Representation Theory and Spaces of Polynomials

E. Mukhin计划继续研究代数Bethe Ansatzmethod与非齐次Gaudin、XXX、XXY等模型的关系。要解决的主要问题是：建立一种基于Bethe Ansatz方程解总体的代数Bethe Ansatz方法；用总体显式地求解bethe Ansatz方程；用种群来研究Bethe Ansatz方程的解的个数；证明了（改进的）BetheAnsatz猜想，该猜想声称在sl(N)的情况下，Bethe向量在参数的一般值的状态空间中形成了一组基。许多数学物理模型的哈密顿量的对角化可以通过求解相应的代数方程组（Bethe - Ansatz方程组）来实现。结果表明，贝特·安萨茨方程的解自然地形成称为种群的族。在与l(N)相关的Gaudin模型的情况下，总体与适当的Schubert循环的交点和只有多项式解的N阶标量微分算子呈一一对应关系。因此，sl(N)总体由单变量多项式的N维空间给出，这些多项式具有指定的奇异点和指数。E. Mukhin希望多项式空间的研究可以使Bethe Ansatz方法更容易处理，并解决该领域的几个老问题。

E. Mukhin plans to continue his studies of the Algebraic Bethe Ansatzmethod in the relation to the non-homogeneous Gaudin, XXX, XXY andother models. The main problems to be addressed are: develop anapproach to the Algebraic Bethe Ansatz based on populations ofsolutions of Bethe Ansatz equations; use the populations to solve theBethe Ansatz equations explicitly; use populations to study the numberof solutions of the Bethe Ansatz equations; prove the (modified) BetheAnsatz Conjecture which claims that the Bethe vectors form a basis inthe space of states for generic values of parameters, in the case ofsl(N).Diagonalization of Hamiltonians of many models of mathematical physicscan be performed if the corresponding system of algebraic equationscalled Bethe Ansatz equations is solved. It was shown that thesolutions of the Bethe Ansatz equations naturally form families whichare called populations. In the case of the Gaudin model associated tosl(N), the populations are in one-to-one correspondence with thepoints of intersection of appropriate Schubert cycles and with thescalar differential operators of order N which have only polynomialsolutions. Thus the sl(N) populations are given by N-dimesnionalspaces of polynomials in one variable with prescribed singular points andexponents. E. Mukhin hopes that the study of the spaces ofpolynomials can make the Bethe Ansatz approach more tractable and resolveseveral old problems in this area.