Let $G$ be a connected reductive algebraic group. We prove that the string parameterization of a crystal basis for a finite-dimensional irreducible representation of $G$ extends to a natural valuation on the field of rational functions on the flag variety $G/B$, which is a highest-term valuation corresponding to a coordinate system on a Bott–Samelson variety. This shows that the string polytopes associated to irreducible representations, can be realized as Newton–Okounkov bodies for the flag variety. This is closely related to an earlier result of Okounkov for the Gelfand–Cetlin polytopes of the symplectic group. As a corollary, we recover a multiplicativity property of the canonical basis due to Caldero. We generalize the results to spherical varieties. From these the existence of SAGBI bases for the homogeneous coordinate rings of flag and spherical varieties, as well as their toric degenerations, follow recovering results by Alexeev and Brion, Caldero, and the author.

We prove that an $n\times n$ random matrix $G$ with independent entries is completely delocalized. Suppose that the entries of $G$ have zero means, variances uniformly bounded below, and a uniform tail decay of exponential type. Then with high probability all unit eigenvectors of $G$ have all coordinates of magnitude $O\left(n^{-1/2}\right)$, modulo logarithmic corrections. This comes as a consequence of a new, geometric approach to delocalization for random matrices.

For an arbitrary associative unital ring $R$, let $J_1$ and $J_2$ be the following noncommutative, birational, partly defined involutions on the set $M_3\left(R\right)$ of $3\times 3$ matrices over $R$: $J_1\left(M\right)=M^{-1}$ (the usual matrix inverse) and $J_2(M{)}_{jk}=(M_{kj}{)}^{-1}$ (the transpose of the Hadamard inverse).

We prove the surprising conjecture by Kontsevich that $(J_2\circ J_1{)}^3$ is the identity map modulo the ${Diag}_L\times {Diag}_R$ action $(D_1,D_2)\left(M\right)=D_1^{-1}M{D}_2$ of pairs of invertible diagonal matrices. That is, we show that, for each $M$ in the domain where $(J_2\circ J_1{)}^3$ is defined, there are invertible diagonal $3\times 3$ matrices $D_1=D_1\left(M\right)$ and $D_2=D_2\left(M\right)$ such that $(J_2\circ J_1{)}^3(M)=D_1^{-1}M{D}_2$.

In this work we establish a connection between two classical notions, unrelated so far: harmonic functions on the one hand and absolutely monotonic functions on the other hand. We use this to prove convexity-type and propagation of smallness results for harmonic functions on the lattice.

Let $K$ be a complete discrete valuation field of characteristic $0$, with valuation ring ${\mathcal{O}}_K$ and perfect residue field $k$ of positive characteristic $p$. We prove that a proper and smooth curve over $K$, admitting a semistable model over ${\mathcal{O}}_K$, has good reduction if and only if its unipotent $p$-adic étale fundamental group is crystalline.