## Expansion in Linear Groups and Applications

This lecture was part of the Institute for Advanced Study’s celebration of its eightieth anniversary, and took place during the events related to the Schools and Mathematics and Natural Sciences.

Jean Bourgain, Professor, School of Mathematics

Institute for Advanced Study

September 24, 2010

This lecture was part of the Institute for Advanced Study’s celebration of its eightieth anniversary, and took place during the events related to the Schools and Mathematics and Natural Sciences.

Vladimir Voevodsky, Professor, School of Mathematics

Institute for Advanced Study

September 25, 2010

This lecture was part of the Institute for Advanced Study’s celebration of its eightieth anniversary, and took place during the events related to the Schools of Mathematics and Natural Sciences.

Avi Wigderson

Institute for Advanced Study

May 25, 2010

The Stepanov method is an elementary method for proving bounds on the number of roots of polynomials. At its core is the following idea. To upper bound the number of roots of a polynomial f(x) in a field, one sets up an auxiliary polynomial F(x) , of (magically) low degree, which vanishes at the roots of f with high multiplicity. That appropriate F exits is usually proved by a dimension argument.

Jean Bourgain

Institute for Advanced Study

June 16, 2010

Avi Wigderson

Institute for Advanced Study

June 15, 2010

Avi Wigderson

Institute for Advanced Study

June 15, 2010

Robert MacPherson

Institute for Advanced Study

April 7, 2010

The ordinary homology of a subset S of Euclidean space depends only on its topology. By systematically organizing homology of neighborhoods of S, we get quantities that measure the shape of S, rather than just its topology. These quantities can be used to define a new notion of fractional dimension of S. They can also be effectively calculated on a computer.

Peter Sarnak

Institute for Advanced Study

April 6, 2010

Helmut Hofer

Institute for Advanced Study

March 10, 2010

The mathematical problems arising from modern celestial mechanics, which originated with Isaac Newton’s Principia in 1687, have led to many mathematical theories. Poincaré (1854-1912) discovered that a system of several celestial bodies moving under Newton’s gravitational law shows chaotic dynamics. Earlier, Euler (1707–83) and Lagrange (1736–1813) found instances of stable motion; a spacecraft in the gravitational fields of the sun, earth, and the moon provides an interesting system of this kind. Helmut Hofer, Professor in the School of Mathematics, explains how these observations have led to the development of a geometry based on area rather than distance.

Didier Fassin

Institute for Advanced Study

February 17, 2010

Humanitarianism, which can be defined as the introduction of moral sentiments into human affairs, is a major component of contemporary politics—locally and globally—for the relief of poverty or the management of disasters, in times of peace as well as in times of war. But how different is the world and our understanding of it when we mobilize compassion rather than justice, call for emotions instead of rights, consider inequality in terms of suffering, and violence in terms of trauma? What is gained—and lost—in this translation? In this lecture, Didier Fassin, James D. Wolfensohn Professor in the School of Social Science, attempts to comprehend humanitarian government, to make sense of its expansion, and to assess its ethical and political consequences.