  Let $\mathscr{C}_n=\{-1,1\}^n$ be the discrete hypercube equipped with the
uniform probability measure $\sigma_n$. Talagrand's influence inequality (1994)
asserts that there exists $C\in(0,\infty)$ such that for every
$n\in\mathbb{N}$, every function $f:\mathscr{C}_n\to\mathbb{C}$ satisfies
$$\mathrm{Var}_{\sigma_n}(f) \leq C \sum_{i=1}^n
\frac{\|\partial_if\|_{L_2(\sigma_n)}^2}{1+\log\big(\|\partial_if\|_{L_2(\sigma_n)}/\|\partial_i
f\|_{L_1(\sigma_n)}\big)}.$$ In this work, we undertake a systematic
investigation of this and related inequalities via harmonic analytic and
stochastic techniques and derive applications to metric embeddings. We prove
that Talagrand's inequality extends, up to an additional doubly logarithmic
factor, to Banach space-valued functions under the necessary assumption that
the target space has Rademacher type 2 and that this doubly logarithmic term
can be omitted if the target space admits an equivalent 2-uniformly smooth
norm. These are the first vector-valued extensions of Talagrand's influence
inequality. We also obtain a joint strengthening of results of Bakry-Meyer
(1982) and Naor-Schechtman (2002) on the action of negative powers of the
hypercube Laplacian on functions $f:\mathscr{C}_n\to E$, whose target space $E$
has nontrivial Rademacher type via a new vector-valued version of Meyer's
multiplier theorem (1984). Inspired by Talagrand's influence inequality, we
introduce a new metric invariant called Talagrand type and estimate it for
Banach spaces with prescribed Rademacher or martingale type, Gromov hyperbolic
groups and simply connected Riemannian manifolds of pinched negative curvature.
Finally, we prove that Talagrand type is an obstruction to the bi-Lipschitz
embeddability of nonlinear quotients of the hypercube $\mathscr{C}_n$, thus
deriving new nonembeddability results for these finite metrics.
