The pentagram map takes a planar polygon $P$ to a polygon $P'$ whose vertices are the intersection points of consecutive shortest diagonals of $P$. This map is known to interact nicely with Poncelet polygons, that is, polygons which are simultaneously inscribed in a conic and circumscribed about a conic. A theorem of Schwartz states that if $P$ is a Poncelet polygon, then the image of $P$ under the pentagram map is projectively equivalent to $P$. In the present paper, we show that in the convex case this property characterizes Poncelet polygons: if a convex polygon is projectively equivalent to its pentagram image, then it is Poncelet. The proof is based on the theory of commuting difference operators, as well as on properties of real elliptic curves and theta functions.

We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete ellipticequations. In particular, we consider the simplest case possible: An elliptic equation on the d-dimensional lattice $\mathbb{Z}^d$with independent and identically distributed conductivities on the associated edges.Recent results by Otto and the author quantify the error made by approximatingthe homogenized coefficient by the averaged energy of a regularizedcorrector (with parameter T) on some box of finite size L. In this article, we replace the regularizedcorrector (which is the solution of a problem posed on $\mathbb{Z}^d$) by some practically computable proxy on some box of size R ≥ L, and quantify the associated additional error.In order to improve the convergence, one may also consider N independentrealizations of the computable proxy, and take the empirical average of the associated approximate homogenized coefficients.A natural optimization problem consists in properly choosing T, R, L and N in order toreduce the error at given computational complexity.Our analysis is sharp and sheds some light on this question.In particular, we propose and analyze a numerical algorithm to approximate the homogenized coefficients, taking advantage of the (nearly) optimal scalings of the errorswe derive. The efficiency of the approach is illustrated by a numerical study in dimension 2.

In this paper, we investigate the value distribution of difference polynomials and prove some difference analogues of results of Hayman and the Brück conjecture.

Differential operators ${{D}_{x,}}\,{{D}_{y}}$, and ${{D}_{z}}$ are formed using the action of the 3-dimensional discrete Heisenberg group $G$ on a set $S$, and the operators will act on functions on $S$. The Laplacian operator $L\,=\,D_{x}^{2}+D_{y}^{2}+D_{z}^{2}$ is a difference operator with variable differences which can be associated to a unitary representation of $G$ on the Hilbert space ${{L}^{2}}\left( S \right)$. Using techniques from harmonic analysis and representation theory, we show that the Laplacian operator is locally solvable.

We introduce and describe the characteristic class of a difference operator over the difference field (k((t)),τ). Here k is an algebraically closed field of characteristic zero and τ is the k-linear automorphism of k((t)) defined by τ(t)=t/(1+t). The approach is based on the characterization of simple difference operators in terms of their eigenvalues.