Combinatorics and commutative algebra of algebraic varieties with group actions

代数几何是数学的中心领域。它不仅在纯数学和应用数学中具有重要意义，而且在自然科学、工程学和其他领域也具有重要意义(例如。经济学)。代数几何的核心是研究多元多项式集合的公共零点。该领域的一个普遍主题是将关于这些零集或代数簇的几何问题转化为等价的代数问题。这类几何问题的例子包括“这个零点集有多个分量吗？”和“有没有什么奇点？”

Algebraic geometry is a central area of mathematics. It has importance not just in pure and applied mathematics, but also in the natural sciences, engineering, and beyond (eg. economics). At its core, algebraic geometry is the study of common zeros of collections of polynomials in multiple variables. A general theme in the field is to translate geometric questions about these zero-sets, or algebraic varieties, into equivalent algebraic questions. Examples of such geometric questions include "does this zero-set have multiple components?" and "are there any singularities?".