Related papers: Integrable Systems and Discrete Geometry
Some aspects of the connection between differential geometry and multidimensional soliton equations are discussed.
A brief exposition of the point of higher topos theory in (mathematical) physics, commissioned for the Encyclopedia of Mathematical Physics 2nd ed.
This expository article written for the Notices of the American Mathematical Society provides an overview of transcendental functions arising as solutions of the discrete Painlev\'e equations, for which the developments of the last two…
This is an expanded version of the lecture notes for a minicourse that I gave at a summer school called "Advanced Course on Geometry and Dynamics of Integrable Systems" at CRM Barcelona, 9--14/September/2013. In this text we study the…
Proposal for contribution to the quantum field theory section in "Encyclopedia of Mathematical Physics".
This paper focuses on polynomial dynamical systems over finite fields. These systems appear in a variety of contexts, in computer science, engineering, and computational biology, for instance as models of intracellular biochemical networks.…
This is a comment on the article "Integrable Systems in Stringy Gravity" by D. V. Gal'tsov, Phys. Rev. Lett. 74, 2863, (1995).
This book aims to provide a self-contained introduction to the regularity theory for integro-differential elliptic equations, mostly developed in the 21st century. Such a class of equations often arises in analysis, probability theory,…
We introduce a novel formulation for geometry on discrete points. It is based on a universal differential calculus, which gives a geometric description of a discrete set by the algebra of functions. We expand this mathematical framework so…
An introductory theory of frames on finite dimensional quaternion Hilbert spaces is demonstrated along the lines of their complex counterpart.
We show that the integral \int e^{S(x_1,...,x_n)}dx_1...dx_n, for an arbitrary polynomial S, satisfies a generalized hypergeometric system of differential equations in the sense of I. M. Gelfand et al.
A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete…
This expository article discusses recent advances in understanding 3-dimensional mirror symmetry and the mathematical definitions of the Higgs and Coulomb branches. This is a slightly expanded version of an article appearing in the Notices…
The geometric theory of Lie systems is used to establish integrability conditions for several systems of differential equations, in particular some Riccati equations and Ermakov systems. Many different integrability criteria in the…
This text is an introduction to algebraic enumerative geometry and to applications of tropical geometry to classical geometry, based on a course given during the X-UPS mathematical days, 2008 May 14th and 15th. The aim of this text is to be…
This article describes some aspects of Cauchy integrals and related geometry of sets and measures in Euclidean spaces, etc.
The subjects in the title are interwoven in many different and very deep ways. I recently wrote several expository accounts [64-66] that reflect a certain range of developments, but even in their totality they cannot be taken as a…
Further formulas are presented involving quantum mechanics, thermodynamics, and integrable systems. Modifications of dispersionless theory are developed.
This review article discusses recent progress in understanding of various families of integrable models in terms of algebraic geometry, representation theory, and physics. In particular, we address the connections between soluble many-body…
This is an expository paper in which we define projective GIT quotients and introduce toric varieties from this perspective. It is intended primarily for readers who are learning either invariant theory or toric geometry for the first time.