Fall 2016
Published by the Courant Institute of Mathematical Sciences at New York University Courant Newsletter Modeling the Unseen Earth with Georg Stadler Volume 13, Issue 1 p4
p12 p2 IN MEMORIAM: ENCRYPTION FOR A JOSEPH KELLER POST-QUANTUM WORLD WITH ODED REGEV
p8 FOR THE LOVE OF MATH: CMT nurtures mathematically talented, underserved kids
ALSO IN THIS ISSUE:
AT THE BOUNDARY NEW FACULTY OF TWO FIELDS: p11 A COURANT STORY p6 PUZZLE: FIND ME QUICKLY CHANGING OF p13 THE GUARD p7
IN MEMORIAM: ELIEZER HAMEIRI p10 Encryption for a post-quantum world With Oded Regev
by April Bacon
assumed secure because the problem on which it is based — in this case, factoring very large numbers — is assumed very hard. “The key to almost all quantum algorithms is something called the quantum Fourier transform,” says Oded. “It allows quantum computers to easily identify periodicity [occurrences at regular intervals], which, as it turns out, allows to solve the integer factorization problem.” By using Shor’s algorithm, quantum computers will be able to break systems used today very quickly — it is thought in a matter of seconds. The speed is “not because the computer is fast – this is a common misconception,” says Oded. “It just does
©NYU Photo Bureau: Asselin things in different ways.” Whereas a regular computer has bits that alternate between (From left to right) Oded Regev with Ph.D. students Noah Stephens-Davidowitz and state “0” and “1,” a quantum computer – Alexander (Sasha) Golovnev. by putting photons, electrons, and other Oded Regev established a landmark lattice- Public-key encryption is “one of the quantum objects to work as “qubits” – can based encryption system, which could help main conceptual discoveries of the 20th take advantage of a quantum phenomenon keep the internet secure in a post-quantum century,” says Oded. Discovered in the 70s, called superposition. Superposition allows computing world. it allows information to be securely sent each qubit to be in state “0,” “1,” or a across the otherwise insecure internet and combination of both, a phenomenon that Oded Regev has been fascinated by phone lines. “Everyone is listening. Still, we “stacks” when multiple qubits operate lattices ever since they were first introduced can communicate securely.” How is that together, enabling “a huge superposition of to him in the final semester of his Ph.D. possible? “The key is that what’s in my brain all those qubits together, coherently doing program at Tel Aviv University, fifteen years is something only I know. I randomly choose calculations,” explains Oded. ago. A lattice is “a regular arrangement of a secret. I will tell you something about the Now in the prototype phase, fully points in space, as formed by atoms in a secret, and what I tell you will allow you to operable quantum computers would be crystal, when packing oranges in a crate, send me encrypted information,” he says. the scientific achievement of a lifetime. It’s or by a honeycomb in a beehive,” explains “We’ve gotten used to it perhaps — but that an advance that many experts say could be Oded, who has been a Professor of Computer this exists is amazing.” a reality within one to two decades and a Science at the Institute for four years. Lattices Cryptographic systems pave prospect that keeps cryptographers up at first became a subject of mathematical the way for many daily activities in our night. “We need alternatives,” says Oded. “If inquiry in some early work in number theory technologically advanced society. They verify in a year or two someone finally makes this in the 19th century. The real foundation, signatures on documents; secure our emails, breakthrough, chaos will ensue, because though, was built in the early 1900s by ATMs, and chip cards; block or allow access no one will have any way to encrypt. We Hermann Minkowski, who also gave the area to websites and cable TV; and authenticate a would have no way to communicate securely of study its name: geometry of numbers. business’s website. “Cryptography also offers anymore.” Cryptographic applications of lattices were us lots of advanced tools beyond encryption first proposed by Miklós Ajtai in the 90s, and authentication. E.g., it offers solutions Learning with Errors and it is within that context that Oded has for anonymity, deniability, and even online The answer, it turns out, is in error. found many deep questions regarding voting,” says Oded. The key to lattice-based cryptography their mathematical and computational But a breakthrough discovery made is in introducing error and thereby properties. One of his most foundational by Peter Shor (MIT) in the mid-90s showed disrupting periodicity, something not easily scientific contributions to the area is known that systems like RSA (which stands for accomplished with RSA and other number as Learning with Errors, a problem that Rivest-Shamir-Adleman), which are currently theoretic schemes. To show how it works, serves as the basis for a multitude of efficient securing these everyday activities, could Oded offers a simplified demonstration of lattice-based cryptographic constructions, be broken if quantum computers are built. Learning with Errors (LWE): and in particular public-key encryption. Like other systems of encryption, RSA is
2 — Courant Newsletter, Fall 2016 Bob wishes to send Alice a dimension of the lattice,” says Oded. primitive known as fully homomorphic secret yes or no message. To initiate the “One surprising and unique aspect encryption. Craig Gentry (IBM) discovered communication process, Alice chooses a of this security proof is that it requires “encryption in the cloud” – its trendier alias large, odd number, ideally over 100 digits. ideas from quantum computation,” he – which allows companies to work on users’ For sake of ease, we’ll use something smaller adds. “This is totally unexpected since the encrypted data without having to decrypt here: 1,001. This number itself remains cryptosystem itself has nothing quantum it. Inefficiencies are still being worked out, a secret. She sends Bob a list of numbers in it — just adding and dividing numbers.” but once it is ready for prime time, it will which are multiples of 1,001 with some allow a company to, for example, analyze small, even numbers (the “error”) added in: Encryption in the cloud someone’s DNA in encrypted form without 65,069 (1,001 times 65 plus 4), 17,023 (1,001 LWE is a landmark contribution the company being able to look inside the times 17 plus 6), 144,146 (1,001 times 144 which brought lattice-based encryption file. “It sounds like a contradiction,” says plus 2), and a couple dozen or so other such systems into the realm of the possible. Oded. “But it’s not, because the answer they numbers. Bob then randomly chooses some By now the most advanced and deeply- send is also encrypted – they can’t even numbers from this list. For simplicity, say researched scheme for encryption in a read the answer. Only you can later open he chooses the first and the last included post-quantum world, it could also prove the envelope. For many decades, people above. He adds these together to get 209,215 useful for other important cryptographic didn’t even know this exists. Now we know and if his message is “yes,” he sends that applications such as digital signatures. But these things can be constructed based on number as-is. If his message is “no,” he adds perhaps the most surprising application Learning with Errors.” n a one and sends 209,216 instead. Alice then of LWE is in constructing a cryptographic divides by her secret key – 1,001 – which will yield an even remainder (6) for “yes” or an odd remainder (7) for “no.” “If you don’t add the small, even A famous theorem in reverse numbers in the first step, it’s not secure because all these numbers will be divisible Hermann Minkowski’s 1910 paper “Geometry of Numbers” proved the most by 1,001. In this case, you could easily celebrated theorem of the area, known as Minkowski’s first fundamental theorem. determine the secret,” says Oded. “Euclid It proves that if you look at a lattice’s cloud of points from afar and there are a lot of knew that. It’s a 2,000-year-old technique.” points, and you then zoom in, “there will be a small area around the origin where Part of the beauty of LWE is in its you will also have a lot of points,” says Oded. In other words: Global density implies simplicity. “Learning with Errors is probably local density. the easiest encryption scheme you can While thinking about this classic paper, Daniel Dadush, a former postdoc imagine,” Oded says. “It’s mainly adding at the Institute who is now a researcher at the National Research Institute for numbers and dividing. There’s not more Mathematics and Computer Science (CWI) in the Netherlands, asked if the theorem to it than that. There’s no deep math in the would hold in reverse: Does local density also imply global density? system itself — the math appears in the “It’s a fantastic question that was waiting to be asked for over a century; I security proof.” know of very few researchers that can come up with such conjectures,” says Oded. Motivated by the beauty of the problem, Oded and Daniel began working about five Security without Q.E.D. years ago to prove that the reverse version of the famous theorem is true. The con- It is always possible that a brilliant jecture found some applications in cryptography, as well as in additive combina- mathematician or computer scientist will torics, in analysis of Brownian motions, and in the integer programming problem. devise an algorithm powerful enough to “That’s nice,” says Oded. “It shows that Daniel’s conjecture is natural.” break a code. In other words, as Oded says: It is especially useful in problems for which the global picture is known but “In cryptography, there is no Q.E.D. We the local one is not – by analogy think of something that appears sharp to the naked can never be absolutely sure that a system eye but blurs under a magnifying glass, says Oded. “In the application to additive is secure.” Instead, we can substantiate combinatorics, we need to understand the structure of random lattices. There are a system’s level of security by proving some beautiful theorems that tell us what the global picture looks like, but when that breaking a system implies solving it comes to the local behavior, it seems very hard. This conjecture tells us locally a particular intractable (i.e. very hard) you will not have too many points, because if you had too many points locally, you problem. would also have too many points globally, and we know this doesn’t happen. So Oded has shown that if LWE we’re using it in the contrapositive.” is broken, we would then have a fast Building on work done earlier with Daniel, Oded and his Ph.D. student Noah algorithm for finding short vectors in Stephens-Davidowitz proved the conjecture just this November. lattices, a classical hard problem in the “With this work, it was magical,” says Oded. “If you asked me a few months geometry of numbers. The sheer size of ago, I would say there’s no chance. We’ll never prove it, because it seems so hard. high-dimensional space makes them hard Then someone comes with the right idea. It’s fun. It doesn’t happen that often in life to find, a phenomenon referred to as the – but sometimes it happens.” curse of dimensionality. “We believe this should take time that is exponential in the
3 Modeling the Unseen Earth With Georg Stadler
by April Bacon
over 7.5 miles) and, in total, is about 1,800 Often, solving mathematical miles deep. The mantle moves by convection. equations that describe physical processes That is, it is heated from below, causing requires explicit solvers, which are relatively plumes of warmed mantle to rise toward the easy to use on supercomputer clusters. surface, in much the same process as water But with problems like mantle convection set to boil on a stove. Mantle convection that have a huge range of time and space occurs very slowly — over millions of years. scales, explicit solvers don’t work. “To do a The Earth’s tectonic plates are formed when simulation of waves that travel through some
©NYU Photo Bureau: Asselin mantle rock cools at the Earth’s surface. medium — through the Earth, for instance Convection and buoyancy forces cause — [explicit solvers are] enough,” says Georg. Georg Stadler (seated) with Ph.D. students them to press down and sideways through However, “for these mantle convection fluid Yair Daon and Karina Koval. the mantle in a process called subduction, dynamics problems, they cannot be used as moving no more than 10 cm per year, and the physics simply does not allow that.” Georg Stadler and collaborators have other rock with very different viscosities Implicit solvers require solving linear developed fast algorithms to simulate flows at different speeds. “We have some and nonlinear tightly coupled matrix systems, the flow of Earth’s mantle on the global basic idea about how [convection in the which is difficult for very large problems. scale, and inverse methods to improve mantle] works, but there are fundamental Our desktop computers and even cell our knowledge of the mantle from surface uncertainties because we can’t look inside,” phones are capable of quickly solving matrix observations. These methods also turn out to says Georg. systems when the equations number into the help in understanding Antarctic ice sheets. In order for Georg and collaborators hundreds and thousands. But very complex, to build an accurate model, they had to huge systems — like those describing What we know about the structure develop algorithms that could handle the the flow of the Earth’s mantle — demand of the Earth, we know largely because of highly heterogeneous contents of the mantle, equations numbering into the billions. It seismic waves. Earthquakes, which are as well as resolve features at both local and is very difficult to efficiently use cluster caused by the sudden movement of tectonic global scales. While the Earth’s circumference computers to solve such systems. “These plates, have played a necessary and starring is approximately 24,900 miles, the model systems are so large and so difficult that the role in helping us to see what we otherwise also had to account for the subduction challenge, in some way, is to figure out how could not: the shape, density, depth, and dynamics of plates, which occurs in the to solve them using the computer hardware approximate composition of the Earth and narrow plate boundary regions that are only we have today,” says Georg. The genius of its layers. Much of this work was done by a few miles thick. the method devised by the team amounts to geologists and geodynamicists, with first “Algorithms have to focus on small- figuring out how to design implicit solvers strides made in the early 1900s, but not really scale features when you need them to, but that are scalable — that is, they get faster taking hold and shaping what we know today on large-scale features when you don’t,” says with each addition of a new computer in until mid-century. The field is relatively Georg. “It’s very useful if you can put down the cluster. young — and is providing rich questions a fine mesh — many degrees of freedom, Their work made interdisciplinary for mathematicians like Georg Stadler, who, many unknowns — where you need them. In gains in mathematics, geodynamics, along with collaborators, received the 2015 other regions, you don’t want to do this. This and computing. As the ACM’s Bell Prize Association for Computing Machinery’s is what’s called adaptive mesh refinement.” citation states, “The group’s submission Gordon Bell Prize for the team’s “trailblazing The computational mesh in figure 1, a demonstrates that, contrary to conventional approach to modeling Earth’s geological zoomed-in section of the mantle, consists of wisdom, implicit solvers can be designed processes.” cubes, each of which represents a nonlinear that enable efficient global convection Using the IBM Sequoia — one of the equation. “We have 500 (and more) million modeling of the Earth’s interior, allowing fastest supercomputers in the world — and of those equations, which we cannot fit on researchers to gain new insights into the the team’s innovative and carefully designed, a single computer, so we have to fit them on geological evolution of the planet.” And IBM scalable algorithms, they developed a thousands of computers – a cluster. It’s not reports that “the team’s code reached an groundbreaking simulation of the flow of trivial to use these supercomputers. They unprecedented 97 percent parallel efficiency the Earth’s mantle and the corresponding are composed of many regular computers in scaling the solver to 1.6 million cores, a motion of tectonic plates. The mantle is the connected to each other; they each have new world record.” A large group from IBM layer of the Earth just under the tectonic separate memory, and algorithms have to Zurich helped to tune the algorithms for the crust. It begins at just over 20 miles down coordinate how computers talk to each other supercomputer, to make them faster. They (the deepest we have physically drilled is just to exchange information.” were one part of the team, contributing their
4 — Courant Newsletter, Fall 2016 Figure 1: Cross- sections of the Earth show tectonic plates (in blue) running into the mantle. Areas with smaller cubes require more equations to properly simulate the flow. Colors illustrate different mantle viscosities.
Figure 2: The three maps show three different possibilities for how strongly the ice is connected to the ground, with blue indicating the strongest connection, red the weakest expertise along with Georg and other applied connection. We can be more confident that our basal map is accurate in areas that are mathematicians, and geophysicist Michael consistent across the images. Gurnis, director of the Seismological Lab at the California Institute of Technology. things we can observe. In the Antarctic ice would give you many possible maps that problem, this means finding the boundary describe the connection between the ice and Shedding light on a cold case conditions at the ice’s base from satellite rock. This reflects the level of uncertainty Some of Georg’s colleagues at research observations of surface flow velocities. For inherent in inverse problem solutions, due labs are using similar methods to develop the mantle flow problem, we can observe to observation and model errors, and the fact a model that can predict, under different plate motion, mountain building, and the that inverse problems are so-called ill-posed climate conditions, how much land ice location of Earthquakes, each of which mathematical problems – that is, different might be lost into oceans and contribute to allows us a glimpse of what is happening maps can lead to very similar observations.” sea-level rise. “In terms of the mathematical inside our planet. The inverse problem uses The mathematics behind this equations, this [forward modelling] looks that information to constrain mechanisms approach is, of course, more complex, but very similar to mantle convection,” he and forces in the mantle. Medical imaging Georg is developing methods to make it says. “One is very hot; one is very cold. techniques such as MRI or PET are other more feasible. “Mathematically it is very The timescales are very different. One is examples of inverse problems.” interesting because several fields come hundreds of millions of years; one is maybe There are different approaches for together,” he explains. “Your model is usually tens of years – but mathematically, it’s really solving inverse problems. A deterministic a partial differential equation. Then aspects not that different. It’s a very similar PDE model yields an Occam’s razor kind of of optimization and probability theory [Partial Differential Equation] called the solution — a single best guess of conditions. come in. Finally, numerical methods and Stokes equation. That’s nice, because that This method is still the one most employed computing are required to approximate the shows that math gives you tools that can be today and is used, for example, by oil problem solution.” applied universally.” companies to make decisions about where “Some science goals are so One barrier to improving the to drill. “A deterministic inverse-problem challenging that we might not be able estimation of sea-level rise is that we lack approach [for ice sheets] would show a single to achieve them any time soon. For a complete model for Antarctic ice sheet map that gives you an estimate of how much instance, probabilistic inverse problems dynamics. The challenge is that there is no resistance the ice experiences when sliding that combine various observational data way to directly observe what’s happening over the bedrock,” says Georg. “It’s your best with fully resolved, three-dimensional, at the boundary between land ice and the guess, but it’s not going to be the reality.” complex models —such as time-dependent ground on which it sits, but Georg has A newer, probabilistic approach mantle flow or ice sheet dynamics — are developed a way to apply sophisticated to inverse problems called “uncertainty likely to remain grand challenge problems,” inverse methods to uncover what’s likely to quantification” provides a more complete concludes Georg. “But I can make two or be happening at that interface. answer because, in addition to predictions, three steps towards these goals. That’s a “Many problems are inherently inverse it includes a measure of how confident we great motivation for me — to develop and problems because they involve things we can be in this prediction (see figure 2). “Good analyze the mathematics and algorithms cannot observe directly,” explains Georg. simulation results should come with some that will be useful for specific applications, “But we can infer what we can’t observe measure of uncertainty,” Georg says. “This and hopefully also for many others.” n by combining mathematical models with probabilistic approach to an inverse problem
5 At the boundary of two fields: A Courant story
by April Bacon
that have been out of the reach of current “From there on, it was just trying mathematical techniques,” says Aukosh. to analyze this variational question,” says “Using their rather ingenious methods Aukosh. they’ve been able to develop beautiful There was a difference in theories for how to solve these problems.” mathematical languages to sort out, but “we Mathematicians in this area are working to had the advantage of being good friends,” add mathematical rigor to those theories and says Ian. “We had been talking about math to describe the nature of spin glasses. for some time, working on homework Aukosh, advised by Professor Gérard assignments together and studying for the Ben Arous, presented his first project as a oral exams and so on.” Ph.D. student on spin glass-related work at The next big step came one night the Banff International Research Station. when the two were doing calculations on Afterwards, attendees suggested that he the board. Ian wrote an expression that should think about the stronger version of his Aukosh recognized. “I went and dug up an underlying theory. “So now the question was, old textbook, and I realized that what he ‘Can you quantify this conjecture?’” he says. had written was the equation for this very ©NYU Photo Bureau: Asselin “I realized with time that there was a more famous conjectured phase boundary in these Aukosh Jagannath and Ian Tobasco concrete question to ask, that sounded more problems called the de Almeida-Thouless manageable, about studying the properties line.” For a classic model of spin glass, de of free energy, rather than their fluctuations. Almeida and Thouless were able to describe Friends and recent Courant Ph.D. I beat my head against this problem for a a curve (the “AT line”) which laid out the grads stumbled upon a problem during while, and then I realized that I wouldn’t be boundary between “replica symmetry” and their graduate studies that made them able to answer it until I understood how to “replica symmetry breaking.” If a spin glass collaborators. solve a certain variational calculus question.” system is found within replica symmetry, Aukosh Jagannath and Ian Tobasco Lucky for him, his best friend and peer was it was then predicted to be ordered; and in both joined Courant as Ph.D. students in Fall studying in that very area. replica symmetry breaking, disordered. 2011 and became fast friends. The former Aukosh found the point of entry for “Very early in my training with studies probability theory—more specifically, the collaboration in a paper by Antonio Gérard,” continues Aukosh, “he mentioned mathematical questions arising from Auffinger (a former student of Ben Arous’s) that a big driver of research in this field was statistical physics—and the latter calculus of and Wei-Kuo Chen. And so in early fall 2014 in trying to prove that this line was actually variations and partial differential equations over lunch in Warren Weaver Hall’s 13th floor the correct phase boundary in these systems. (PDEs), especially those arising in elasticity lounge, Aukosh presented a particular PDE In the end, nobody really got a satisfactory theory. Neither expected that in just a few to Ian and asked: “Do you recognize this answer. But Ian and I got a new foothold on years they would be collaborators, tackling formula?” the problem.” problems that arise in ‘spin glass’ systems— “The formula involved solving what’s “We were able to prove that a certain work that just so happens to require the joint called the Hamilton-Jacobi-Bellman PDE natural generalization of the AT Line was expertise of their two disciplines. by a stochastic optimal control approach,” correct in all of phase space except for a Spin glasses are a kind of highly says Ian. “At the time we weren’t using that compact set,” says Ian. “Meaning, practically disordered magnet first studied by physicists. language, because we hadn’t identified it speaking, a set that you might attack on a Unlike a simple magnet where all of the as such. Auffinger and Chen were solving computer. For the classical mean field spin “arrows” of magnetism point in predictable this PDE by writing down an optimization glass model—the Sherrington Kirkpatrick directions, each atom in a spin glass is formula. I told Aukosh that it reminded me model—this means that we now know that magnetized in a randomized direction. of this Hamilton-Jacobi theory that I had the AT Line is the correct phase boundary This is pretty useless as a magnet, but the studied with my advisor, [Professor] Bob everywhere that our methods apply.” mathematical models used to understand Kohn. Naturally, I went to Bob and asked, ‘Is The pair wrote an initial paper its behavior turn out to have applications there a version of this theory for the elliptic thinking of new ways to look at the result in real-world problems like scheduling, PDE instead of just the first order PDE?’ And from Auffinger and Chen’s paper, a second on message encoding and pattern recognition. he said, ‘Yes, that’s stochastic optimal control the above-mentioned work with the AT line, They are also core to our understanding of rather than the usual optimal control. That’s and two more looking at replica symmetry solid matter itself, and the models are deeply the difference. You add some randomness.’ breaking in spherical spin glasses, a type of interesting to mathematicians. “There are If you go back to the paper that Auffinger spin glasses which physicists invented to be a many beautiful problems of pure probability and Chen wrote you can see traces of this simpler form of a disordered system. that statistical physicists have encountered connection. It’s just not enunciated in the same language.”
6 — Courant Newsletter, Fall 2016 Setting bounds on complexity After a short period, Ian sent Aukosh Ian, “and specify further where amongst all In metallurgy, annealing is the process a message: “I found the dual. You’re really the bumpiness these components can lie.” of heating up and then cooling a sword going to like it when you see it.” The dual “In the end we were able to resolve in order to beat it into shape. As Aukosh is in a general class of problems known as this picture,” says Aukosh. “It was a very explains, the “mathematical ramifications obstacle problems. Ian offers an example beautiful, old question, this obstacle of this physically intuitive idea can be very which comes from his field: Think of a problem.” A big moment for the pair, he profound. The idea is that if you want to tablecloth pulled tight over a flat or spherical adds, was learning from Professor Sylvia study the smallest the energy could ever be table. One can describe mathematically Serfaty, an expert in the area, that using for a system, you heat the system up and where the cloth meets the table, which is the an obstacle problem as a dual is not only a then cool it down to zero temperature. I had obstacle, and also what happens in-between. natural move, but also one which has driven this idea that maybe you could formalize this With the dual in hand, Aukosh a lot of progress in her area of the variational move by using a technique in the calculus of and Ian studied spherical spin glasses at community. variations called ‘Gamma convergence,’ at zero temperature and then broadened Aukosh and Ian graduated in 2016 least for the kinds of models we study.” what they learned from that work to and their collaboration continues across The two set out to study spherical spin other temperatures. “What we found was borders—Aukosh is now an NSF postdoctoral glasses at zero temperature, quickly turning something very astounding, unexpected,” fellow at the University of Toronto and Ian is the probabilistic question into a variational says Ian. “For the spherical spin glass models, the James Van Loo Post-Doctoral Fellow at one, but then were unsure of how to get a even though replica symmetry breaking the University of Michigan. hold on it so searched instead for a ‘dual’ happens, there is a way once and for all to “I was really lucky that it turned out problem. limit the complexity of the disordered phase that my best friend is an expert exactly in “We fell back to what we had learned and limit it in a very precise way.” the field in which I needed help,” concludes under Bob,” explains Aukosh. “Oftentimes, Going back to the table and the Aukosh. “If the cards played out differently, when you study a problem in mathematics, tablecloth, Ian completes the analogy for how we never would have stumbled upon these especially in variational calculus, what you they were able to set limits on the complexity. really synergistic connections.” find is that there’s the problem that’s given It’s simple to understand where the cloth “This has been a very fruitful to you by the world, by physics, and the meets a flat or spherical table. But now collaboration because it is a variational question is very natural, but trying to get a imagine the cloth pulled taut across a bumpy problem, that’s what my speciality is,” says foothold on it is very difficult. The idea is that table—the points of contact become more Ian. “But it says lots of very deep things you find a problem that’s the mirror image complex. “What a bound on this complexity about a probabilistic system, that’s what of this problem. If you can solve the dual would mean is that, given the shape of the Aukosh’s specialty is. I’d say we’re very problem, it is exactly the same as solving the table, you are able to limit the number of fortunate. At the same time, maybe that’s the original problem.” disjoint components of the contact set,” says magic of Courant.” n
Changing of the Guard
Gérard Ben Arous steps down; Richard Cole Director from 1994 to 2002 and NYU’s Provost Dhabi and Shanghai. steps in as Interim Director. from 2002 through this past summer. Additionally, Gérard “initiated the With a growing faculty and a steep very successful Center for Data Science at Gérard Ben Arous has stepped down as increase in students, Gérard was also a NYU, as an outgrowth from the excellence Director of the Courant Institute to continue champion for resources for the Institute. “He in machine learning in Computer his service as Professor of Mathematics. He focused on the development of Computer Science,” says Dave. Math and computer held the directorship at Courant from fall Science at Courant, first by finding additional science are at the foundation of the 2011 through August 2016, having served an new space for members of Computer Science Center, which brings together researchers additional year as Acting Director from 2009- and Data Science in the Forbes building,” says and professors from 18 schools and 2010. Gérard made significant strides for the Dave. The building, located on Fifth Avenue colleges across NYU. Institute and University during his ardent and and 12th street, is now in final preparations to As Dave concludes, “In Gérard, influential directorship. receive its new tenants. Courant had an outstanding Director.” “To me, directorships are best In addition to Director of the Institute, Richard Cole, Silver Professor judged through the recruitment of excellent Gérard served as Vice Provost for Science and of Computer Science at the Institute, faculty, and during Gérard’s time as Engineering Development during a time that graciously agreed to serve as Interim Director of Courant, the Institute’s faculty saw significant growth in those areas at NYU, Director until a new permanent Director is improved greatly, with many outstanding including the 2014 merger with Polytechnic named. Richard has provided leadership in mathematicians and computer scientists University in Brooklyn, now NYU’s Tandon several capacities in his over thirty years joining the Institute,” says Dave McLaughlin, School of Engineering, and the establishment at the Institute, and will be filling this Professor at Courant, who served as Courant’s of new global research centers at NYU in Abu particular role for the second time. n
7 For the love of math: CMT nurtures mathematically talented, underserved kids
by April Bacon
©NYU Photo Bureau: Asselin
Ph.D. student Morten V. Pedersen (pointing to a math problem) and CMT summer program students.
Through its in-house activities and for mathematically talented students without In-house Programs partnerships, Courant’s Center for access to opportunities, has continued The CMT also runs its own math Mathematical Talent has reached thousands mentoring alumni from their summer circles for talented students. James Fennell, a of mathematically talented students across programs at Courant, about two hundred to fourth-year Ph.D. student, was first brought New York City since its inception in 2010. date. All of these programs have free access on to co-lead the Center’s math circles in to the space under the auspices of Courant’s both semesters of academic year 2014-15. No classroom goes unused at Center for Mathematical Talent (CMT), Over ten weeks, he taught transformational Courant’s Warren Weaver Hall, even when funded by the Alfred P. Sloan Foundation, geometry—with a brief foray into non- regular classes are finished — during whose generous support has helped Euclidean geometry—to students recruited summers, weekends, and off hours, primary encourage others to invest in the program from two of New York City’s strongest schools and secondary school students enter its and the students it serves. for math and science. And this past spring, halls, eager to deepen their knowledge of “We’re the hub of a lot of math James ran a math circle on the mathematics mathematics. More than three hundred enrichment programs in New York,” says of games (focusing on invariance) with 23 students come to Courant over the weekends Berna Falay Ok, who joined Courant in alumni from BEAM’s summer program. throughout the year for New York Math Circle November 2015 as the new Director for The change in recruitment strategies from groups. Over one-hundred students with the CMT. The Center, now in its sixth the former to the latter is part of the CMT’s the NYC Math Team also meet regularly at year, pursues its mission of reaching broadened mission to meet the needs not the Institute to prepare for competitions mathematically talented pre-college students only of the city’s most mathematically all around the United States. And BEAM in a great variety of ways, and providing advanced students, but also of talented (Bridge to Enter Advanced Mathematics), a space for these programs is just one of them. students from underserved communities. nonprofit organization that runs programs
8 — CIMS Newsletter, Fall 2016 “There was a core group of fifteen “You should see the way the kids Deputy Director of the Capacity Building students [BEAM alumni] who came in interact with Fred,” says Berna. “I’m so glad Unit at DYCD and a recent addition to voluntarily over ten weeks to take the he’s in the program. These are their first the CMT Advisory Board. She has worked program,” says James. “These students interactions with a working mathematician in with the Center for several years, originally were very capable in mathematics but their lives. Math teachers they know—math forging the connection by meeting its clearly hadn’t had many opportunities to do teachers are pressured for time to prepare former director, Mark Saul, at the New York something like this before. They enjoyed it you for your upcoming state exam. But Fred Academy of Sciences. She was there seeking a lot. It’s all kind of new to them, so it was is different. He takes them through a journey: ways to introduce more math into low very nice.” At the Sunday classes, students If you fail, you continue, and you come up. income communities, and a partnership was were introduced to games that evolved into That’s what a true mathematician is, to me at promptly formed. math challenges such as: Can you develop a least. It’s all about struggling and staying in The Center now teaches its “Finding strategy in which player one (or player two) there to solve the problem. It’s not just about Math” curriculum to instructors who always wins? “That’s how you transition the right answer.” take that curriculum into DYCD-funded from a fun activity to then talking about “I think the kids do need that role afterschool programs. Just this past spring, mathematics,” adds James. As a capstone model,” says Selin. “Our grad students are the CMT taught 54 instructors. “It’s sort of experience, the kids participated in an very good. They are leading amazing sessions. like a ‘wow moment’ with the staff,” says exhibition day at the end of the program, But the presence of a known professor—it Reyes-Dandrea of the instructors’ reactions and taught the games and accompanying makes a big difference.” to the curriculum. “They’re like, ‘I didn’t strategies to their parents and friends. know this was math! I didn’t realize this is This summer commenced another Partnering Up what math is about!’ And I went through of the CMT’s internal programs: a three- With the vision of becoming a unifying that transition also. It’s really changed my week, non-residential summer program. force for all math enrichment activities perspective. Math is really about asking The goal, explains Selin Kalaycioglu, across New York City, partnerships are a key why and plugging away—it’s about using Clinical Associate Professor at Courant and component of the CMT’s everyday operations. reasoning and logic skills and thinking more Principal Investigator on the grant from the “There are so many organizations—some of deeply about what questions are being Sloan Foundation, is to develop it into a them are quite big and organized and known, asked.” nationwide, credited course that will be the and some of them are just using their own There is special attention given to keep Center’s signature program. resources within their schools,” says Selin. the activities fun, and the students really “We have 21 students right now,” Through alliances with these take to the math games. “We were teaching says Berna. “We find talented kids, some organizations, expertise and resources are elementary kids how to do ciphers at the from underserved communities, and get combined to best serve students, and the school in Harlem,” says Berna, “and one of them to Courant, give them a feel for what Center can keep track of students better the kids in the group would test her parents university life is and show them why we and use referrals to ensure that they are not every day as she learned things.” After the love math. They come here four out of five abandoned at any point in their development. program, the student’s mother wrote to weekdays from ten o’clock to four. They sit “The center’s mission is to create a pipeline Berna to say, “Just a couple of weeks ago, on down with people that they’ve never met that can take a student from a very early mother’s day, [my daughter] hid ‘secret codes’ before, with people from similar grades all age, starting in elementary school, and then everywhere in our apartment and asked me across NYC, and just do math. You should mentor them through various enrichment to find and decipher them to get clue 1, clue see how excited they are; during lunchtime, activities until they finish high school,” 2, clue 3.” After a very full and challenging to take a break from math, they have been says Selin. “For example, if we detect an morning of deciphering the clues, she playing other math games!” elementary school kid who’s very talented, found her way to a hidden treasure from her The students are taking three after completing a math circle cycle a few daughter: “A sweet mother’s day card made courses: Combinatorics and Graph Theory years with us, then we can direct them to New by herself!” are taught by James and another graduate York Math Circle.” In just the fall of 2015 and spring student, Morten Pedersen, and Introduction One partnership, with the NYC of 2016, the CMT and DYCD partnership to Proofs is taught by Courant Professor Department of Youth and Community reached 498 students. In past years they Emeritus Fred Greenleaf. They also enjoy a Development (DYCD), has been ongoing were in middle schools, but this year went few hours of math games and puzzles at the since 2012. Candace Reyes-Dandrea is Continued on page 10 end of each week.
Find out more about the Center for Mathematical Talent at its recently relaunched website: http://cims.nyu.edu/cmt/
9 For the love of math: CMT nurtures mathematically talented, underserved kids (Continued)
in to two elementary schools, one in Courant faculty such as Gérard Ben East Harlem and one in East Elmhurst. Arous, Sylvain Cappell, and Chuck The change is part of a new CMT Newman, has also provided as-needed effort to reach kids at a younger age. advanced instruction for New York “Finding talented students early and Math Circle students from Courant nurturing them throughout the years is Professors; organized math circles for very important,” says Berna, “because BEAM’s summer program alumni to kids get stressed about math very continue “Finding Math”; in spring
early on, and then they hold on to that 2016, subsidized the cost of attending Selin Kalaycioglu and Berna Falay Ok nervousness.” New York Math Circles’ programs “We’re not giving up on the idea for 67 students from low-income of working with middle school kids,” families; and, this October, with the says Reyes-Dandrea, “but are thinking National Association of Math Circles a little more strategically. If we can start and Mathematical Sciences Research working with elementary school kids Institute (MSRI), hosted at Courant and follow them through, then there’s a a three-day National Association of greater possibility of changing attitudes Math Circles meeting, which gathered to learning. So I’m hopeful. And I math facilitators from all around the appreciate the opportunity to just talk U.S. to the Institute. Professor Emeritus Fred Greenleaf with Berna and the team, because I’m All of the above activities learning as I go along. The way I look at approach math outreach as an it is, I have an understanding of the after ecosystem by working toward the school world in public education, Berna Center’s overall mission from many has an understanding of math—our angles and at many levels, and each combined knowledge can work to the is driven by the common heart of the benefit of these kids.” program, as Berna expresses: “The The Center, which has been love we have for math, we want to pass guided and bolstered by support from it on to kids.” n Ph.D. student James Fennell (left) and students.
In Memoriam: Eliezer Hameiri (September 28, 1947 to June 14, 2016)
Eliezer Hameiri spent his entire but exceptionally talented group. Elie first-rate scientist. He was extremely original career at the Courant Institute, first joining belonged to this distinguished group. He was and a very fine scholar in the classical sense… as a Ph.D. student in 1972. After graduating, trained in that tradition, and carried that In the years following getting his Ph.D., he was having written a dissertation under the legacy forward.” able to turn the problem into a major piece of supervision of Harold Weitzner, he was Elie was dedicated to developing work for the field in developing the underlying immediately hired as a Research Scientist a thorough and precise understanding of structure of ideal magnetohydrodynamics.” and then tenure track professor, attaining his the properties and dynamics of plasmas. In their remembrance for Elie, Fusion full professorship in 1988. He was a critical He “repeatedly uncovered previously Power Associates, an educational foundation part of the plasma physics group at Courant, unknown properties of plasmas and that advocates for fusion power, wrote that which was founded in 1956 by Harold Grad corrected numerous misunderstandings Elie’s “studies of the spectrum of linearized and grew to include faculty members Paul concerning plasma dynamics,” said Gérard ideal magnetohydrodynamics was the first Garabedian, Harold Weitzner, Elie, and at Ben Arous, as Director of Courant, in a complete characterization of the problem and times Steve Childress. message after Elie’s passing in June. And had major implications for the understanding “Elie was a very strong embodiment Elie’s work was significant and impactful of flow stability problems and the role of of the Courant tradition,” says Professor both mathematically and for the physical ‘ballooning modes’ in a plasma.” Amitava Bhattacharjee, who is head of the sciences. “He was one of the people who Another significant body of work, Princeton Plasma Physics Theory program, could do both, and he did them extremely around mid-career, resulted in an early and and a collaborator and longtime friend of well,” says Harold Weitzner. “He was a very fairly complete analysis on the relaxation and Elie’s. “Often, fundamental problems arise fine mathematician.” evolution of turbulent plasma states. While for which the Courant group have time When Elie was a graduate student, “I much literature on turbulence problems and again provided rigorous and beautiful gave him a relatively mundane problem,” is “a bit of a mess,” says Harold, Elie and solutions that the whole community has recalls Harold. “He saw how to do much, collaborators released a series of papers with benefited from. It has always been a small much more with it. And that’s the mark of a “very clean, very straight-forward results.
10 — CIMS Newsletter, Fall 2016 Joan Bruna, Assistant Sylvia Serfaty, Professor Professor of Computer of Mathematics, holds a Science with affiliation Ph.D. in Mathematics from in mathematics and in the Université Paris-Sud. association with the Her research interests WELCOME Center for Data Science, holds a Ph.D. revolve around the analysis of partial in Applied Mathematics from l’École differential equations and variational TO THE INSTITUTE’S Polytechnique. Before moving to problems coming from physics, in NEWEST FACULTY! Courant, he was an Assistant Professor particular the Ginzburg-Landau model of Statistics at the University of of superconductivity, and recently the California, Berkeley. Bruna’s research statistical mechanics of Coulomb systems. interests include invariant signal She was a Courant faculty member from Scott Armstrong, representations, pattern recognition, 2001 to 2008, and, most recently, a Associate Professor of harmonic analysis, stochastic processes, Professor of Mathematics at Université Mathematics, received and machine learning. Pierre et Marie Curie-Paris 6 as well as a his Ph.D. in Mathematics Global Distinguished Professor at Courant. from the University Hesam Oveys, Clinical of California, Berkeley. His research Assistant Professor of Fan Ny Shum, Clinical interests are partial differential Mathematics, holds a Assistant Professor of equations, probability theory, and Ph.D. in Mathematics Mathematics, holds a stochastic homogenization. Armstrong from the University of Ph.D. in Mathematics previously held positions at Louisiana Missouri. His research interests include from the University of State University, the University probability theory and stochastic Connecticut. Her research interests are of Chicago, and the University of calculus. He is the recipient of several stochastic analysis, partial differential Wisconsin, Madison. Most recently, he teaching awards. Prior to joining equations, and sub-Riemannian geometry. was a research scientist at Université Courant, Oveys was a Faculty Instructor Prior to joining NYU, she was a research Paris-Dauphine. of Mathematics at the University of supervisor for Math Research Experience
Missouri. He has also taught at Stephens for Undergraduates (REU) at the College in Columbia, Missouri. University of Connecticut.
But this is what I look for in Elie’s work.” As become aware that it is important to have easily,” says Bhattacharjee. “We were lucky Bhattacharjee says, “whatever he published this more complex model for a plasma,” says that somebody of his talents in mathematics was deeply instructive and often definitive.” Harold. “More commonly people use what was as deeply interested in plasma physics as Phil Morrison of the University of Texas, a is called magnetohydrodynamics or ideal or he was. He was a dear friend. One I trusted. I plasma physicist who also specializes in the dissipative magnetohydrodynamics.” There did some of my best work with him.” mathematical side of research, encountered are a good number of people who work on In addition to his interests in Elie through the years at conferences where these one-fluid models, he explains. There mathematics and physics, Elie cared about the two mutually enjoyed one another’s are also a number who work on two-fluid music, as well as the study of religion, presentations and conversation. Morrison models, such as the Hall model, but none especially Judaism, and the development of echoes the celebration of Elie’s papers, which are really “clean and fully consistent” the State of Israel, where he grew up. Elie was saying that they were “distinguished by as Elie’s models have unfailingly been. steadfast in these areas, too, and had great their crispness and clarity, and enriched Though often blunt in dialogue, after depth in the history of religions, especially in our field by maintaining the careful and a little bit of time spent with Elie, one came the pre-Biblical period. His interest led him to mathematically informed style of Harold soon to realize that, as Harold says, he was “a obtain a degree from the Jewish Theological Grad and other early plasma researchers.” very kind-hearted and dear soul.” Genuine Seminary. “It was always very instructive to At the time of his death, Elie was and caring toward others and in his work, he listen to him because he knew so much,” says continuing with work that began in the early drew a recurring group of visitors from all Harold. “It was quite an exceptional thing.” 2000s to determine the basic physics and over, ranging from postdocs to senior faculty. “Elie’s collaborators and coworkers phenomena of Hall magnetohydrodynamics “It was clear that they saw Elie as somebody consistently looked to him for new with which he could build a model that lived that had something substantial to offer that insights,” concludes Gérard. “He will up to his standards of accuracy, robustness, one couldn’t easily get from other people or be sorely missed by his many students, and elegance both mathematically and Institutions,” says Harold. collaborators, colleagues and friends physically. “In recent years, people have “The likes of Elie don’t come about at Courant.” n
11 In Memoriam: Joseph Keller (July 31, 1923 – September 7, 2016)
Joseph Keller rest of my career I’ve done complex multi- there was a seat available at his table,” says was one of the scale asymptotic modeling of phenomena.” Dave. There, Joe would present the group leading applied Joe used asymptotic analysis as with mathematical challenges found from mathematicians of his main tool for studying a vast range of everyday life. Andy remembers Joe asking: his generation. “He problems and phenomena. “He would Why does old paint curl up on a wall? Charlie showed us how apply that to PDEs, to integral equations, recalls the question: When you put a drop powerful applied to ordinary differential equations; it was of water on paper, why does it spread out a mathematics throughout the field of analysis,” says Dave. certain distance and then stop? Professor could be,” says “It’s an art, and Joe was the highest Emeritus Steve Childress remembers Photo of Joe Keller at Dave McLaughlin, practitioner of the art,” says Charlie Peskin, discussing how lichen grows on a rock at that the Courant Institute, Silver Professor of Silver Professor of Mathematics and Neural round table, which, he says, “was the social taken in the early 70s. Mathematics and Science at Courant. “Joe had an incredibly event of the day. It occupied us for years, and Neural Science at Courant. “If you think distinctive style. You could suggest a Joe was always at the heart of it.” about the breadth of his work throughout methodology in shorthand just by referring Andy remembers attending Joe’s the sciences, the social sciences, the health to his name.” holiday lecture on another such question: sciences — it is truly remarkable. His Joe’s lasting impact can be What is the optimal way to run a mile? “He curiosity was without equal.” understood not only in terms of his set it up as a control problem,” says Andy. “Joe Keller was equally knowledgeable mathematical achievements, but also of “The answer was that you should keep in mathematics and physics, and he was his mathematical heritage. “He was such running faster and faster until the end of willing to look at a very wide variety of a great mentor to generations of young your mile, accelerating constantly at a fixed problems,” says Courant Professor Emeritus mathematicians and scientists,” says rate, and then you should die at the end of Peter Lax. Joe in fact received both of his Dave. “He worked with so many people, your run! It made for a lot of laughs at the graduate degrees in physics (B.A. in math essentially showing them how to use holiday lecture.” Two other “everyday life” and physics in ’43; M.S. and Ph.D. in physics applied mathematics.” Joe had 60 students problems Joe worked on – one describing in’ 46 and ’48, all from NYU). He then joined — 40 while he was at NYU — according to the motion of a runner’s ponytail and the faculty at Courant, which was just under the Mathematics Genealogy Project. But another showing how to make a teapot that fifteen years old. He played an integral as Dave notes, this large number doesn’t doesn’t drip — earned him Ig Nobel awards, part in building applied math at the young include the many mathematicians, such given for work that makes readers both Institute over the next thirty years, including as himself, whom Joe mentored when they think and laugh. leading a large group at NYU’s former were postdocs or junior faculty. In 1978, Joe moved to Stanford for the Heights campus uptown. The round table is emblematic of remainder of his career. He continued to visit While at the Institute, Joe developed Joe and the personable way in which he the Institute each year. “I think Joe really had his geometric theory of diffraction, inspired others to delight in mathematics. a great affinity for the Courant Institute,” says contributions for which he later received In his years at Courant, he could be found at Steve. “I think he still considered it his home.” a National Medal of Science. The work lunchtime at one of the round tables in the Joe’s returns did feel like homecomings. He analyzed how waves propagate. As stated in 13th floor lounge, a group gathered eagerly would light up doorways, enliven seminars, Stanford’s obituary for Joe, “The theory can around him. “Many of the junior people and rekindle conversations with friends as if be applied whether the waves are acoustic, would go up early, just to make sure that no time had passed. n electromagnetic, elastic or fluid, and has become an indispensable tool for engineers and scientists working on applications The Joseph B. and Herbert B. Keller Professorship in Applied Mathematics such as radar, stealth technology and With a generous gift, Joseph Keller established a professorship in applied antenna design.” mathematics in his and his brother’s name, for “a noted scholar, researcher and In 1974, Andy Majda, now Professor teacher in the field of applied mathematics.” Upon hearing of Professor Keller’s of Mathematics and the Samuel F. B. Morse bequest, then Director Gérard Ben Arous said that “the Joseph B. Keller and Herbert B. Professor of Arts and Science at Courant, Keller Professorship is a special tribute to these brothers’ formative years at NYU and sat in on Joe’s random wave propagation Courant, their many distinguished years of service to applied mathematics, and their course. “I saw for the first time that you can great accomplishments in the field. It will be an inspiration to the faculty members do very complex problems where there’s no who hold it through the years ahead of us.” hope of doing rigorous analysis for the next Herbert Keller, who passed away in 2008, earned his M.A. and Ph.D. at the century,” says Andy, who was then a Courant Courant Institute in ’48 and ’54, respectively, and then joined the faculty. While at Instructor. “It could be done by very concise, Courant, he was Associate Director of the Atomic Energy Commission’s Computing mathematically-based formal asymptotic and Applied Mathematics Center, under Peter Lax. In 1967, he moved to the California analysis. I was taken with that topic. For the Institute of Technology for the remainder of his career.
12 — Courant Newsletter, Fall 2016 PUZZLE FALL 2016
complexity of n/2 minutes but also a better average-case time Find Me Quickly complexity than the go-to-a-common-node strategy? by Dennis Shasha Solution to warm up. Player A can always move clockwise (given Professor of Computer Science a map of the graph for which clockwise makes sense), and player B can always move counter-clockwise. They will meet each other in at most n/2 minutes in the worst case, with an expected value less than the go-to-a-common-node strategy. 3 2 A graph consisting of a single cycle is, of course, a special case. For an arbitrary graph of size n, where each player knows his or 1 her own position and the topology of the graph and where every 4 node has a unique identifier, is there a solution that will take no more than n/2 minutes in the worst case? Solution. Go to the centroid of the graph, or the node to which 9 the maximum distance from any other node is minimized. If there are several such nodes, go to the one with the lexicographically 5 minimum node id. Note that such a centroid cannot have a distance greater than n/2 to any other node. 8 We are just getting started. Now consider situations in which 6 each player knows the topology but not where he or she is placed Figure 1 7 and the nodes have no identifiers. Start by considering a graph consisting of a single path. If player A stays put and player B moves in one direction and bounces In this cooperative game, two players on a graph want to meet back from the end if player B does not find A, the worst-case each other as quickly as possible. Meeting each other means both time could be 2n–3 minutes. Is there a strategy that takes no players are at the same node at the same time or cross each other more than n minutes in the worst case? on an edge. Each player moves or stays put each minute. A move takes one player from one node across an edge to a neighboring Solution. Yes, each player goes in some direction, and when node in the given undirected graph. For a graph consisting of a that player hits an end he or she bounces back. In the worst single cycle and where each player knows his or her position but case, this strategy takes n–1 minutes, with an expected value of not the position of the other player, which strategy—from among approximately 3n/4. the following—is best? One player stays put and the other moves Now here are two questions I don’t know the answers to. We’ll around the cycle; both agree to move to some specific node; or call them upstart questions. some third option. UPSTART 1. Better than staying put. When both players do not Warm-up: Suppose the two players are in a graph consisting of a know where they are placed, nodes are unlabeled and the graph cycle of n nodes (see Figure 1). The nodes are numbered, and each has at least one cycle, find a strategy that is better in the worst player knows both the topology and the number of the node where case than the one-player-stays-put strategy. he or she is placed. If both players move, say, clockwise, they may never meet. If player A does not move (the “stay-put” strategy) UPSTART 2. Also better than staying put. In the same setting and player B moves in one direction, player B will find player A as Upstart 1, say we allow both players to leave notes on nodes in n–1 minutes in the worst case. Alternatively, if both agree to they have visited. Is there an approach that takes n/2 minutes move as quickly as possible to some node, say, node 4, and stay for the two players to meet up in the worst case? If not, is there there, then the latter of the two will arrive at node 4 in n/2 minutes an approach that takes 3n/2 minutes in the worst case? Please at most. Is there any other strategy that has a worst-case time specify whichever approach you come up with.
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13 GIVING TO THE COURANT INSTITUTE 2016 ACADEMIC YEAR
The Courant Institute recognizes with gratitude the following alumni, faculty, parents, and friends who made gifts during the 2016 fiscal year (September 1, 2015 - August 31, 2016). Special thanks to members of our Sigma Society—indicated by a ∑ – donors who have given in any amount in each of the last five years.
Gifts of $5,000 or more Bruce A. and Paula B. Esposito ∑ Robert F. Brodsky Kenneth G. Grossman (NYU President’s Council, Global Andrew B. Flint ∑ Ryan H. Brothers ∑ Edward H. Grossman Circle, Chairman’s Circle) Craig A. Friedman Thomas V. Brown, II Pierre-Yves M. Guillo Glen de Vries Daniel B. Gordon Irina Brudaru Richard Haenssler Lynnel Marg Garabedian Philip Greenwald ∑ Harry Bullen Milton Halem ∑ Arthur Levitt, Jr. Yongming Hong David Cape Zheng-Chao Han Estate of Donald I. MacDavid Carol Hutchins ∑ Paul J. Capobianco Richard D. Hannes David K.A. Mordecai and Kazuhiro Iwasawa Felipe Carino, Jr. David Blake Harrison ∑ Samantha Kappagoda Lucien F. Kraner ∑ John Morrison Carpenter Keith and Madeline Trieger Charitable Trust ∑ Peter V. Lessek Lyzelle Carreon Harrow ∑ Elon Reeve Musk Shasha Liao Danielle Denice Cauthen Dalia Navon Hartman ∑ Joshua N. Newman ∑ Yue-Sheng Liu Man-Yee V. Chan Miriam Hausman ∑ Jan and Joan Popkin Yu Lu Robert Paul Chase Suzanne A. Hecht Kurt S. Riedel Enkeleida Lushi Sindhura Chava David Hertzberg May and Samuel Rudin Kevin B. McGrattan ∑ Jen-Lung Chiu Bernhard Hientzsch ∑ Foundation ∑ Morris J. Meisner ∑ Wai Choy Frederick C. Hixon William C. Rudin ∑ Neeta S. Nabar Melvyn Ciment Pearl R.S. Hochstadt ∑ Louis K. Salkind and Deborah J. Anthony Nicholas Palazotto Jason S. Clague Brian P. Hotaling Rennels ∑ Enlin Pan ∑ Daniel A. Cline Douglas R. Howell Simons Foundation ∑ Jianbo Peng Richard J. Cole ∑ Devjani Huggins Lawrence Sirovich and Carole Ivan P. Polonsky Brandon Alec Cypel Vivek B. Hungund Hochman Sirovich ∑ Jihai Qiu Paul Vartan Socrates Ioannidis Thomas C. Spencer Paul H. Rabinowitz ∑ Czkwianianc ∑ Barry E. Jacobs ∑ Benjamin S. White ∑ Philip S. Shaw Rachel O. Dale Steven Jaffe Zegar Family Foundation ∑ Zeyu Shen Teymour T. Darkhosh ∑ Jaroslaw Jaracz Sascia Yuan Brian G. Smith Indy Y. DeLeon Stephen A. Jarowski Yunyue Zhu Leon H. Tatevossian Thomas K. DeLillo ∑ Frank P. Jones ∑ Homer F. Walker Anastasios Demetrakopoulos Allan Akira Kaku The Director’s Circle Alan A. Weiss ∑ Igor Desyatnikov Vasilios Elias Kalomiris Gifts of $1,000 to $4,999 Brad Wilson ∑ Sidney P. Diamond ∑ Susanne E. Karam Steve Allen and Caroline E. Michael Yanowitch ∑ Pierre Louis Edouard Drogoul Muriel Karlin ∑ Thompson ∑ Bernard Kwangsoo Yoo ∑ Gordon L. Ebbitt Miss Dora F. Kearsley Clark Barrett Jan S. Edler ∑ Robert F. Kelly Ernest Battifarano ∑ Up to $499 Michael E. Eigen Eric Y. Kemperman Joseph A. Cerniglia Frances Armada Adamo Raymond Stuart Ennis Mary B. Kerins ∑ Gilbert Alan Clark Salvatore Anastasio Charles L. Epstein Caroline Davitte Kerr Douglas P. Doucette Sara Landis Angrist Alina Espada Barbara Lee Keyfitz Carol Ann Fugosich Paul J. Atzberger Tseng Ming Fan B. Melvin Kiernan Robert B. Gordon George E.R. Ault Marco A. Fanti Hong W. Kim David G. and Susan L. Korn ∑ Victoria Z. Averbukh Daniel J. Farkas ∑ Morris M. Klein Patrick P. Lin ∑ Nadim Awad Philip J. Feinsilver Paula Shafran Koerte ∑ Paris T. Pender Evan C. Ayala ∑ Xiaosong Feng Roman Kosecki Kristina Joanne Penfold Paul M. Bailyn Richard D. Feuer ∑ Yoon-Wook Kwon Sashi P. Reddi Jonathan J. Baker ∑ Martin Feuerman Julian D. Laderman Charles M. Robins David Baker Norman J. Finizio ∑ Julie E. Landstein Chris Rorres Bernard C. Baldwin Nicholas Fiori Arnold Lapidus ∑ Robin Roy Arash Baratloo Thomas R. Flack ∑ Eugene Manual Laska ∑ Jeffrey S. Saltzman ∑ Evan Barba Daniel B. Forger, Jr. Nam Le Joseph Spinden ∑ Wayne W. Barrett Robert Freundlich, M.D. Irene Huen Lee ∑ Guy A. Story, Jr. Sylvia Baruch Donna Y. Furutani-Aksay Lei Lei Michael K. Tippett Dr. and Ms. Joseph D. Becker Carol Joy Geisler Opher Lekach ∑ Peter Alexander Vasseur ∑ Juan F. Bellantoni Bijoy M. George Thomas James Lepore Anne Xin Xiong Evan Benshetler Andrew W. and Amy B. Gideon Marlene Ader Lerner Robert A. Young ∑ Neil E. Berger ∑ Vadim Goldenberg Brian R. Lessing ∑ Minghua Zhang Daniel Berkowitz Dmitry Goykhman C. David Levermore Xiaojian Zhao Michael E. Bernstein Arvin Grabel Bernard W. Levinger Miaomin Zhu Geoffrey C. Berresford ∑ Nancy Mila Gracin Robert M. Levy Jalal Besharati Betty J. Grad Dorothy Mussman Levy $500-999 Nurit Binenbaum Samuel M. Graff ∑ Bing Li Babak Atri Jay B. and Ellen Bitkower Barry Granoff ∑ Chao Li Stanley M. Benson Albert A. Blank ∑ John Grant Burton B. Lieberman Evelyn Berezin Rachel Joy Blumberg Frank A. Greco Yuanrun Lin Robert Buff Andrew E. Borthwick Eric Z. Grey William J. Lindorfer ∑ Joel L. Cunningham ∑ Craig Seth Brass ∑ Norman G. Griffin Arieh Listowsky ∑ Dean G. Diongson Robert C. Brigham ∑ Edna Grossman Jim S. Litsas
14 — Courant Newsletter, Fall 2016 Your gift makes a difference.
Walter Littman Jonathan Schwartz ∑ Corporations and Foundations ∑ Donald W. Loveland Miss Abby Schwartz Corporations, foundations and other organizations providing research and Yuan Samson Lui Cayman Seacrest philanthropic support and/or employee matching gifts. You can double Mihaela M. Mack Thomas I. Seidman or triple the value of your gift by participating in your employer’s gift Kurt Maly Kent Seinfeld matching program. Please visit http://www.matchinggifts.com/nyu/ to ∑ Edward Man Nickhil Sethi see if your employer matches your gifts to education. Paul R. Manning Arnold Shak Charles A. Marino Seena Sharp Max Martinez Seymour Sherman Aarhus University International Business Machines Karen D. Martinez, Esq. Arthur S. Sherman Alfred P. Sloan Foundation ∑ Corp. Anne-Claire McAllister ∑ Joseph K. Shraibman AstraZeneca Pharmaceuticals LP May and Samuel Rudin Family Victor S. McBarnette Alex Shvartser AT&T Foundation Fdn., Inc. Donald J. McCarthy Bertram Siegel Bank of America Charitable Fund Microsoft Corporation Lt. Colonel Vincent P. McGinn Carole E. Siegel Bank of America Foundation Mobileye Gregory P McNulty ∑ Murray H. Siegel Benevity Community Impact New York Life Foundation Miriam Melnick Barry M. Singer Fund Northrop Grumman Foundation Lakshmi Ajitkumar Menon Ramlala P. Sinha Black Rock Novartis US Foundation Charles A. Miglierina Max Elliot Sklar Canadian Institute for Advanced Nvidia Corporation ∑ Joseph S. Miller Yury A. Skobov Research Open Networking User Group David L. Millman Richard D. Soohoo David K A Mordecai & Samantha Schwab Charitable Fnd. Stanislav Miltchev Mintchev John A. Sottile Kappagoda Charitable Trust Silicon Valley Community Thomas Mitchel Michael St. Vincent ExxonMobil Foundation Foundation James W. Moeller ∑ Edward Stateman Facebook, Inc. Simons Foundation ∑ Van John Molino ∑ Eric Lawrence Stedfeld Fidelity Charitable Tag Online, Inc. Raymond Enrique Moradel Alan H. and Marsha E. Stein ∑ Fujitsu Limited, Legal Division The Boeing Company Carlos Julio Moreno Michael B. Stoeckel ∑ Google DeepMind The D. E. Shaw Group Burt J. Morse Salvatore Stolfo Google, Inc. ∑ The Winston Foundation, Inc. Tanvi Pravin Nabar Keith L. Stransky Gordon & Betty Moore Vanguard Charitable Sara S. Naphtali-Krumbein Noelle Suaifan Foundation Verizon Wireless Mr. and Mrs. Harlan E. Sundry Donors -WS Gordon Foundation Viewpoints Research Institute Nebenhaus ∑ Longjun Tan Grainger, Inc. Wells Fargo Foundation Joyce A. and Stuart B. Newman Jean Taylor Hudson River Trading LLC Yahoo! Kam Chuen Ng ∑ Edward H. Theil Human Frontier Science Program YourCause LLC Louis Nirenberg Margie H. Thompson IBM International Foundation Zegar Family Foundation ∑ Duk-soon Oh Maja-Lisa Wessman Thomson ∑ Insurance Services Office Harold Z. Ollin ∑ Charles Tier ∑ Thomas J. Osler Abby Toomey Gregory Piatetsky Wai Hon Tsui ∑ 2016 Alumni Matching Challenge Evan C. Picoult Peter Ungar In June, 2016, we launched the 2016 Alumni Matching Challenge. Several William Pierce Julius J. Vandekopple ∑ alumni who wished to remain anonymous offered to match any new gifts Stuart S. Piltch Elizabeth L. Vandepaer to the Annual Funds for the remainder of the giving year. Huge thanks to Stanley Preiser Kesava C. Vatti all those who made their first gift ever(N) , renewed their giving to Courant Julian F. Prince ∑ Rohin R. Vazirani (R), made their initial gift for 2016, increased their gift (I) or made an Jin Qian Charles A. von Urff additional 2016 gift (A) — totaling $12,400.31 — that was matched 1:1 by the Sahinur Rahman Zhichao Wang generous alumni matchers! Hedley K.J. Rainnie Henry S. Warren, Jr. ∑ Ajay Rajkumar Donald R. Warren Frances Armada Adamo (R) Arnold Lapidus (A) Kanwarpreet Singh Randhawa Andrew H. Weigel Paul J. Atzberger (R) Lei Lei (N) Ramya Rangarajan Henry Weinberg ∑ Paul M. Bailyn Peter V. Lessek (A) Elbert H. Redford Abraham S. Weishaus ∑ John Morrison Carpenter Arieh Listowsky (A) ∑ Charles Andrew Reich Stewart N. Weiss Teymour T. Darkhosh (A) Edward Man Alexander Retakh Bonnie Katz Weiss Indy Y. DeLeon (R) Mr. and Mrs. Harlan E. Nebenhaus ∑ Charles Wesley Rice Elia S. Weixelbaum Martin Feuerman (A) Enlin Pan (I) Robert N. Rico Benjamin P. Wellington Carol Joy Geisler (N) Sahinur Rahman (A) John M. Rinzel ∑ Michael Williams ∑ Tag Online (Amy and Andrew Hedley K. J. Rainnie Andrew Ronan Karl R. Wittig Gideon) Sashi P. Reddi Hope A. Ropke Peter Wolfe Arvin Grabel (R) Roger S. Rutter (N) Nicholas J. Rose Kurt C. Wulfekuhler Betty J. Grad (A) David E. Scherr Rochelle S. Rosenfeld Marvin Yablon ∑ John Grant Zeyu Shen (A) Harry Rosenthal Raymond J. Yatko Norman G. Griffin (R) Murray H. Siegel (N) Roger S. Rutter Donald Poy Yee Richard D. Hannes (R) Michael K. Tippett (R) Lawrence Thomas Ryan Jacqueline Michele Yee Douglas R. Howell (N) Elizabeth L. Vandepaer (N) ∑ Chelluri C. A. Sastri Phua K. Young Socrates Ioannidis (R) Kesava C. Vatti ∑ Richard E. Schantz Kathleen A. Zammit Kazuhiro Iwasawa (R) Abraham S. Weishaus David E. Scherr Yanni Zeng Muriel Karlin Donald Poy Yee Brooklyn Schlamp Anthony J. Zeppetella ∑ Paula Shafran Koerte Xiaoqing Zhang (A) John F. Schmeelk Miss Thalia Zetlin David G. Korn (A) Xiaojian Zhao (A) ∑ James Ernest Schmitz Xiaoqing Zhang Yoon-Wook Kwon (A) Miaomin Zhu (N) Lorin Schneider Wei F Zheng Ernest Zhu 15 New York University Courant Institute of Mathematical Sciences Warren Weaver Hall 251 Mercer Street New York, NY 10012
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Correction: Changyang Ryoo was incorrectly listed as the Please make sure your contact information is up to date by recipient of the Math Master’s Thesis Prize in our spring issue. visiting the NYU Alumni website at www.nyu.edu/alumni or Though Ryoo was selected for the prize, he declined to accept it. send your email and postal address, phone or employment changes to [email protected] and we’ll take care of the rest. Simulated tectonic plate motion in There are benefits to being an NYU alumnus/a. Check out the NYU the North Eastern Pacific. Arrows Alumni webpage for campus and library access, insurance, entertainment show plate velocities and colors Earth structure. Read more about Georg and dining, university club memberships in your area, and much more, Stadler’s work on mantle convection including NYU alumni networking events wherever you go! and plate tectonics on page 4.