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

可积微分方程和函数方程，阿贝尔簇的表征问题

• 批准号: 0405519
• 负责人: Igor Krichever
• 金额: $ 14万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2007-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405519&HistoricalAwards=false
• 关键词: Integrable; differential; functional; equations; chracterization

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.

AbstractAward：DMS-0405519首席研究员：Igor Krichever本项目的主要目标是进一步发展孤子方程的代数几何理论，旨在整合非线性方程、固态物理模型和量子场论模型。本文的直接目标是建立变代数曲线上的零曲率和Lax方程的理论，这对于构造新的可积模型和研究全纯向量丛的模空间几何是有用的，特别是离散的等单值方程的Hamilton理论和B\“acklund变换。将致力于Baker-Akhiezer函数的函数方程及其在阿贝尔几何中的应用。经典代数几何主要是一种分析理论，它与世纪的Abel、Riemann、Weierstrass、Poincare、Clebsch、Jacobi等杰出数学家的名字密不可分。在上个世纪，拓扑学、交换代数的方法和思想丰富了它，使它成为最基本的数学学科之一。 代数几何的传统折衷主义（从最好的意义上来说）一直是其在数学其他分支中众多应用的来源。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上有了新的应用。上个世纪70年代，孤子的发现已经永远地改变了可积系统在数学和物理发展中所起的作用。孤立子理论适用于具有可推广普适性的方程。它们出现在等离子体物理学、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。可积系统的这种独特性以及它们背后的美丽结构使人们对这一领域的兴趣日益浓厚。几何与代数几何、泛函方程与特殊函数、李代数与群都在现代可积系统理论中融合在一起。这种看似无关的数学和物理学分支的结合为创建新的跨学科教育模式提供了机会。