Moduli Spaces and Differential Equations

项目编号DMS - 0205643研究者Emma Previato ep@math.bu.edu模空间和微分方程这个项目结合了投影几何和微分代数。几十年来，矢量束over曲线的模空间在数学物理中占有重要地位。几个相互关联的开放问题是本研究的对象。在射影空间中，向量束的模模型仍然没有明确的描述，无论是用方程还是用分类。广义theta函数（高阶向量束的部分）没有发展到可以描述完全可积层次的流或执行量子场论的必要计算（例如配分函数）的地步，最后，偏微分算子的交换环的流（广义KP流）没有被明确描述。在射影几何方面，我们将利用表示理论和格拉斯曼对应的方法计算高阶束的模空间方程和Brill-Noether轨迹的维数。在解析方面，Kleinian函数的微分方程（它推广了weerstrass p函数）将被导出并应用于整合新的哈密顿系统和广义KP流。贯穿该研究的主题是经典的约简问题：Weierstrass的学生Koenigsberger和Kowalevski分别描述了2和3属的阿贝尔积分，它们以“多重性”2约简为椭圆积分；从那时起，一般很少发现（例如，更高的属或多重性）。这个问题目前正在取得进展，部分原因是对克莱因函数的重新审视，部分原因是计算机代数的帮助。约简问题与具有自同构的曲线问题相联系，该项目包括将结果应用于微分伽罗瓦理论，常微分方程的单一性，以及代数（Goppa）码的解码算法。用维格纳的话来说，椭圆函数具有“不合理的有效性”。在模拟谐振子、打台球、测量海浪振幅、计算量子场论的配分函数时，都会出现这种现象。“椭圆”一词指的是函数周期的数量（实数的一个周期，复数的两个周期）。函数是椭圆函数的多周期模拟。尽管像Klein， H.F. Baker和O. Bolza这样的古典大师已经得到了2类（4个复周期）函数的方程，但是函数的大部分性质仍然是不明确的，包括它们对周期格的依赖。这些函数研究的新动力来自可积偏微分方程理论，如Kadomtsev-Petviashvili方程，以及自20世纪70年代以来被广泛研究的代数上完全可积系统。本项目将结合投影几何和微分代数来识别由(Kleinian) θ函数满足的微分方程，并将其应用于哈密顿系统和非线性偏微分方程的精确解。与此同时，该项目将通过将模空间嵌入射影空间来追求另一个主要方向，即theta函数的“非阿贝尔”泛化。可以简化为椭圆函数表达式的Theta函数将被几何表征并用于有效的解码算法。

Proposal Number DMS - 0205643 Investigator Emma Previato ep@math.bu.edu Moduli spaces and differential equationsThis project combines projective geometry and differential algebra. For a few decades now, moduli spaces of vector bundlesover curves have been prominent in mathematical physics. A few interrelated open problems are the object of this research. In projective space, models of moduli of vector bundles still don't have explicit description, by equations or by classification. The generalized theta functions (sections of higher-rank vector bundles) are not developed to the point that one can describe flows of completely integrable hierarchies or perform the necessary calculations of quantum field theory(partition function, e.g.) Finally, flows of commutative rings of partial differential operators (generalized KP flows) have not been described explicitly. In the proposed work: on the projective-geometry side, equations for moduli spaces of higher-rank bundles and dimensions of Brill-Noether loci will be calculated by methods of representation theory and correspondences between Grassmannians. On the analytic side, differential equations for the Kleinian functions (which generalize the Weierstrass p-function) will be derived and applied to integrate new Hamiltonian systems and the generalized KP flows. A theme that runs through the proposed research is the classical problem of reduction: Weierstrass' students Koenigsberger and Kowalevski, respectively, characterized the abelian integrals of genus 2, 3 respectively, that reduce with "multiplicity" 2 to an elliptic integral; since then, very little was found in general (e.g., higher genus or multiplicity). Progress on this problem is now under way, partly due to results that revisit the Kleinianfunctions, and partly to the aid of computer algebra. The problem of reduction is linked with the problem of curves with automorphisms, and the project includes applications of the results to differential Galois theory, monodromy of ordinary differential equations, and decoding algorithms for algebraic (Goppa) codes. Elliptic functions have an "unreasonable effectiveness",to borrow E.P. Wigner's phrase. They occur when modeling harmonic oscillators, shooting billiards, measuring the amplitude of ocean waves, computing the partition functions of quantum field theory. The word "elliptic" refers to the number of periods of the functions (one over the real, two over the complex numbers). Theta functions are the multi-periodic analog of elliptic functions. Although Old Masters such as Klein, H.F. Baker and O. Bolza,obtained equations for genus-2 (4 complex periods) theta functions, most properties of theta functions are still inexplicit, including their dependence on the period lattice. New impetus in the study of such functions came from the theory of integrable PDEs, such as the Kadomtsev-Petviashvili equation, and the attendantalgebraically completely integrable systems that have been intensively studied since the 1970s. This project will combine projective geometry and differential algebra to identify differential equations satisfied by the (Kleinian) theta functions, and apply them to find exact solutions of Hamiltonian systems and non-linear PDEs.At the same time the project will pursue the other major, "non-abelian" generalization of theta functions, by embedding moduli spaces into projective space. Theta functions that can be reduced to expressions in elliptic functions will be characterized geometrically and used in effective decoding algorithms.