Mathematical Sciences: Research in Algebraic Geometry

这个项目涉及多个主题。关于正则有限表示分次非交换代数的研究包括：(1)3维代数与椭圆曲线的自同构之间的对应；(2)对偶模；(3)借助于计算机发现一些新的具有两个生成元的4维代数，(4)代数是有限的且在其中心具有n阶的条件的可构造性，以及(5)通过几乎分裂序列和倾斜等价对这些代数进行分类。在几何方面，项目包括：(1)希尔伯特格式的映射的多点理论，(2)相对紧化雅可比的结构和自对偶，(3)光滑平面三次曲线的经典计数几何，(4)对称簇的Halphen交环，(5)切空间的退化和Whitney分层，(6)通过交同调的Kazdan-Lusztig多项式系数的正性和Schubert簇的几何；(7)作为WALL的EF-函子壳的二维超曲面奇点的极小变形空间的层次；(8)作为投影的曲面的研究，作为三次小次的子簇，以及(9)函数域上某些椭圆曲线的L函数的晶体上同调研究。这个项目涉及代数几何中的各种主题的研究。在经历了一段长时间的休眠后，这一数学领域蓬勃发展，数百年前的问题得到了解决。这项研究的结果将对许多不同的数学领域产生兴趣。

This project involves a variety of topics. Those concerning regular finitely presented graded noncommutative algebras include the investigation of these topics: (1) the correspondence between algebras of dimension 3 and automorphisms of elliptic curves, (2) the dualizing module, (3) the discovery, with the aid of a computer, of some new algebras of dimension 4 having two generators, (4) the constructability of the condition that an algebra be finite and of rank n over its center, and (5) the classification of those algebras via almost split sequences and via tilting equivalences. In geometry, the projects include the investigation of these topics: (1) the multiple-point theory of maps via the Hilbert scheme, (2) the structure and autoduality of the relative compactified jacobian, (3) the classical enumerative geometry of smooth plane cubics, (4) the Halphen intersection ring of a symmetric variety, (5) the degeneration of the tangent space and Whitney stratification, (6) the positivity of the coefficients of Kazdan-Lusztig polynomials via intersection homology and the geometry of Schubert varieties (7) the stratum of the miniversal deformation space of a 2-dimensional hypersurface singularity as the hull of Wall's EF-functor, (8) the study of surfaces as projections, as subvarieties of 3-folds of small degree, and as the carriers of important double structures, and (9) the study of L-functions of certain elliptic curves over function fields via crytalline cohomology. This project concerns research on a variety of topics in algebraic geometry. After a long period of dormancy, this area of mathematics has flourished with problems that are centuries old being resolved. The results of this research will be of interest to many different areas of mathematics.