nominated by 443 academic institutions well-spoken and attentive to the needs of with assisting a professor having serious of certain topological spaces. "These for the award, which provides up to $7,500 the class," Newton Xie wrote about Gravina. health issues because "we had complete topological spaces can sometimes be per undergraduate year for tuition, mandatory "His instruction set a high bar for excellence faith in him to handle the class on his own visualized as a certain graph, and with fees, books, room and board. and helped ease my transition academically if necessary." the graph theoretic interpretation a lot of during my first semester at the university." interesting questions can be asked." "His office hours were always well attended, and our students often mentioned his name
26 27 SURF spotlight Summer 2019 Math SURF Research Projects
Numerical Max Aires Xiang Si Edge Density of Maximally Minimizing Curve and Distribution Analysis Linkless Graphs of Average Network Commute Advisor: Florian Frick Time Problem Advisor: Dejan Slepčev of Liquid Zhijie Chen Finite Difference Scheme for Noah Stevenson Crystals Solving the Allen-Cahn Equation Steady-State Stability Analysis Coupled with an Elliptic Equation of Periodic Micropolar Fluids Liquid crystals occur naturally in cell membranes Advisor: Franziska Weber Advisor: Ian Tice and are perhaps most famous for their application in liquid crystal displays. Liquid Varun Gudibanda Jianming Wang crystals are an intermediate state between a Numerical Analysis of Horizontal-Vertical Geometry liquid and a crystal. In a liquid, all the particles Liquid Crystals for Signal Comparison are randomly oriented while in a crystal, all the Advisor: Franziska Weber Advisor: Dejan Slepčev particles are completely aligned. In liquid crystals, (Image sequence to the left) particles are thought of as being "vaguely Yuepeng Yang oriented" in that the particles in a small section An Estimator of Point/Manifold- of the liquid crystal will usually be pointing in Scott Harvey-Arnold 1D Free Boundary Problems Manifold Distance Based on the same direction. Physicists have produced Local PCA a system of partial differential equations (PDEs) Advisor: Giovanni Leoni Advisor: Dejan Slepčev that models a system of liquid crystals and predicts their alignment in space and time. Zhiyang He These equations are very nonlinear, which makes New Upper Bound on Extremal finding exact solutions nearly impossible. Number for Even Cycles Meeting of the Minds Advisor: Boris Bukh In his SURF project Varun Gudibanda, now Poster Competition a senior, worked with Assistant Professor Philip Lamkin Franziska Weber on a numerical scheme that David Simmons Prize for Undergraduate Log-convexity of Moments computes an approximation to the solution of Research in Mathematics of Averages these PDEs. The sequence of images shows Emily Zhu Advisor: Tomasz Tkocz the evolution of this numerical approximation. Multicolor Ramsey Numbers for The small lines represent the orientation of the Small Hypergraphs liquid crystal molecules. The orientations evolve Yile Liu Multilevel Monte Carlo according to a gradient flow with respect to a Honorable Mention Advisor: Yu Gu certain energy potential towards equilibrium. Alp Müyesser The energy potential is composed of an elastic Unavoidable Colorful Patterns in Graphs part and a bulk part (thermotrophic). The Hannah Milano simulation was done with a finite difference Learning Transport Velocity Xinyu Wu scheme that represents a discrete version of Fields from Data A Log-Sobolev Inequality for the continuous gradient flow and preserves the Advisor: Hayden Schaeffer the Multislice, with Applications structure of the energy and its decay over time, roughly speaking. Fei Peng Annie Xu What Can You Draw? Traces for Homogeneous Sobolev Gudibanda is currently working on extending Spaces in Infinite Cylindrical Domains the scheme to more complicated energy Advisor: Florian Frick potentials and adding additional physics, (Image on cover) Young Research Award including fluid flow and external forces. Laura Li & Lily Qiao He is also working on rigorously proving Matthew Shi convergence of the numerical method. Inequalities in Convex Geometry Portfolio Optimization with Advisor: Raghavendra Venkatraman Order Size Impact
28 29 Classclass of of 2016 2019
Graduate Emily Zhu Receives Judith A. Resnik Award
Emily Zhu received the 2019 Judith A. in coding theory and Resnik Award, an award named after the contributed to a paper Challenger astronaut and CMU alumna, titled "Multicast Triangular during Commencement Weekend. The Semilattice Network." award recognizes an exceptional, senior woman graduating with an undergraduate In addition to research, technical course of study who will be Zhu wanted to help others pursuing graduate or professional training develop their skills and in a technical field. increase their appreciation of mathematics. She became a Zhu graduated with a bachelor of math tutor for other Carnegie science in mathematical sciences, with Mellon students in her first a concentration in discrete mathematics year. She also volunteered and logic, and a master of science in with Western Pennsylvania mathematical sciences. She earned the American Regions two degrees concurrently as part of Mathematics League team to her participation in the department's mentor high school students honors program. for math competitions. She became a teaching assistant "Emily's intellect is unsurpassed by (TA) for her department in other math majors in her cohort. She is 2017 and served as a Head incredibly talented mathematically and TA for Mathematical Concepts she has a work ethic to match," Jason and Proofs. Howell, director of undergraduate studies Class of 2019 Destinations for the Department of Mathematical "At CMU, I worked with a variety of people Sciences, said. in math and gained a sense of how to Firms that hired more than explain things in an effective way," Zhu said. Zhu's research is in combinatorics. She one member of the class: Doctoral Programs: completed work described as "cutting While at Carnegie Mellon, Zhu was involved edge" by Howell for her honors thesis on in Math Club and was an organizer for multicolor Ramsey numbers for 3-uniform an undergraduate summer math seminar Berlin Mathematics School hypergraphs with 3 edges, advised by and Carnegie Mellon's Informatics and Alexander M. Knaster Professor and Mathematics Competition. She participated Dartmouth, CS Department Head Tom Bohman. in designing and building the Math Club Illinois, Math Booth for Spring Carnival three years in a Interest in her thesis topic developed in row, winning Best Blitz Booth in 2018. Tennessee, Math her first year when she took Bohman's course in discrete mathematics and Zhu graduated with university and college McGill, Math continued to grow when she studied at honors and is pursuing doctoral studies at the Budapest Semesters in Mathematics the University of California, San Diego. Penn, Math Program after her first year and through her later coursework. "I will remember CMU's amazing network of Pitt, Physics supportive professors and my collaborative After her sophomore year, Zhu studied peer group," Zhu said. UCSD, Math at Clemson University as part of the Research Experience for Undergraduates Image. Emily Zhu with Dr. Amy Burkert, program, where she worked on research vice provost for education.
30 31 Franziska Weber Assistant Professor, Department of Mathematical Sciences
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