We exhibit a strong connection between cover times of graphs, Gaussian processes, and Talagrand's theory of majorizing measures. In particular, we show that the cover time of any graph G is equivalent, up to universal constants, to the square of the expected maximum of the Gaussian free field on G, scaled by the number of edges in G.This allows us to resolve a number of open questions. We give a deterministic polynomialtime algorithm that computes the cover time to within an O(1) factor for any graph, answering a question of Aldous and Fill (1994). We also positively resolve the blanket time conjectures of Winkler and Zuckerman (1996), showing that for any graph, the blanket and cover times are within an O(1) factor. The best previous approximation factor for both these problems was O((log log n)2 ) for n-vertex graphs, due to Kahn, Kim, Lovász, and Vu (2000).

We show that an infinite weighted tree admits a bi-Lipschitz embedding into Hilbert space if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. We also introduce a new metric invariant called Markov convexity, and show how it can be used to compute the Euclidean distortion of any metric tree up to universal factors.

We exhibit a strong connection between cover times of graphs, Gaussian processes, and Talagrand's theory of majorizing measures. In particular, we show that the cover time of any graph G is equivalent, up to universal constants, to the square of the expected maximum of the Gaussian free field on G, scaled by the number of edges in G.This allows us to resolve a number of open questions. We give a deterministic polynomial-time algorithm that computes the cover time to within an O(1) factor for any graph, answering a question of Aldous and Fill (1994). We also positively resolve the blanket time conjectures of Winkler and Zuckerman (1996), showing that for any graph, the blanket and cover times are within an O(1) factor. The best previous approximation factor for both these problems was O((log log n)2 ) for n-vertex graphs, due to Kahn, Kim, Lovász, and Vu (2000).

We show that the multi-commodity max-flow/min-cut gap for series-parallel graphs can be as bad as 2, matching a recent upper bound [8] for this class, and resolving one side of a conjecture of Gupta, Newman, Rabinovich, and Sinclair.This also improves the largest known gap for planar graphs from 3 2 to 2, yielding the first lower bound that doesn't follow from elementary calculations. Our approach uses the coarse differentiation method of Eskin, Fisher, and Whyte in order to lower bound the distortion for embedding a particular family of shortest-path metrics into L 1 .