Homological Techniques in Commutative Algebra

交换代数在发展代数几何的基础方面是至关重要的。这些领域研究的两个中心主题是重数和模的同调行为。这项提案涉及与这些专题有关的问题，特别是它们之间的相互作用。第一个项目涉及研究希尔伯特-昆兹多重性，这是一种有趣但不被很好理解的正特征奇点复杂性的度量，在特征零中是否有一种解释，在那里几何解释可能会更简单或更清楚(一些已知的例子表明，随着特征走向无穷大，可能存在一个极限，而且似乎是一个更简单的数字)。第二个涉及Koszul复形的同调，虽然在这种情况下，这些方法并不完全来自完全交环域，但由于Koszul复形是构造Tate归结的第一步，因此产生了许多相同的想法。人们正在研究许多方向，Huneke、Simis、Vasconelos和其他许多人研究了几种解决旧问题的新方法。第三个涉及有理同伦理论中一个更古老的猜想，在这个领域中，来自完全交的分解和来自差分分次分解的思想已经有很长的相互作用的历史，但还没有被用于这个特定的问题；这种方法的初步结果看起来很令人鼓舞。第四个问题是关于非正则环上的Serre交重数。这些项目位于交换代数的一些主要方向的中心，但着眼于邻近的数学领域或代数的子领域。例如，第一个也是主要的项目涉及多重性，它是曲线、曲面或更高维度对象上非平滑(即，尖锐)点的复杂性的数字度量。研究人员特别感兴趣的是与物体的物理几何没有直接关系的代数或数论环境中的这些。已知的结果显示了这些数字相当神秘的行为，研究人员希望通过将背景与可能应用经典几何技术的更几何的背景联系起来，来揭示一些情况。这是一个微妙的过程，也是一项正在进行的工作。同样，第三个项目涉及一个问题，将她抽象的代数领域的思想应用到拓扑学的一部分，即处理物理空间的领域。总而言之，尽管项目的主题相当不同，但它们持有一个可能不会立即显现的共同主题：即，调查者建议将她在自己领域的经验应用于进一步远离该领域、传统上不以这种方式研究的各种问题。

Commutative algebra is crucial in developing the foundations of algebraic geometry. Two central topics of research in these fields are multiplicities and the homological behavior of modules. This proposal concerns questions related to these topics and especially their interplay. The first project involves studying whether Hilbert-Kunz multiplicity, an intriguing and not well-understood measure of the complexity of a singularity in positive characteristic, has an interpretation in characteristic zero where it could be simpler or clearer to interpret geometrically (some known examples show that a limit as the characteristic goes to infinity might exist and seems to be a simpler number). The second concerns the homology of Koszul complexes, and although in this case the approaches are not exclusively from the realm of complete intersection rings, many of the same ideas arise since the Koszul complex is the first step in constructing a Tate resolution. Many directions are being examined, with several new approaches to old problems studied by Huneke, Simis, Vasconcelos and many others. The third concerns an older conjecture in rational homotopy theory, a field with which the ideas from resolutions of complete intersections and from differential graded resolutions have a long history of interaction, but have not yet been used for this particular problem; preliminary results of this approach look encouraging. The fourth concerns Serre's intersection multiplicity over non-regular rings. These projects are at the center of some of the main directions in commutative algebra, but with a view towards neighboring fields of mathematics or subfields of algebra. For example, the first and main project concerns multiplicities, which are numerical measures of the complexity of a non-smooth (that is, sharp) point on a curve, surface, or higher dimensional object. The investigator is especially interested in these in an algebraic or number theoretic setting that is not directly related to the physical geometry of an object. Known results show quite mysterious behavior of these numbers and the investigator hopes to shed some light on the situation by relating the setting to a more geometric one where classical geometric techniques might be applied. This is a delicate procedure and work in progress. Likewise, the third project involves a problem that would apply ideas from her abstract field of algebra to a part of topology, a field which deals with physical spaces. In summary, although the subjects of the projects are quite varied, they hold a common theme that may not be immediately evident: Namely, the investigator proposes to apply her experience in her own area to various problems further removed from this area and not traditionally studied in this way.