Mathematical Sciences: Quadratic Regular Algebras

该奖项支持M.Vancliff教授致力于非交换环理论的研究，特别强调由分次正则代数理论引起的问题。特别地，Vancliff教授将试图对全局维二次正则代数进行分类，其Hilbert级数与映射到n-1维射影空间中二次曲面的某个扭曲齐次坐标环上的n元多项式环相同。她希望这将导致构造具有新行为的Artin-Schelter代数的例子的方法。这是代数子领域的研究，称为非对易环论。代数可以看作是对抽象对称性的研究。因此，代数在物理和化学领域有直接的应用。事实上，许多非对易环理论概括了量子力学中发现的结构，该提议继续探索与现代量子场论直接相关的领域。

This award supports the research of Professor M. Vancliff to work in non-commutative ring theory with special emphasis on problems arising from the theory of graded regular algebras. In particular, Professor Vancliff will try to classify the quadratic regular algebras of global dimension n with Hilbert series the same as that of the polynomial ring on n variables that map onto a certain twisted homogeneous coordinate ring of a quadric lying in n-1 dimensional projective space. She hopes that this will lead to methods of constructing examples of Artin-Schelter algebras with new behavior. This is research in the subfield of algebra called non-commutative ring theory. Algebra can be thought of as the study of symmetry in the abstract. As such, algebra has direct applications to areas of physics and chemistry. In fact, much of non-commutative ring theory generalizes structures found in quantum mechanics and this proposal continues to explore areas that connect directly with modern quantum field theory.