We show that any embedded minimal torus in S³ is congruent to the Clifford torus. This answers a question posed by H. B. Lawson, Jr., in 1970.

We study the critical Ising model on the square lattice in bounded simply connected domains with + and free boundary conditions. We relate the energy density of the model to a discrete fermionic correlator and compute its scaling limit by discrete complex analysis methods. As a consequence, we obtain a simple exact formula for the scaling limit of the energy field one-point function in terms of the hyperbolic metric. This confirms the predictions originating in physics, but also provides a higher precision.

We prove the so-called unitary isotropy theorem, a result on isotropy of a unitary involution. The analogous previously known results on isotropy of orthogonal and symplectic involutions as well as on hyperbolicity of orthogonal, symplectic, and unitary involutions are formal consequences of this theorem. A component of the proof is a detailed study of the quasi-split unitary grassmannians.

We prove the BMV (Bessis, Moussa, Villani, [1]) conjecture, which states that the function $t\mapsto \mathrm{Tr}exp(A-tB)$ , $t⩾0$ , is the Laplace transform of a positive measure on [0,∞) if A and B are $n\times n$ Hermitian matrices and B is positive semidefinite. A semi-explicit representation for this measure is given.

Let u_n be the nth term of a Lucas sequence or a Lehmer sequence. In this article we shall establish an estimate from below for the greatest prime factor of u_n which is of the form n exp(log n/104 log log n). In doing so, we are able to resolve a question of Schinzel from 1962 and a conjecture of Erdős from 1965. In addition we are able to give the first general improvement on results of Bang from 1886 and Carmichael from 1912.

For any natural number m(>1) let P(m) denote the greatest prime divisor of m. By the problem of Erdős in the title of the present paper we mean the general version of his problem, that is, his conjecture from 1965 that $$\frac{P(2^n-1)}{n}\to \infty \mathrm{as}n\to \infty$$(see Erdős [10]) and its far-reaching generalization to Lucas and Lehmer numbers. In the present paper the author delivers three refinements upon Yu [40] required by C. L. Stewart for solving completely the problem of Erdős (see Stewart [25]). The author gives also some remarks on the solution of this problem, aiming to be more streamlined with respect to the p-adic theory of logarithmic forms.