In this paper we develop the method of finding sharp estimates by using a Bellman function. In such a form the method appears in the proofs of the classical John-Nirenberg inequality and L p estimations of BMO functions. In the present paper we elaborate a method of solving the boundary value problem for the homogeneous Monge-Ampère equation in a parabolic strip for sufficiently smooth boundary conditions. In such a way, we have obtained an algorithm for constructing an exact Bellman function for a large class of integral functionals on the BMO space.

We prove a duality theorem the computation of certain Bellman functions is usually based on. As a byproduct, we obtain sharp results about the norms of monotonic rearrangements. The main novelty of our approach is a special class of martingales and an extremal problem on this class, which is dual to the minimization problem for locally concave functions.

This volume contains the proceedings of the conference Local and Global Methods in Algebraic Geometry, held from May 12-15, 2016, at the University of Illinois at Chicago, in honor of Lawrence Ein's 60th birthday. The articles cover a broad range of topics in algebraic geometry and related fields, including birational geometry and moduli theory, analytic and positive characteristic methods, geometry of surfaces, singularity theory, hyper-Kähler geometry, rational points, and rational curves. Contents: R. Lazarsfeld, Some remarks on the work of Lawrence Ein; A. Calabri and C. Ciliberto, Contractible curves on a rational surface; F. Catanese, On the canonical map of some surfaces isogenous to a product; C. Ciliberto, F. Flamini, C. Galati, and A. L. Knutsen, Degeneration of differentials and moduli of nodal curves on 3 surfaces; I. Coskun and J. Huizenga, Weak Brill-Noether for rational surfaces; R. Datta and K. E. Smith, Excellence in prime characteristic; M. González Villa, A. Libgober, and L. Maxim, Motivic zeta functions and infinite cyclic covers; C. Hacon, M. Popa, and C. Schnell, Algebraic fiber spaces over abelian varieties: Around a recent theorem by Cao and Păun; S. Ishii and W. Niu, A strongly geometric general residual intersection; J. Kollár, Quadratic solutions of quadratic forms; S. J Kovács, Non-Cohen-Macaulay canonical singularities; N. Mok, Full cones swept out by minimal rational curves on irreducible Hermitian symmetric spaces as examples of varieties underlying geometric substructures; M. Mustaţă and Y. Nakamura, A boundedness conjecture for minimal log discrepancies on a fixed germ; E. Sernesi, The Wahl map of one-nodal curves on K3 surfaces; Y.-T. Siu, Skoda's ideal generation from vanishing theorem for semipositive Nakano curvature and Cauchy-Schwarz inequality for tensors; C. Voisin, Hyper-Kähler compactification of the intermediate Jacobian fibration of a cubic fourfold: The twisted case.

We study a wide class of metrics in a Lebesgue space, namely the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the ε-entropy of a measure space with an admissible metric, etc. These notions and related results are applied to the theory of transformations with invariant measure; namely, we study the asymptotic properties of orbits in the cone of admissible metrics with respect to a given transformation or a group of transformations. The main result of this paper is a new discreteness criterion for the spectrum of an ergodic transformation: we prove that the spectrum is discrete if and only if the ε-entropy of the averages of some (and hence any) admissible metric over its trajectory is uniformly bounded. MSC:37A05, 11J83, 37C85