Non-commutative Algebraic Geometry

首席研究员和他的同事研究非对易代数测距，特别强调投射表面。发展了一般的抽象方法，并将其应用于特定的重要类，如量子直纹面和射影平面的非对易类的爆破。对这些曲面上曲线的正锥进行了研究，目的是为了更好地理解交理论，并使用Picard群作为不变量。使用循环同调作为新的不变量的可能性也在研究中。非对易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根本。现代的起源可以追溯到笛卡尔提出的坐标轴，但直到最近，对代数的几何观点一直局限于交换代数。然而，现代物理学表明，自然界在量子水平上的行为是非对易的(海森伯格的测不准原理说，动量和位置的测量是不可交换的--也就是说，人们测量它们的顺序会影响所获得的值)。出现在物理学中的其他基本数学工具，如李群和微分算子，也是非对易的。因此，非对易几何学的发展是理解现代物理学数学结构背后的几何学的一部分。

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.