Research Highlights

Visualizing PML
Visualizing 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
During 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
It 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
Researchers 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
By 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
Optimal 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
ALEX 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, […]

Limits of Permutations
In a wellshuffled deck of cards, about half of the pairs of cards are out of order. Mathematically, we say that in a permutation of there are about inversions, that is, pairs for which . Suppose we are interested in studying permutations for which the number of inversions is exceptionally large […]

A chromatic look at the homotopy groups of spheres
A hallmark of algebraic topology is its collection of decadesold computational problems that have attracted considerable attention and yet remain unsolved. The most famous is the computation of the homotopy groups of spheres, which asks for a classification of all continuous functions Sk → Sn from the kdimensional sphere to the ndimensional sphere up […]

The Topology of Algebraic Varieties
During the academic year 20142015, the School of Mathematics conducted a special program on The Topology of Algebraic Varieties. The program was led by Distinguished Visiting Professor Claire Voisin, from CNRS and Institut de Mathématiques de Jussieu, and Member Burt Totaro from UCLA. The central themes of the program were Hodge theory and algebraic cycles. […]