School of Mathematics
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.
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.
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.
We shall discuss new pseudorandom generators for regular read-once branching programs of small width. A branching program is regular if the in-degree of every vertex in it is (either 0 or) 2. For every width d and length n, the pseudorandom generator uses a seed of length $O((log d + log log n + log(1/p)) log n)$ to produce $n$ bits that cannot be distinguished from a uniformly random string by any regular width $d$ length $n$ read-once branching program, except with probability $p > 0$
In this survey I will present several extremal problems, and some solutions, concerning convex lattice polytopes.
A typical example is to determine the smallest area that a convex lattice polygon can have if it has exactly n vertices.