30 years of finite-gap integration theory

Vladimir B Matveev

doi:10.1098/rsta.2007.2055

30 years of finite-gap integration theory

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2007.2055 ◽

2007 ◽

Vol 366 (1867) ◽

pp. 837-875 ◽

Cited By ~ 29

Author(s):

Vladimir B Matveev

Keyword(s):

Algebraic Geometry ◽

Riemann Surfaces ◽

Theta Functions ◽

Theoretical Physics ◽

Integration Theory ◽

Difference Operators ◽

Strong Impact ◽

Riemann Theta Functions ◽

Compact Riemann Surfaces ◽

Modern Mathematics

The method of finite-gap integration was created to solve the periodic KdV initial problem. Its development during last 30 years, combining the spectral theory of differential and difference operators with periodic coefficients, the algebraic geometry of compact Riemann surfaces and their Jacobians, the Riemann theta functions and inverse problems, had a strong impact on the evolution of modern mathematics and theoretical physics. This article explains some of the principal historical points in the creation of this method during the period 1973–1976, and briefly comments on its evolution during the last 30 years.

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A Hodge theoretic projective structure on compact Riemann surfaces

Journal de Mathématiques Pures et Appliquées ◽

10.1016/j.matpur.2021.02.005 ◽

2021 ◽

Vol 149 ◽

pp. 1-27

Author(s):

Indranil Biswas ◽

Elisabetta Colombo ◽

Paola Frediani ◽

Gian Pietro Pirola

Keyword(s):

Riemann Surfaces ◽

Projective Structure ◽

Compact Riemann Surfaces

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Convergence of Discrete Period Matrices and Discrete Holomorphic Integrals for Ramified Coverings of the Riemann Sphere

Mathematical Physics Analysis and Geometry ◽

10.1007/s11040-021-09394-2 ◽

2021 ◽

Vol 24 (3) ◽

Author(s):

Alexander I. Bobenko ◽

Ulrike Bücking

Keyword(s):

Error Estimate ◽

Branch Point ◽

Riemann Surfaces ◽

Riemann Sphere ◽

Holomorphic Functions ◽

Convergence Result ◽

Edge Length ◽

Ramified Covering ◽

Compact Riemann Surfaces ◽

Ramified Coverings

AbstractWe consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere $\hat {\mathbb {C}}$ ℂ ̂ . Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.

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