Abstract:
The lower semi-continuity of best approximation operators from Banach lattices on to closed ideals is investigated. Also the existence of best approximation to sub-function modules of function modules is proved. The order intersection properties of cells are studied and used to prove the above results.

Abstract:
We prove two continuity theorems for the Lyapunov exponents of the maximal entropy measure of polynomial automorphisms of $\mathbb{C}^2$. The first continuity result holds for any family of polynomial automorphisms of constant dynamical degree. The second result is the continuity of the upper exponent for families degenerating to a 1-dimensional map.

Abstract:
We give a necessary and sufficient condition for non-local functionals on vector-valued Lebesgue spaces to be weakly sequentially lower semi-continuous. Here a non-local functional shall have the form of a double integral of a density which depends on the function values at two different points. The characterisation we get is essentially that the density has to be convex in one variable if we integrate over the other one with an arbitrary test function in it. Moreover, we show that this condition is in the case of non-local functionals on real-valued Lebesgue spaces (up to some equivalence in the density) equivalent to the separate convexity of the density.

Abstract:
This paper establishes uniform bounds in characteristic $p$ rings which are either F-finite or essentially of finite type over an excellent local ring. These uniform bounds are then used to show that the Hilbert-Kunz length functions and the normalized Frobenius splitting numbers defined on the Spectrum of a ring converge uniformly to their limits, namely the Hilbert-Kunz multiplicity function and the F-signature function. From this we establish that the F-signature function is lower semi-continuous. We also give a new proof of the upper semi-continuity of Hilbert-Kunz multiplicity, which was originally proven by Ilya Smirnov.

Abstract:
In this note, we report some progress we made recently on the automorphisms groups of K3 surfaces. A short and straightforward proof of the impossibility of Z/(60) acting purely non-symplectically on a K3 surface, is also given, by using Lefschetz fixed point formula for vector bundles.

Abstract:
In the present paper we describe the K3 surfaces admitting order 11 automorphisms and apply this to classify log Enriques surfaces of global index 11.

Abstract:
In this paper we study the automorphisms group of some K3 surfaces which are double covers of the projective plane ramified over a smooth sextic plane curve. More precisely, we study some particlar case of a K3 surface of Picard rank two.

Abstract:
We determine all posible orders of automorphisms of finite order of complex K3 surfaces or of K3 surfaces in characteristic $p>3$. E.g., a positive integer $N$ is the order of an automorphism of a complex K3 surface if and only if $\phi(N)\le 20$ where $\phi$ is the Euler function. In particular, 66 is the maximum finite order in each characteristic $p\neq 2,3$. As a consequence, we give a bound for the orders of finite groups acting on K3 surfaces in characteristic $p>7$.

Abstract:
In the present paper we prove that finite symplectic groups of automorphisms of manifolds of k3^[n] type can be obtained by deforming natural morphisms arising from K3 surfaces if and only if they satisfy a certain numerical condition.

Abstract:
This is a systematic exposition of recent results which completely describe the group of automorphisms and the group of autoequivalences of generic analytic K3 surfaces. These groups, hard to determine in the algebraic case, admit a good description for generic analytic K3 surfaces, and are in fact seen to be closely interrelated.