## Measuring Shape With Homology

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.

## Mobius Randomness and Dynamics

## Compressing Bounded-Round Communication

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

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.