We prove a conjecture of Dipendra Prasad on Ext-branching from ${\mathrm{GL}}_{n+1}(F)$ to ${\mathrm{GL}}_n(F)$, where F is a p-adic field, and we give a projectivity criterion, resulting in some interesting consequences.

Let X be an arbitrary smooth hypersurface in $\mathbb{C}{\mathbb{P}}^n$ of degree d. We prove the de Jong–Debarre conjecture for $n\ge 2d-4$: the space of lines in X has dimension $2n-d-3$. We also prove an analogous result for k-planes: if $n\ge 2\left(\genfrac{}{}{0.0pt}{}{d+k-1}{k}\right)+k$, then the space of k-planes on X will be irreducible of the expected dimension. As applications, we prove that an arbitrary smooth hypersurface satisfying $n\ge 2^{d!}$ is unirational, and we prove that the space of degree-e curves on X will be irreducible of the expected dimension provided that $d\le \frac{e+n}{e+1}$.

Loop-erased random walk (LERW) is one of the most well-studied critical lattice models. It is the self-avoiding random walk that one gets after erasing the loops from a simple random walk in order or, alternatively, by considering the branches in a spanning tree chosen uniformly at random. We prove that planar LERW parameterized by renormalized length converges in the lattice-size scaling limit to ${\mathrm{SLE}}_2$ parameterized by $5∕4$-dimensional Minkowski content. In doing this, we also provide a method for proving similar convergence results for other models converging to SLE. Besides the main theorem, several of our results about LERW are of independent interest. We give, for example, two-point estimates, estimates on maximal content, and a “separation lemma” for two-sided LERW.

The p-adic Littlewood conjecture due to De Mathan and Teulié asserts that for any prime number p and any real number α, the equation

$$\underset{|m|\ge 1}{inf}|m|\cdot |m{|}_p\cdot |⟨m\mathit{\alpha }⟩|=0$$

holds. Here $|m|$ is the usual absolute value of the integer m, $|m{|}_p$ is its p-adic absolute value, and $|⟨x⟩|$ denotes the distance from a real number x to the set of integers. This still-open conjecture stands as a variant of the well-known Littlewood conjecture. In the same way as the latter, it admits a natural counterpart over the field of formal Laurent series $\mathbb{K}((t^{-1}))$ of a ground field $\mathbb{K}$. This is the so-called t-adic Littlewood conjecture (t-LC).

It is known that t-LC fails when the ground field $\mathbb{K}$ is infinite. The present article is concerned with the much more difficult case when this field is finite. More precisely, a fully explicit counterexample is provided to show that t-LC does not hold in the case that $\mathbb{K}$ is a finite field with characteristic 3. Generalizations to fields with characteristic other than 3 are also discussed.

The proof is computer-assisted. It reduces to showing that an infinite matrix encoding Hankel determinants of the paperfolding sequence over ${\mathbb{F}}_3$, the so-called number wall of this sequence, can be obtained as a 2-dimensional automatic tiling satisfying a finite number of suitable local constraints.