Mathematical Sciences: Research in Algebraic Geometry

这个项目涉及代数几何、非对易环的几何理论、复代数簇的微分拓扑和算术代数几何中的一些问题。这些问题包括：(1)像上二元映射的多点计数；(2)Whitney均衡性和Buchsbaum-Rim重数；(3)光滑完备相互作用的Gauss映射的不可分性；(4)4维一般五次超曲面上有理正规曲线的计数，以及有理正规n-ICS的Hilbert格式分量的奇异性，(5)紧化雅可比的自对偶性，(6)剩余理论和对偶理论中的过渡映射理论，(7)量子射影平面，(7)量子射影平面，(5)紧化雅可比的自对偶，(6)剩余理论和对偶理论中的过渡映射理论，(7)量子射影平面；(8)非对易射影几何的基础；(9)1维非对易射影格式；(10)特征p中具有p重分支的极大阶；(11)特殊多面体滤子；(12)作为派生函子的高阶Chow群；(13)双变层和动机；(14)几何在数域类群研究中的应用。这是对代数几何领域的研究，代数几何是现代数学中最古老的部分之一。在它的起源中，它处理的是可以在平面上用最简单的方程定义的图形，即多项式。如今，这个领域不仅使用了代数的方法，还使用了分析和拓扑学的方法。它在这些数学领域以及理论计算机科学和机器人学中得到了应用。

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.