Mathematical Sciences: Research in Algebraic Geometry

这个项目涉及各种各样的主题。关于正则有限表示的分级非交换代数的研究包括以下主题：(1) 3维代数与椭圆曲线自同构的对应关系，(2)对偶模，(3)借助于计算机发现了一些具有两个生成子的4维新代数，(4)有限代数在其中心上为n秩的条件的可构造性，(5)通过几乎分裂序列和倾斜等价对代数进行分类。在几何方面，项目包括以下主题的调查：(1) Hilbert格式映射的多点理论，(2)相对紧化雅可比矩阵的结构和自对偶性，(3)光滑平面立方体的经典枚举几何，(4)对称变化的Halphen交环，(5)切空间的退化和Whitney分层，(6) Kazdan-Lusztig多项式通过交同调和Schubert变分的几何性质的系数的正性(7)作为Wall's ef函子壳的二维超曲面奇点的通用变形空间的地层，(8)作为投影的曲面的研究，作为小次3折叠的子变种，(9)利用晶体上同调研究函数场上某些椭圆曲线的l函数。这个项目涉及代数几何中各种主题的研究。经过长时间的沉寂之后，这一数学领域开始蓬勃发展，几个世纪前的问题得到了解决。这项研究的结果将引起许多不同数学领域的兴趣。

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.