Planar Ising model at criticality: state-of-the-art and perspectives

In this essay, we briefly discuss recent developments, started a decade ago in the seminal work of Smirnov and continued by a number of authors, centered around the conformal invariance of the critical planar Ising model on $\mathbb{Z}^2$ and, more generally, of the critical Z-invariant Ising model on isoradial graphs (rhombic lattices). We also introduce a new class of embeddings of general weighted planar graphs (s-embeddings), which might, in particular, pave the way to true universality results for the planar Ising model.Comment: 19 pages (+ references), prepared for the Proceedings of ICM2018. Second version: two references added, a few misprints fixe

Parametrization of the isospectral set for the vector-valued Sturm-Liouville problem

We obtain a parametrization of the isospectral set of matrix-valued potentials for the vector-valued Sturm-Liouville problem on a finite interval

The inverse problem for perturbed harmonic oscillator on the half-line with Dirichlet boundary conditions

We consider the perturbed harmonic oscillator $T_D\psi=-\psi''+x^2\psi+q(x)\psi$, $\psi(0)=0$, in $L^2(\R_+)$, where q\in\bH_+=\{q', xq\in L^2(\R_+)\} is a real-valued potential. We prove that the mapping q\mapsto{\rm spectral data}={\rm \{eigenvalues of\}T_D{\rm \}}\oplus{\rm \{norming constants\}} is one-to-one and onto. The complete characterization of the set of spectral data which corresponds to q\in\bH_+ is given