Research Highlights

Limits of Permutations
October 2015In 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
September 2015A 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
September 2015During 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
July 2015With 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?
November 2014When 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
August 2014During 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
July 2014Red 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
July 2014Author: 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 […]

KadisonSinger Resolved
June 2014In 2013, the KadisonSinger problem was solved by Adam Marcus, Dan Spielman, and Nikhil Strivastava. As Dana Mackenzie said in the January/February 2014 issue of SIAM News, the KadisonSinger problem was like a “great subterranean river of mathematics, mysteriously disappearing below the surface and reappearing where you least expect it.” The problem […]

Topology Provides Tools for Intrusion Detection with Robot Swarms
June 2014Robotic swarms are increasingly being designed for tasks deemed too difficult, timeconsuming, or hazardous for humans, such as underwater exploration, wildlife sampling, surveillance, and search and rescue. These cheap, simple robots move using collective behaviors that emerge from local interactions with each other and their environment, functioning much like social insects. One use of robot […]