Research Highlights

Trisections
October 2017Understanding four dimensional structures or manifolds is both a mysterious and complex task. The goal of the AIM workshop ‘Trisections and lowdimensional topology’ held in March of 2017 was to study these four dimensional objects with a new perspective introduced by David Gay and Robion Kirby in 2012. In three dimensions it has been known […]

Waves, Kakeya sets, and Diophantine equations
June 2017Ciprian Demeter A central problem in Physics is to understand the complicated ways in which waves can interact with one another. The field of Harmonic Analysis grew out of the observation that any natural signal that is periodic in time can be built by superimposing simple waves with whole number frequencies. In geometric optics, […]

SAMSI Brings Astronomers and Statisticians Together to Study Universe
June 2017Contributed by: Jim Barrett, Ph.D. student, School of Physics & Astronomy, University of Birmingham, UK Maya Fishbach, Ph.D. student, Department of Astronomy and Astrophysics, University of Chicago, USA Bo Ning, Ph.D. candidate, Department of Statistics, North Carolina State University, USA Daniel Wysocki, […]

Visualizing PML
September 2016Visualizing PML David Dumas and François Guéritaud On the surface of a sphere, every simple closed curve (that is, a curve that starts and ends at the same point and which does not cross itself) forms the boundary of a disk, i.e. a contiguous region without holes. In this sense, the sphere has […]

Special Year on Geometric Structures on 3Manifolds
September 2016During the 201516 academic year, the School of Mathematics conducted a special program on Geometric Structures on 3Manifolds. The program was led by Distinguished Visiting Professor Ian Agol of the University of California at Berkeley. The theme of the program was classication of geometric structures on 3manifolds. Twenty members took part in the program. Senior […]

NET Maps
July 2016It is human nature to try to classify things—that is, to sort them into organized types. Many of the central problems in mathematics are problems of classification of various types of related mathematical objects. The classification of finite groups, for example, was a landmark accomplishment of the last century, and the classification of manifolds continues […]

Big Data meets Number Theory
May 2016Researchers from ICERM’s special semester “Computational Aspects of the Langlands Program” are creating new datadriven models for collaborative research in number theory, culminating in the May 10, 2016 official release of the Lfunctions and Modular Forms Database (LMFDB) at www.lmfdb.org. Computation is not new to number theory – in Babylon huge tablets […]

Identifying Links Between the S&P500 and VIX Derivatives
April 2016By Andrew Papanicolaou The technique of volatility trading has been common practice since the 1970’s. Typically, a long (short) position in volatility included a long (short) position in options. In 2003 there came a more standardized way of trading volatility, as the VIX formula became the universallyaccepted predictor of volatility. The VIX […]

Overcoming the Curse of Dimensionality for Control Theory
November 2015Optimal control problems lead to HamiltonJacobi Bellman (HJB) differential equations in many space variables for finding the cost function to be optimized. This beautiful connection has not generally led to effective numerical methods because grid based solutions of partial differential equations in \( n \) variables generally have memory requirements and complexity that […]

Illumination and Security
October 2015ALEX WRIGHT AND KATHRYN MANN Imagine that you are in a room with walls made out of mirrors. The room may be very oddly shaped and have corridors and nooks, but all the walls are at mirrored planes. If you light a single lamp, must every point in the room be illuminated? Perhaps surprisingly, […]