Grigor Giorgadze

We consider a system of differential equations of Pfaffian type Df = ω on a Riemann surface and develop for it an analog of the monodromy theory of Fuchsian systems of differential equations. Namely, let X be a compact Riemann surface and G a compact Lie group. Let f0 be a solution of system (1) in a neighborhood U ⊂ X of the point z0. After continuation of f0 along a path circling around a singular point zi system (1) determines a monodromy representation ρ : π1(X \ S, z0) → GC. For such system, it is possible to formulate Riemann-Hilbert monodromy problem as follows: whether, for a given homomorphism: ρ : π1(X \ S) → G, there exists a G-system monodromy of which coincides with ρ. It is known that solution of this problem depends on group G. The following theorem holds. Theorem. 1) If π1(M) is a free group and GC is connected, then ρ is a monodromy homomorphism for a system (1). 2) If π1(M) is a free abelian group and G is a connected compact Lie group with torsion free cohomology, and if Imρ ⊂ G, then ρ is a monodromy homomorphism for some system of type (1). Above theorem is a solution of the Riemann-Hilbert problem for holomorphic systems of type Df = ω

algebraic approach to solve the Riemann-Hilbert boundary value problem and extend this technique for solving the problem in closed form in the case of piecewise constant transition function. 

It is known, that the Euler integral of the second kind converges for each z of the complex plane under the condition: Rez>0. According to recent studies, the above-mentioned integral defines a generalized function G(-α+iγ) in the singularity region (α≥0) In this abstract, the analytical form of the functional G(-α+iγ) is proposed at the po¬ints α=0.

Recent studies show that the Euler integral of the first kind for some pairs of parameters is integrable in the sense of generalized functions in the singularity domain. In this abstract, an analytical functional B(iγ-m, -iγ-n) is proposed, which is determined by the above integ¬ral.   

Every holomorphic vector bundle   on Riemann sphere splits into the direct sum of line bundles and the total Chern number of this vector bundle is equal to sum of Chern numbers of line bundles. The integer-valued vector with components Chern number of line bundles is called splitting type of holomorphic vector bundle and is analytic invariant of complex vector bundles. 

There exists a one-to-one correspondence between the H\”older continues matrix function and the holomorphic vector bundles described above, wherein the splitting type of vector bundles coincides with partial indices of matrix functions. It is known that every holomorphic vector bundle equipped with meromorphic (in general) connection with logarithmic singularities at finite set of marked points and corresponding meromorphic 1-from have first order poles in marked points and removable singularity at infinity. 

The Fucshian system of equations induced from this 1-form gives the monodromy representation of the fundamental group of Riemann sphere without marked points. The monodromy representation induces trivial holomorphic vector bundles with connection. The extension of the pair (\texttt{bundle, connection}) on the Riemann sphere is not unique and defines a family of holomorphically nontrivial vector bundles.   

In the talk we present about the following statements:    

1. From the solvability condition (in the sense Galois differential theory) of the Fuchsian   system follows formula for computation of partial indices of piecewise constant matrix function.    

2. All extensions of vector bundle on noncompact Riemann surface correspond to   rational matrix functions algorithmically computable by monodromy matrices of Fucshian system.

We consider the generalized CauchyRiemann equation (Carleman-Bers-Vekua equation in complex form) ∂w/¯z = aw + b¯w, where a ∈ Lp^loc (C) and b ∈ Lp,2(C), p > 2. Introduce subclasses of the class L loc p (C), p > 2, elements of which have ∂/ ∂¯z primitives and satisfying certain additional asymptotic conditions. In the talk we define these classes and prove their properties.

The solutions of the problem of linear conjugation in a closed form, solving a system of singular integral equations in quadratures, determining the splitting type of holomorphic vector bundle on the Riemann sphere is reduced on the computation of the partial indices of matrix-functions of Holder class on the unit circle. From the solutions of one of listed problems the solution of others follows. On the other hand, the linear conjugation problem for piecewise constant matrix functions reduces to monodromic Riemann-Hilbert problem for the class of regular systems of differential equations. Recently, an algorithm for solving the linear conjugation problem for piecewise constant boundary matrix functions and an algorithm for calculation splitting type of holomorphic vector bundle induced from Fuchsian system of differential equations on the Riemann sphere has been proposed. In the talk, we give an algorithm that permit for any set of marked points on the Riemann sphere and for any quadratic matrices constructed a rational matrix function, the set of singular points of which is contained in the set of marked points and the left partial indices of this matrix function coincide with splitting type of the holomorphic vector bundle induced from corresponding Fuchsian system. 

We discuss equilibrium configurations of the Coulomb potential of positive point charges with positions satisfying certain quadratic constraints in the plane and three-dimensional Euclidean space. The main attention is given to the case of three point charges satisfying a positive definite quadratic constraint in the form of equality or inequality. For a triple of points on the boundary of convex domain, we give a geometric criterion of the existence of positive point charges for which the given triple is an equilibrium configuration. Using this criterion, rather comprehensive results are obtained for three positive charges in the disc, ellipse, and three-dimensional ball. In the case of the circle, we strengthen these results by showing that any configuration consisting of an odd number of points on the circle can be realized as an equilibrium configuration of certain nonzero point charges and give a simple criterion for existence of positive charges with this property. Similar results are obtained for three point charges each of which belongs to one of the three concentric circles. Several related problems and possible generalizations are also discussed.

We discuss equilibrium configurations of point charges with Coulomb interaction on the circle, line segment and system of three concentric circles, and obtain characterization of stable electrostatic configurations with a few points.

In the case of the circle, we show that any configuration consisting of odd number of points on the circle can be realized as an equilibrium configuration of certain non-zero point charges and give a simple criterion for existence of positive charges with this property.

Several related problems to Fuchsian differential equations on complex plane and possible

generalizations are also discussed.

Certain physically interesting Schrödinger equations can be rewritten as Fuchsian systems of Okubo form. It is known that every accessory parameter free system of differential equations of Okubo normal form is rigid and has an integral representation of solutions; therefore, such systems are used to describe real physical processes. Thus Schrödinger equations was written in Okubo form with explicitly known coefficient matrices, which enables one to compute the monodromy and asymptotics of solutions.

We consider the moduli space of conformal structures of the manifold manifold Xm=C/{z1, z2,…zm} in case when the distances between the points z1, z2,…zm are constants and prove that this space is isomorphic to the moduli space of conguration spaces of m-gons with the vertices at points z1, z2,…zm. When m= 4; we use the conformal moduli of quadrilaterals for the classification Heun equations by location of four singular points. In this approach, the main tool are topological and analytic properties of quadratic maps.

We discuss the possibility of stabilizing a given conguration of point charges in convex planar domain D using the electrostatic potential of point charges placed on the boundary B of D. An algebraic approach to this problem will be described and illustrated by typical results of three types. First, we prove that, for any nite conguration X of points in D, there exists a system S of charges on B such that all points of X are critical points of the electrostatic potential of S. Second, a similar result will be established in the case where X consists of equal point charges and one considers the total electrostatic potential of all charges in X and S. Finally, we will present some situations where the arising equilibrium conguration is stable with respect to small displacements of any charge considered. In conclusion we will indicate possible applications of our approach to constructions of abstract analogs of Paul electromagnetic traps and perspectives of using computer algebra for calculating the values and positions of stabilizing charges in concrete situations.

In this paper we consider the relations between the Riemann–Hilbert monodromy problem and the matrix Riemann–Hilbert boundary-value problem with piecewise continuous coefficient and construct the so-called canonical matrix for the boundary-value problem for a piecewise continuous matrix-function. The formula for the calculation of the index is also obtained.

We consider the impact generating triple for the following irregular Carlemann-Bers-Vekua System (see I.N. Vekua, Generalized analytic functions, Nauka, Moscow, 1988, and V. Jikia, Georgian Math. J., vol. 17, (2010)) on the classes of functions induced by irregular Carlemann-Bers-Vekua equations.

In this paper we investigate the relationship between the holomorphic and conformal structures and the spaces of generalized analytic functions induced from conformal structures.

In the talk we discussed on the global theory of Riemann-Hilbert boundary value and monodromy problems and consider relations between this problems similar to classical one. We use algebraic topological methods for solve this classical problems in invariant forms. 

We discuss a number of topics concerned with certain boundary-value problems in the context of generalized analytic functions. Solution of the classical Riemann-Hilbert problem and the linear conjugation problem for analytic functions is described in appropriate function classes and the same scheme is applied to generalized analytic functions and vectors. In particular, we describe solution of the Riemann-Hilbert problem for generalized analytic functions and obtain an explicit analytic presentation of monodromy matrices in the case of generalized analytic vectors.

The Riemann-Hilbert monodromy problem (21st Hilbert problem) is to construct the Fuchsian system by the marked points and given non-degenerate matrices for which the marked points will be poles of system and the monodromy matrices coincide with given matrices. The Riemann-Hilbert boundary value problem (the problem of linear conjugation) was posed by Riemann in the same work as one of the methods of the solution of the monodromy problem. Notwithstanding the fact that the problem of linear conjugation is studied for the generalized analytic functions (vectors) and the progress achieved in research of singular elliptic systems, the connection between RiemannHilbert problem with singular elliptic systems and problem of linear conjugation for the generalized analytic functions was not noted until now. We consider the analogous problems and apply the methods of algebraic topology. The analog of Fuchsian system in our approach will be Carleman-Bers-Vekua system with sindular points. Research object will be the space of multiplevalued solutions, the analog of the monodromy theorem, Chen iterated integral,Riemann-Roch theorem and Riemann-Hurwitz formula.

We consider second order Fuchsian systems as Schrödinger type equation and shown that some values of parameters this systems describes completely controllable quantum systems. 

This article is dedicated to the investigation of the density problem of monodromy groups of Fuchsian systems on complex manifolds in linear groups. We consider the so-called inverse problem. We apply the Riemann-Hilbert monodromy problem and we show that there exists a Fuchsian system with dense monodromy subgroup in the special unitary group.

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