Abelian functional equations, planar web geometry and polylogarithms

We study some abelian functional equations (Afe). They are equations in the F_i’s of the form F₁(U₁) + ... + F_N(U_N) = 0 where the U_i’s are real rational functions in two variables. First we prove that the local measurable solutions are actually analytic and we characterize their components as solutions of linear differential equations constructed from the U_i’s. Then we propose two methods for solving Afe. Next we apply these methods to the explicit resolution of generalized versions of classical (inhomogeneous) Afe satisfied by low order polylogarithms. Interpreted in the framework of web geometry, these results give us new nonlinearizable maximal rank planar webs. Then we observe that there is a relation between these webs and certain configurations of points in which leads us to define the notion of web associated to a configuration: these webs seem of high rank and could provide numerous new exceptional webs. Finally, we use the preceding results to show that, under weak regularity assumptions, the trilogarithm is the only function which satisfies the Spence–Kummer equation.     

Additive polylogarithms and their functional equations

Let ${k[\varepsilon]_{2}:=k[\varepsilon]/(\varepsilon^{2})}Let k[e]₂:=k[e]/(e²){k[\varepsilon]_{2}:=k[\varepsilon]/(\varepsilon^{2})} . The single valued real analytic n-polylogarithm L_n: \mathbbC ? \mathbbR{\mathcal{L}_{n}: \mathbb{C} \to \mathbb{R}} is fundamental in the study of weight n motivic cohomology over a field k, of characteristic 0. In this paper, we extend the construction in ünver (Algebra Number Theory 3:1–34, 2009) to define additive n-polylogarithms li_n:k[e]₂? k{li_{n}:k[\varepsilon]_{2}\to k} and prove that they satisfy functional equations analogous to those of L_n{\mathcal{L}_{n}}. Under a mild hypothesis, we show that these functions descend to an analog of the nth Bloch group B_n￠(k[e]₂){B_{n}' (k[\varepsilon]_{2})} defined by Goncharov (Adv Math 114:197–318, 1995). We hope that these functions will be useful in the study of weight n motivic cohomology over k[ε]₂.     

Abelian equations and rank problems for planar webs

We find an invariant characterization of planar webs of maximum rank. For 4-webs, we prove that a planar 4-web is of maximum rank three if and only if it is linearizable and its curvature vanishes. This result leads to the direct web-theoretical proof of the Poincaré theorem: A planar 4-web of maximum rank is linearizable. We also find an invariant intrinsic characterization of planar 4-webs of rank two and one and prove that in general such webs are not linearizable. This solves the Blaschke problem “to find invariant conditions for a planar 4-web to be of rank 1 or 2 or 3.” Finally, we find invariant characterization of planar 5-webs of maximum rank and prove than in general such webs are not linearizable. The text was submitted by the authors in English.     

Abelian Hermitian geometry

We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct product of several copies of the real hyperbolic plane by a Euclidean factor. Moreover, we show that if a left invariant Hermitian metric on a Lie group with an abelian complex structure has flat first canonical connection, then the Lie group is abelian.     

Abelian monopoles and complex geometry

Three remarks concerning the constructive properties of smooth 4-manifolds with nontrivial Seiberg-Witten invariants are presented. Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 420–428, March, 1999.     

Some recent applications of functional equations to geometry

The results of applications of solutions of functional equations or of methods used in the theory of functional equations to the following subjects are discussed in this paper. Determination of all Cremona transformations which reduce linear transformations with triangular matrices to translations. One-parameter subsemigroups of affine transformations and their homomorphisms. Extensions of homomorphisms from sub-semigroups to groups generated by them. Determination of all collineations on subsets of general projective planes and their extensions to the entire plane.     

On the geometry of a four-parameter rational planar system of difference equations

In this paper, we consider geometric aspects of a rational, planar system of difference equations defined on the open first quadrant and whose behaviour is governed by four independent, non-negative parameters. This system, indexed as (23, 23) in the notation of Ladas (Open problems and conjectures, J. Differential Equ. Appl. 15(3) 2009, pp. 303–323), is one of the 200 systems from Ladas about which little is known. Using geometric techniques, we answer several questions concerning the behaviour of this system.     

Generalized planar curves and quaternionic geometry

Motivated by the analogies between the projective and the almost quaternionic geometries, we first study the generalized planar curves and mappings. We follow, recover, and extend the classical approach, see e.g., (Sov. Math. 27(1) 63–70 (1983), Rediconti del circolo matematico di Palermo, Serie II, Suppl. 54 75–81) (1998), Then we exploit the impact of the general results in the almost quaternionic geometry. In particular we show, that the natural class of ℍ-planar curves coincides with the class of all geodesics of the so called Weyl connections and preserving this class turns out to be the necessary and sufficient condition on diffeomorphisms to become morphisms of almost quaternionic geometries.     

On a general principle in geometry that leads to functional equations

Summary Suppose that an invariant (or an invariant notion) of some geometry is given, like the distance between two points, the cross ratio of four points, the tangential distance between two spheres (or like the notion of orthogonality, of order, of a circle). One may ask what are the functions preserving (or preserving partially) that invariant (invariant notion). Originating from this principle some functional equation problems are formulated, namely the functional equations of distance, of area, of angle preservance.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

Baxter's equations and algebraic geometry

Functional Analysis and Its Applications -     

geometry of differential equations

This paper contains a survey of papers on the geometry of differential equations, which appeared no earlier than 1972, continuing the general survey (RZhMat, 1974, 11A800), and considers in more detail a special cycle of investigations of the geometry of systems of partial differential equations, distinguished by the presence of practical applications. Then we continue the survey of new results on the geometry of an ordinary differential equation of arbitrary order, started in (RZhMat, 1978, 1A645). There is constructed a general theory of invariants of equations, and classes of equations admitting a simplified coordinate representation are invariantly distinguished.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 12, pp. 127–164, 1981.     

Convolution equations and nonlinear functional equations

The survey is devoted to applications of nonlinear integral equations to linear convolution equations, their discrete analogues, and also the connection of these equations with problems of radiative transfer, in particular, with the Ambartsumyan equations.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 22, pp. 175–244, 1984.     

Heun equations coming from geometry

We give a list of Heun equations which are Picard-Fuchs associated to families of algebraic varieties. Our list is based on the classification of families of elliptic curves with four singular fibers done by Herfurtner. We also show that pull-backs of hypergeometric functions by rational Belyi functions with restricted ramification data give rise to Heun equations.     

Evolution equations in Riemannian geometry

A fundamental question in Riemannian geometry is to find canonical metrics on a given smooth manifold. In the 1980s, R.S. Hamilton proposed an approach to this question based on parabolic partial differential equations. The goal is to start from a given initial metric and deform it to a canonical metric by means of an evolution equation. There are various natural evolution equations for Riemannian metrics, including the Ricci flow and the conformal Yamabe flow. In this survey, we discuss the global behavior of the solutions to these equations. In particular, we describe how these techniques can be used to prove the Differentiable Sphere Theorem.     

Conformally invariant differential equations and projective geometry

We combine harmonic analysis on certain pseudo-Riemannian symmetric spaces with results on conformally invariant linear and non-linear differential equations. This gives in many cases part of the decomposition of certain representations of the conformal group of a manifold when restricted to the isometry group.     

Abelian Lie symmetry algebras of two-dimensional quasilinear evolution equations

We carry out the classification of abelian Lie symmetry algebras of two-dimensional second-order nondegenerate quasilinear evolution equations. It is shown that such an equation is linearizable if it admits an abelian Lie symmetry algebra that is of dimension greater than or equal to 5 or of dimension greater than or equal to 3 with rank-one.     

Working with differential,functional and difference equations using functional networks

In this paper we first analyze the problem of equivalence of differential, functional and difference equations and give methods to move between them. We also introduce functional networks, a powerful alternative to neural networks, which allow neural functions to be different, multidimensional, multiargument and constrained by link connections, and use them for predicting values of magnitudes satisfying differential, functional and/or difference equations, and for obtaining the difference and differential equation associated with a set of data. The estimation of the differential or difference equation coefficients is done by simply solving systems of linear equations, in the cases of equally or unequally spaced or missing data points. Some examples of applications are given to illustrate the method.