# School of Mathematics

## Hardness of Approximately Solving Linear Equations Over Reals

## Can Complexity Theory Ratify the Invisible Hand of the Market?

*It is not from the benevolence of the butcher, the brewer, or the baker, that we expect our dinner, but from their regard for their own interest. Each participant in a competitive economy is led by an invisible hand to promote an end which was no part of his intention.*

Adam Smith, 1776

## Critical Slowdown for the Ising Model on the Two-Dimensional Lattice

## Cover Times, Blanket Times, and Majorizing Measures

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.

## 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.