Research Highlights

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. […]

Modeling the Nation’s Food System
With over seven billion people living on our planet, how do we ensure a secure food system? How do we manage to produce, harvest, process, transport, sell, and eventually prepare the food we consume and do this better so that over one billion people are not hungry? Food system activities, like those described above comprise […]

Topology of Shapes, Persistent Homology and Point Clouds: Where Does it Take Us?
When viewed from the outside, a human brain appears as a volume with a highly wrinkled surface having numerous long crevices. Sulcal fundi are 3D curves that lie in the depths of the cerebral cortex; informally, the fundus of a sulcus is the curve of maximal average depth that spans the length of the sulcus. […]

Nonequilibrium Dynamics and Random Matrices
During the academic year 20132014, the School of Mathematics conducted a special program on Nonequilibrium Dynamics and Random Matrices. This program was led by Distinguished Visiting Professor HorngTzer Yau from Harvard University, and Thomas Spencer (IAS). Yau is a leading expert on both random matrix theory and nonequilibrium dynamics. He was joined by senior participants […]

The Secret Life of Red Blood Cells Revealed Through Topological Data Analysis
Red blood cells serve the critical biological function of delivering oxygen to all parts of the human body. Yet the cell’s ability to perform its job undergoes many changes as it ages. It will typically shrink in volume by 30 percent, while the membrane becomes increasingly stiff. The loss in elasticity of the membrane results […]

The Kakeya Problem
Author: Larry Guth (MIT) The Kakeya problem is an open problem in Euclidean geometry, and it was a central theme in the 2014 IPAM long program “Algebraic Techniques for Combinatorial and Computational Geometry”. It is elementary to state, but it’s also a model for deep open questions in Fourier analysis. Different approaches […]