  In this paper, we obtain the maximal estimate for the Weyl sums on the torus
$\mathbb{T}^d$ with $d\geq 2$, which is sharp up to the endpoint. We also
consider two variants of this problem which include the maximal estimate along
the rational lines and on the generic torus. Applications, which include some
new upper bound on the Hausdorff dimension of the sets associated to the large
value of the Weyl sums, reflect the compound phenomenon between the square root
cancellation and the constructive interference. In the Appendix, an alternate
proof of Theorem 1.1 inspired by Baker's argument in [1] is given by Barron,
which also improves the $N^{\epsilon}$ loss in Theorem 1.1, and the
Strichartz-type estimates for the Weyl sums with logarithmic losses are
obtained by the same argument.
