Комплексные и кватернионные функции в решеточных моделях и теории циркулярных поверхностей тема диссертации и автореферата по ВАК РФ 00.00.00, доктор наук Скопенков Михаил Борисович

• Скопенков Михаил Борисович
• доктор наук
• 2023

Оглавление диссертации доктор наук Скопенков Михаил Борисович

Contents

Introduction

References

Appendix A. Article 1: Tiling by rectangles and alternating current

Appendix B. Article 2: Ruled Laguerre minimal surfaces

Appendix C. Article 3: Darboux cyclides and webs from circles

Appendix D. Article 4: A surface containing a line and a circle through each point is a quadric

Appendix E. Article 5: The boundary value problem for discrete analytic functions

Appendix F. Article 6: Discrete Riemann surfaces: linear discretization and its convergence

Appendix G. Article 7: Surfaces containing two circles through each point

Appendix H. Article 8: Characterizing envelopes of moving rotational cones and applications in CNC machining

Appendix I. Article 9: Lattice gauge theory and a random-medium Ising model

Appendix J. Article 10: Feynman checkers: towards algorithmic quantum theory

Введение диссертации (часть автореферата) на тему «Комплексные и кватернионные функции в решеточных моделях и теории циркулярных поверхностей»

Introduction

This thesis is devoted to the study of the functions of a complex and quaternion variable and their applications, mainly in geometry. We consider functions both on the plane and the Riemann surfaces, and on the lattices.

In theory of functions of a complex variable, especially on Riemann surfaces, the evidence for the existence of basic objects is often non-constructive, which makes it difficult to calculate them in practice. One of the promising approaches to computation in this theory is its discretization and the subsequent limit transition.

Various constructions of complex analysis on planar graphs were introduced by R.Isaacs, R.Duffin, C.Mercat [19, 17, 31, 32], I.Dynnikov-S.Novikov [18], A.Bobenko-C.Mercat-Yu.Suris [4], and A.Bobenko-U.Pinkall-B.Springborn [5]. The first of these theories, the so-called linear discretization of complex analysis on a quadrilateral lattice, is currently being developed especially actively. This is due to its applications in statistical physics (S.Smirnov-D.Chelkak [46]; M.Khristoforov, S.Smirnov, and the applicant [25]), numerical analysis [22], and combinatorial geometry [24]. See the surveys by S.Smirnov and L.Lovasz [29, 46].

Complex analysis on graphs, like classical complex analysis, is related to potential theory. Therefore, physical terminology is often used: a weighted graph is called a direct-current (respectively, alternating current) circuit, if the weight of each edge is positive (respectively, it has a positive real part).

Figure 1: Physical interpretation of tilings: of a rectangle by squares, of a polygon by squares, of a square by similar rectangles (from the left to the right).

Historically, among the first applications of discrete complex analysis (in the appearance of electric networks) was the problem of tiling a given polygon by rectangles. A celebrated physical interpretation of such tilings by R.Brooks, K.Smith, K.Stone, and U.Tutte uses direct-current networks [6]; see the left and the middle parts of Figure 1. This interpretation allowed to completely solve a number of problems on cutting polygons into squares. Recently this idea is experiencing a new peak of popularity, thanks to the application to models of quantum gravity, due to S.Sheffield and coauthors.

A more general problem of cutting a polygon into polygons of a given shape is also related to the discrete functions of a complex variable. To solve it, further development of the theory of such functions is required.

Further, we consider functions of a quaternion variable, mainly polynomials and rational functions. We study their applications in the classification of surfaces containing

Figure 2: Circular arc structures

several circles through each point (those circles completely lie on the surface). This is motivated, in particular, by potential applications in architecture.

In contemporary architecture, there is a trend towards freeform structures which is very clearly seen in the works by star architects such as F.Gehry or Z.Hadid. While digital models of architectural freeform surfaces are easily created using standard modeling tools, the actual fabrication and construction of architectural freeform structures remains a challenge. In order to make a freeform design realizable, an optimization process known as rationalization has to be applied [35, 3]. This means replacing a smooth surface by a lattice composed of separate panels with special properties.

As a contribution towards rationalization of architectural freeform structures, P.Bo and coauthors [3] have recently suggested so-called circular arc structures. A circular arc structure is a mesh whose edges are realized as circular arcs instead of straight line segments and which possesses congruent nodes with well-defined tangent planes. Figure 2 shows two examples of circular arc structures with quad-mesh and triangle-mesh combinatorics, respectively. In the first case, the construction consists of two discrete sets of arc splines which intersect each other under the right angle. In the second case, we see three sets of arc splines which intersect each other under the 60 degree angle. In the latter case we have an example of a so-called hexagonal web, a classical object of geometry; see Figure 3 and [1, 50, 51, 40, 38, 39, 41].

The following fundamental problem is closely related to the applied one we are discussing. Keeping aside the requirement of equality of angles, let us require that the structure is formed by whole arcs of circles, not just splines from them. The natural question is what constructions can be achieved from two families of circles. It has remained open for a long time despite partial advances tracing back to the works by G.Darboux from the 19th century. Although the solution of this problem has no direct applications

Figure 3: Webs from circles on Darboux cyclides: hyperboloid and its image under inversion, Dupin cyclide, canal cyclide, general cyclide with different webs (from the left to the right).

in architecture, there are reasons to believe that the developed methods could be useful for subsequent applied research.

The resulting question naturally leads to the question when and to what extent the factorization of quaternion polynomials in two variables into irreducible factors is unique. This question was studied earlier for the case of one variable, and the case of two variables was not amenable to the previously known methods.

Discrete complex analysis

Let us give a brief overview of the results known earlier and obtained in the thesis.

Classical methods allowed to construct a discretization of complex analysis and prove its convergence to the continuum theory for square and rhombic lattices. For example, the convergence of the solution of the Dirichlet problem on a square lattice approximating a domain to the solution of the Dirichlet problem in this domain was proved by R.Courant-K.Friedrichs-H.Lewy [13], and for the case of rhombic lattices — by D.Chelkak-S.Smirnov [8] and implicitly by P.Ciarlet-P.Raviart [10]. For more general quadrilateral lattices, the classical methods do not work (the reason for this is that the property of the discrete analyticity of the integral of a discrete analytic function disappears). They allow to obtain convergence results only in a very weak sense, not related to solutions of boundary value problems. We propose new methods that allow to prove convergence for the so-called orthogonal lattices (article 5 in the list of publications), as well as on triangulated Riemannian surfaces (article 6 in the list of publications). This solves a problem posed by S.Smirnov [46, Question 1].

A related question on the convergence of the Galerkin finite element method has been studied thoroughly; see, for example, textbook [9]. However, the question of the convergence of the finite element method for the Delaunay triangulations of Riemann surfaces remained open until our joint work with A.Bobenko (article 6 in the list of publications).

Historically, among the first applications of discrete complex analysis was the problem of tiling a given polygon by rectangles of given shapes. A celebrated physical interpretation of such tilings by R.Brooks, K.Smith, K.Stone, and U.Tutte [6] is exposed in our elementary introductions [42, 43]. In a joint work with M.Prasolov (article 1 in the list of

publications) we develop a new approach to this tiling problem using alternating-current networks and electrical impedance tomography [7, 11, 12, 14, 15]; see Figure 1 to the right. In particular, we describe all polygons which can be tiled by squares. This solves a problem of C.Freiling et al. [21]. This was known in the particular cases when the polygon is an L-shaped hexagon (R.Kenyon [24]).

Close ideas are used by us to analyze various lattice models (articles 10 and 9 in the list of publications); see also reviews [23, 27, 48]. Note that article 9 from the list of publications uses functions on the lattice with both complex and quaternion values.

Functions of a quaternion variable and circular surfaces

Let us give a brief overview of the results known earlier and obtained in the thesis.

Examples of surfaces (in 3-dimensional space) containing several circles through each point have been known for a long time. For brevity, such surfaces are called circular. G.Darboux introduced a special class of 4th degree surfaces, which are called cyclides. Darboux cyclides include quadrics and Dupin cyclides as special cases. Other examples are shown in Figure 3 to the right. G.Darboux showed that up to 10 circles pass through each point of a cyclide, but he did not distinguish between real and complex circles. R.Blum [2] constructed examples of cyclides with 6 real circles through each point.

Figure 4: Laguerre minimal surfaces: helicoid, cycloid, the Plucker conoid, general ruled surface, general surfaces enveloped by families of cones (from the left to the right).

There are very few known results on the classification of circular surfaces. N.Takeuchi showed that a surface of genus 0 or 1, different from the sphere, cannot contain more than 6 circles through each point. It was known that a surface containing 2 cospheric or 2 orthogonal circles through each point is necessarily a cyclide. However, there are circular surfaces that are different from cyclides: for example, a surface obtained by the translation of one circle along another circle. Surfaces containing two conics (respectively, a conic and a straight line) through each point were classified by J.Schicho [37] (respectively, H.Brauner).

In a joint work with F.Nilov (article 4 in the list of publications), all surfaces containing a conic and a line through each point were described.

In a joint work with H.Pottmann and L.Shi (article 3 in the list of publications), all so-called hexagonal webs from circles were found on all cyclides other than a sphere and a plane; see Figure 3. By means of related methods, together with M. Barton, P.Bo, Ph.Grohs, H.Pottmann, we have also described all surfaces containing a family of isotropic

circles and all ruled surfaces that are minimal in the sense of E.Laguerre (articles 8 and 2 in the list of publications, cf. [26, 28, 33, 34, 36, 49]); see Figure 4. Functions of a complex variable played the main role in the proof of the latter result.

Finally, in a joint work with R.Krasauskas (article 7 in the list of publications), the problem of classification of circular surfaces was completely solved using functions of a quaternion variable.

More detailed expositions of the proofs (than in published articles 7 and 2 from the list of publications) are given on the preprint server https://arxiv.org/abs/1011.0272v2 and https://arxiv.org/abs/1512.09062v3.

Список литературы диссертационного исследования доктор наук Скопенков Михаил Борисович, 2023 год

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Mihail B. Skopenkov Received 27/AUG/21