Hodge theory and algebraic cycles

霍奇猜想是代数几何的中心问题之一。如果这是真的，它将在拓扑学和代数几何之间建立强有力的联系。积分霍奇猜想是一个较强的命题，但对于某些代数变量不成立，如Atiyah和Hirzebruch所示。尽管如此，该项目的目的是证明霍奇猜想的积分对一个大类的三维品种。在另一个方向上，对于有限群变量的商，积分霍奇猜想常常失效。本课题将利用多有限群计算商变量代数环的周环。这可以看作是在霍奇猜想失效时找到正确的替代方法。代数几何是研究由多项式方程定义的形状的学科。它是数学的中心领域，介于任意形状（拓扑学）和方程理论（代数和计算）之间。数学的使用者总是不得不解决多项式方程组，这在今天仍然是一个具有计算挑战性的问题。该项目处理的问题是，哪些形状可以连续移动，从而成为多项式方程的定义。有一个看似合理的答案，但似乎很难找到证据。该项目旨在解决最重要的低维情况下的问题。

The Hodge conjecture is one of the central problems of algebraic geometry. If true, it would give a strong connection between topology and algebraic geometry. The integral Hodge conjecture is a stronger statement which fails for some algebraic varieties, as shown by Atiyah and Hirzebruch. Nonetheless, the project aims to prove the integral Hodge conjecture for a large class of 3-dimensional varieties. In another direction, the integral Hodge conjecture often fails for quotients of varieties by finite groups. The project will compute the Chow ring of algebraic cycles for quotient varieties by many finite groups. This can be viewed as finding the correct replacement for the integral Hodge conjecture when it fails.Algebraic geometry is the study of shapes defined by polynomial equations. It is a central area of mathematics, at the border between arbitrary shapes (topology) and the theory of equations (algebra and computation). Users of mathematics have always had to solve systems of polynomial equations, and that remains a computationally challenging problem today. The project deals with the question of which shapes can be moved continuously to become defined by polynomial equations. There is a plausible answer, but a proof seems exceedingly hard to find. The project aims to solve the problem in the most important low-dimensional cases.