A Combinatorial Proof of the Chernoff-Hoeffding Bound, With Applications to Direct-Product Theorems

Valentine Kabanets
Simon Fraser University; Institute for Advanced Study
March 30, 2010

We give a simple combinatorial proof of the Chernoff-Hoeffding concentration
bound for sums of independent Boolean random variables. Unlike the standard
proofs, our proof does not rely on the method of higher moments, but rather uses
an intuitive counting argument. In addition, this new proof is constructive in the
following sense: if the given random variables fail the concentration bound, then
we can efficiently find a subset of the variables that are statistically dependent.

Measuring Shape With Homology

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.

Cover Times, Blanket Times, and Majorizing Measures

James Lee
University of Washington
April 12, 2010

The cover time of a graph is one of the most basic and well-studied properties of the simple random walk, and yet a number of fundamental questions concerning cover times have remained open. We show that there is a deep connection between cover times of graphs and Talagrand's majorizing measure theory. In particular, we prove that the cover time can be characterized, up to universal constants, by the majorizing measure value of a certain metric space on the underlying graph.

Leon Levy Lecture - Mass Higher Education and the Dropout Problem

Paul Attewell
Institute for Advanced Study
April 21, 2010

Paul Atewell, Leon Levy Foundation Member, School of Social Science. In the United States, ever-increasing proportions of high school graduates continue into college, and more and more undergraduates continue into master’s programs. One concern with educational expansion is that many students do not complete their degrees; they “drop out.” Some read this as proof that too many students are going to college, but other scholars argue that not enough Americans are receiving degrees. In this talk, Paul Attewell, Professor of Sociology at the Graduate Center of the City University of New York, will consider the reasons behind the dropout phenomenon, examining individual factors but also highlighting government policies and institutional practices that undercut students’ progress toward graduation.