Integrable differential and functional equations, chracterization problems of the Abelian varieties

摘要奖：DMS-0405519首席研究员：伊戈尔·克里切尔本项目的主要目标是进一步发展旨在整合非线性方程、固体物理模型和量子场论模型的孤子方程的代数几何理论。目前的目标是发展变量代数曲线上的零曲率和Lax方程理论，它可以用于构造新的可积模型和研究全纯向量丛的模空间的几何。特别是离散等差方程的哈密顿理论和B++变换。我们将致力于研究Baker-Akhiezer函数的泛函方程及其在阿贝尔簇几何中的应用。经典代数几何与Abel、Riemann、Weerstrass、Poincare、Clebsch、Jacobi等十九世纪杰出的数学家的名字密不可分，主要是一种分析理论。在上个世纪，它被拓扑学、交换代数的方法和思想所丰富，并具有最基本的数学学科之一的权威。代数几何的传统折衷主义(在最好的意义上)一直是它在其他数学分支中大量应用的来源。在过去的20-25年里，代数几何作为一门“应用科学”的作用得到了极大的发展，因为它在非线性方程和量子场论问题中得到了新的应用。上个世纪七十年代孤子的发现，彻底改变了可积系统在数学和物理发展中所扮演的角色。孤子理论适用于具有显著普适性的方程。它们出现在对等离子体物理、基本粒子理论、超导理论和非线性光学中最多样化现象的描述中。这种无处不在的可积系统以及它们背后的美丽结构导致了人们对这一领域日益增长的兴趣。在现代可积系统理论中，几何和代数几何、函数方程和特殊函数、李代数和群都结合在一起。这种看似互不相关的数学和物理分支的独特结合，为创造新的跨学科教育模式提供了机会。

AbstractAward: DMS-0405519Principal Investigator: Igor KricheverThe main objective of the present project is further developmentof the algebro-geometric theory of soliton equations aimed at theintegration of non-linear equations, models of solid statephysics, and models of quantum field theories. The immediate goalis to develop a theory of zero-curvature and Lax equations onvariable algebraic curves, which can be instrumental inconstruction of new integrable models and in the investigationsof geometry of moduli spaces of holomorphic vector bundles.Particular attention will be paid to the Hamiltonian theory ofthe discrete isomonodromy equations and the B\"acklundtransformations. Efforts will be devoted to functional equationsfor the Baker-Akhiezer functions and their application to thegeometry of the Abelian varieties. Particular attention will bepaid to the characterization problem of the Prim varieties.Classical algebraic geometry, inseparably connected with thenames of Abel, Riemann, Weierstrass, Poincare, Clebsch, Jacobiand other outstanding mathematicians of the XIX-th century hasbeen mainly an analytical theory. In the last century it wasenriched by the methods and ideas of topology, commutativealgebra and has the authority of one of the most fundamentalmathematical disciplines. The traditional eclectism (in the bestsense of the word) of algebraic geometry has always been a sourceof its numerous applications to other branches ofmathematics. The role of algebraic geometry as ``an appliedscience" has grown immensely in the last 20-25 years, when itsnew applications to the problems of non-linear equations andquantum field theory were found. The discovery of solitons in theseventies of the previous century has changed once and foreverthe role which integrable systems play in the development ofmathematics and physics. The soliton theory is applicable toequations which possess the property of remarkableuniversality. They arise in the description of the most diversephenomena in plasma physics, the theory of elementary particles,the theory of superconductivity and in non-linear optics. Thisubiquity of integrable systems together with the beautifulstructures that underlie them has led to ever-growing interest inthis area. Geometry and algebraic geometry, functional equationsand special functions, Lie algebras and groups all come togetherin the modern theory of integrable systems. This uniquecombination of seemingly unrelated branches of mathematics andphysics provides an opportunity to create new interdisciplinaryeducation models.