Non-commutative Algebraic Geometry

非交换代数几何

• 批准号: 0070560
• 负责人: S. Paul Smith
• 金额: $ 11.85万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-15 至 2004-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070560&HistoricalAwards=false
• 关键词: Non; commutative; Algebraic; Geometry

The principal investigator and his colleagues study non-commutativealgebraic goemetry with particular emphasis on projectivesurfaces. General abstract methods are developed and are applied tospecific important classes such as quantum ruled surfaces, and blowups ofnon-commutative analogues of the projective plane. The positive cone ofcurves on these surfaces is studied, with the goal of betterunderstanding the intersection theory and using the Picard group as aninvariant. The possibility of using cyclic homology as a new invariant isalso under examination. Non-commutative algebraic geometry extends the fruitfulinteraction of algebra and geometry that lies at the root of modernalgebraic geometry. The modern origin dates from Descarte's introductionof coordinate axes, but until recently the geometric perspective onalgebra has been restricted to commutative algebra. However, modernphysics shows that nature behaves non-commutatively at the quantum level(Heisenberg's uncertainty priciple says that the measurementof momentum and position do not commute -that is, the order in which onemeasures them affects the values obtained). Other fundamental mathematicaltools that appear in physics,like Lie groups and differential operators,are also non-commutative. Thus, development of non-commutative geometry ispart of understanding the geometry underlying the mathematical structuresof modern physics.

首席研究员和他的同事们研究非交换代数几何，特别强调投影曲面。一般的抽象方法的发展和应用到特定的重要类，如量子直纹面，和blowups ofnon-commutative类似物的射影平面。为了更好地理解相交理论和使用Picard群作为不变量，研究了这类曲面上曲线的正锥。使用循环同调作为一个新的不变量的可能性也正在审查中。非交换代数几何扩展了代数和几何之间卓有成效的相互作用，这是现代代数几何的基础。现代的起源日期从笛卡尔的介绍坐标轴，但直到最近的几何角度对代数一直限于交换代数。然而，现代物理学表明，自然界在量子水平上表现出不可交换性（海森堡的不确定性原理说，动量和位置的测量不会交换-也就是说，测量它们的顺序会影响所获得的值）。其他出现在物理学中的基本代数工具，如李群和微分算子，也是非交换的。因此，非对易几何的发展是理解现代物理学数学结构之下的几何的一部分。