Research Highlights

Visualizing PML

ICERM - September 2016

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 only one “type” of simple closed curve. In contrast, on a surface with a more complicated...

Special Year on Geometric Structures on 3-Manifolds

IAS - September 2016

During the 2015-16 academic year, the School of Mathematics conducted a special program on Geometric Structures on 3-Manifolds. 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 3-manifolds. Twenty members took part in the program. Senior members included David Gabai,...

NET Maps

AIM - July 2016

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 to challenge topologists. The AIM...

Big Data meets Number Theory

ICERM - May 2016

Researchers from ICERM’s special semester “Computational Aspects of the Langlands Program” are creating new data-driven models for collaborative research in number theory, culminating in the May 10, 2016 official release of the L-functions and Modular Forms Database (LMFDB) at www.lmfdb.org. Computation is not new to number theory – in Babylon huge tablets of sines and cosines were created and...

Identifying Links Between the S&amp;P500 and VIX Derivatives

IPAM - April 2016

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 universally-accepted predictor of volatility. The VIX is a 30-day predictor of volatility given by a...

Overcoming the Curse of Dimensionality for Control Theory

IPAM - November 2015

Optimal control problems lead to Hamilton-Jacobi 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 is exponential in \( n...

Illumination and Security

MSRI - October 2015

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, the answer to this problem is “no”, and the reasons are connected with deep mathematical problems whose...

Limits of Permutations

ICERM - October 2015

In a well-shuffled deck of cards, about half of the pairs of cards are out of order. Mathematically, we say that in a permutation $\pi$ of $[n]=\{1,2,\dots,n\}$ there are about $\frac12\binom{n}{2}$ inversions, that is, pairs $i<j$ for which $\pi(j)<\pi(i)$. Suppose we are interested in studying permutations for which the number...

A chromatic look at the homotopy groups of spheres

MSRI - September 2015

A hallmark of algebraic topology is its collection of decades-old 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 k-dimensional sphere to the n-dimensional sphere up to continuous definition. When k < n, it...

The Topology of Algebraic Varieties

IAS - September 2015

During the academic year 2014-2015, 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. A large group of Members took part...