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Spring 2017 Newsletter Spring 2017 Published by the Courant Institute of Mathematical Sciences at New York University Courant Newsletter How to build a heart: Dave McQueen retires Volume 13, Issue 2 p9 p2 AN ANALYST’S VIEW INTO THEORETICAL PHYSICS WITH SYLVIA SERFATY p11 p12 OLOF WIDLUND NEW DIRECTOR: RETIRES RUSSEL CAFLISCH ALSO IN THIS ISSUE: n COURANT AT THE CENTER n RAGHU VARADHAN: 50 n FACULTY HONORS & FOR DATA SCIENCE YEARS ON THE FACULTY STUDENT AWARDS p4 p7 p14 n COMPUTER SCIENCE n BODIES IN FLUID MOTION: n SPRING 2017 PUZZLE MOVES INTO FORMER WITH LISA FAUCI p15 FORBES BUILDING p8 p13 p6 RICHARD COURANT: SCIENTIFIC AMBASSADOR An analyst’s view into theoretical physics With Sylvia Serfaty FACULTY RESEARCH by April Bacon and superconductivity is lost. Type II superconductors can withstand much more powerful magnetic fields than Type I, and are the kind used in technological applications such as those listed above. In the late 1950s, building on the Ginzburg-Landau model of superconductivity, Alexei Abrikosov formulated the theory of Type II superconductors, demonstrating that when vortices form, they will arrange themselves in triangular arrays that are now known as Abrikosov lattices. What he predicted was observed in experiments a decade later. The triangular lattice is thought to be the most energetically efficient arrangement. “The vortices repel each other,” Sylvia explains. © NYU Photo Bureau: Khan “But on the other hand, the magnetic field Sylvia Serfaty teaches Advanced Topics in Analysis. Photo on cover: Stefan Falke confines them. It’s sort of like people who don’t like to be close but are confined together. The best arrangement they find is Mathematicians can have different Sylvia’s work has deepened and refined to put themselves in these triangular lattices.” kinds of intuition, says Professor Sylvia our theoretical understanding of certain In a series of works, Sylvia and Serfaty, and finding the area for which one classic problems derived from physics. A collaborator Etienne Sandier (University is suited may take time. By both trial and significant body of her research has focused of Paris XII) showed that a simplified chance, during her doctoral studies, Sylvia on models related to superconductors— model can be extracted from a physics found what came most natural to her: those materials used to take medical scans model of superconductivity by reducing analysis. Originally attracted to pure math, at of the body, investigate the properties of the vortices in size to points. “We derived a certain point in her studies of algebra, she molecules, accelerate particles in the Large this reduced problem which is a discrete found that “it was starting to be too abstract,” Hadron Collider, confine plasma in magnetic problem on infinite configurations of points she says. In formal analysis courses, however, confinement fusion reactors, and levitate in the plane,” she says. Deriving a reduced the problems at hand felt easy again. “By a high-speed trains. model is essentially reducing the original series of coincidences, I took a PDE course First discovered by Dutch physicist minimization problem to a simpler one. with Fabrice Béthuel and I liked it. It seemed Heike Kamerlingh Onnes in 1911, Looking for a minimum is like, by analogy, natural. In the end, I did my Ph.D. with him superconductivity occurs when certain looking at all the temperatures of a given and he is the one who proposed to me that materials — certain metals, alloys, and day and searching for the lowest point, subject of research [on the Ginzburg-Landau ceramics — are cooled to extreme low says Sylvia. “We believe the minimum is equation of superconductivity].” temperatures, at which point resistance achieved at the triangular lattice, but proving Just two years after attaining her to an electrical current abruptly vanishes. something like that is hard,” and related doctorate, in 2001, Sylvia joined the Courant In a popular experiment demonstrating to some important conjectures in number Institute faculty. Starting in 2008, she spent what is known as the Meissner effect, a theory. 8 years as a Professor at the University of superconductor will float above a magnet From their derived model, “What you Paris 6. During that time she held a Global once it completely repels the magnet’s see is that the point vortices behave […] Distinguished Professorship at Courant, magnetic field. In Type II superconductors like electrostatic charges. That is, their so although she returned full time to the (which can be made from certain alloys), a interaction is logarithmic,” just as are the Institute in 2016, in another way she never magnetic field will penetrate the body of interactions between, say, electrons confined completely left. the superconductor in the form of magnetic to a conducting surface. “Yes, I was never fully out!” she says. “But flux vortices. As the applied magnetic Via a systematic analysis of the it’s a pleasure being back in New York and at field is increased, so are the number of Ginzburg-Landau model, Sylvia and Etienne the Institute.” vortices, until a critical point is reached were able to provide a rigorous mathematical 2 — Courant Newsletter, Spring 2017 foundation for this tendency of magnetic a very large matrix and draw the entries “It’s one of the first hints of that relationship—a vortices to form Abrikosov lattices, as well as to be random, complex iid [independent, clear indication,” says Sylvia. the intensity of a magnetic field required for identically distributed] Gaussians, you She first began working in this area of the vortices to form, and the number that will compute the eigenvalues, and you find that statistical mechanics in 2009. “I wasn’t in appear. the eigenvalues are points in the complex my comfort zone. I had to pick up a whole Sylvia’s work related to these theories plane which repel each other exactly like new literature, meet new people, learn has earned her awards such as the prestigious the vortices.” One of the main questions to new things, new techniques. And I enjoyed Henri Poincaré prize (2012), “for her be answered about these systems is, does a that sense of renewal,” she says. “And now outstanding work on the theory of Ginzburg- critical temperature exist below which the I’m quite occupied with these problems of Landau equations, including remarkable system crystallizes, i.e. below which it gets statistical mechanics. I approach them with progress towards the rigorous proof of the ordered, or even forms lattices? the point of view of a mathematician with onset of the Abrikosov lattice in the theory of Sylvia and Courant Instructor Thomas the tools of partial differential equations, superconductivity.” Leblé looked at these systems of interacting functional analysis, calculus of variations, In a second line of work, Sylvia has points in which temperature is added to and more recently probabilistic tools. I’m described the dynamics of the vortices— create some disorder. “When you zoom sort of an analyst at heart with an interest how they move when allowed to relax over down, these points are very close together,” in physics. And with expanding my toolbox time until they reach a state of equilibrium. says Sylvia, “but if you zoom out, what are into neighboring disciplines, I’m particularly “Instead of describing all the points you just the patterns that you see? What we believe is interested now in the interface between want to understand the force that the vortices that if temperature is very small, then indeed analysis and probability.” Sylvia also notes collectively create,” she says. “It turns out you will see something very ordered, like a that there are several other faculty members— that the equations that govern the motion crystal. We prove that when temperature is Gérard Ben Arous, Paul Bourgade, Percy Deift, are very much like fluid equations.” In 2016, very small, the system is more ordered, and and Ofer Zeitouni—at Courant working on Sylvia proved that in some instances, “the when temperature increases, it is more and random matrices. “They are all experts with Euler equation—the standard equation for more disordered.” different points of view. It’s very interesting for incompressible fluids in a 2-dimensional “With the language of large deviations, me to be here in that sense.” domain—can be derived as a limit of the we have a way of measuring the fact that time-dependent Schrödinger version of the there is more disorder. What we essentially “It’s a great place to be,” she says of the Ginzburg-Landau equation.” prove is that when the temperature tends Institute. “A place with so much history in to infinity in some suitable units, then the math and a particular spirit of collegial work. I Systems of points with logarithmic—or, particles don’t seem to remember that they think people put the interests of the Institute more generally, Coulomb—interactions, turn repel each other. They’re just living their own ahead of their personal interests. We see out to also be important in other problems lives.” there’s a collective endeavor. I also like the fact within theoretical physics, such as the The relationship between increased that compared to many departments, there fractional quantum Hall effect and random temperature and disorder is well-known are no formal boundaries between different matrices. “They arise as particular ensembles within one-dimensional systems, and Sylvia types of math. You always have someone to of statistical mechanics called log gases and Thomas have proven a formal relation talk to on any topic.” or Coulomb gases,” says Sylvia. “You take between the two in a more general setting. This approach to mathematics goes back to Richard Courant and the earliest faculty members he brought to NYU.
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