Mathematical Sciences: Research in Algebraic Geometry

这个项目涉及各种各样的主题。是关于 正则分次非交换代数包括 这些主题的调查：（1）之间的对应关系 3维代数与椭圆曲线的自同构（2） （3）发现，在一个 计算机，一些新的4维代数有两个 生成器，（4）条件的可构造性， 代数是有限的和秩n在其中心，和（5）的 通过几乎可裂序列分类这些代数， 通过倾斜等价物。在几何中，项目包括 这些主题的调查：（1）多点理论的 映射的Hilbert格式，（2）的结构和自对偶 相对紧化Jacobian，（3）经典枚举Jacobian 光滑平面三次体的几何学;（4）Hubrien交 对称簇的环，（5）切线的退化 空间和惠特尼分层，（6）积极的， Kazdan-Lusztig多项式的交系数 同调与舒伯特簇的几何（7）地层 一个二维的极小变形空间 超曲面奇点作为Wall的EF函子的船体，（8） 研究曲面作为投影，作为3重的子变种的学科 小的程度，并作为重要的双重载体 结构的L-函数的研究;（9）一类椭圆函数的L-函数的研究 函数域上的隐上同调曲线 该项目涉及研究各种主题， 代数几何经过长时间的休眠， 数学的繁荣是伴随着几个世纪前的问题 正在解决。这项研究的结果将是有趣的 数学的许多不同领域。

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.