Mathematical Sciences: Research in Algebraic Geometry

这个项目涉及代数中的一些问题 几何，非交换环的几何理论， 复代数簇的微分拓扑，以及 算术代数几何 这些问题包括： (1)一个有限映射的多点枚举， （2）Whitney等奇异性和 Buchsbaum-Rim多重性;（3）高斯的不可分性 光滑完全相互作用的映射;（4） 中一般五次超曲面上的有理正规曲线 4-空间，和奇异性的希尔伯特计划的组成部分， 有理正规n-ics;（5）紧化的 （6）在变分理论中的转移映射理论; 剩余和对偶;（7）量子投影平面;（8） 非交换射影几何的基础;（9） 1维非交换射影概型;（10）极大 特征p中具有p重分支的阶;（11）特殊 多面体滤子;（12）高阶Chow群 函子;（13）双变层和动机;（14）应用 几何学与数域类群的研究有关。 这是代数几何领域的研究， 现代数学最古老的部分之一。 在其起源中，它 处理的图形，可以定义在平面上的最简单的 方程，即多项式。 如今，该领域利用 方法不仅从代数，而且从分析和拓扑。 它在这些数学领域中得到应用， 在理论计算机科学和机器人学中。

This project is concerned with a number of issues in algebraic geometry, the geometric theory of noncommutative rings, the differential topology of complex algebraic varieties, and arithmetic algebraic geometry. These issues include the following: (1) the enumeration of the multiple-points of a finite map that is birational onto its image; (2) Whitney equisingularity and the Buchsbaum-Rim multiplicity; (3) the inseparablility of the Gauss map of a smooth complete interaction; (4) the enumeration of the rational normal curves on the generic quintic hypersurface in 4-space, and the singularity of the Hilbert scheme component of the rational normal n-ics; (5) the autoduality of the compactified Jacobian; (6) the theory of a transition map in the theory of residues and duality; (7) quantum projective planes; (8) the foundations of noncommutative projective geometry; (9) noncommutative projective schemes of dimension 1; (10) maximal orders with p-fold ramification in characteristic p; (11) special polyhedral filtrations; (12) higher Chow groups as derived functors; (13) bivariant sheaves and motives; (14) applications of geometry to the study of class groups of number fields. This is research in the field of algebraic geometry, which is one of the oldest parts of modern mathematics. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but also from analysis and topology. It is finding application in those areas of mathematics as well as in theoretical computer science and in robotics.