Recent Developments in Positive Characteristic Methods in Commutative Algebra: Frobenius Operators and Cartier Algebras

该奖项支持参加2015年3月13日至15日在乔治亚州亚特兰大乔治亚州立大学举行的“交换代数正特征方法的最新发展”会议。会议背后的动机是将交换代数和相关领域的研究人员，包括博士后研究人员和研究生聚集在一起，以交流思想并讨论正特征方法的最新发展。该领域的顶尖研究人员以及博士后和研究生将被邀请在会议上发言。为了使所有与会者都能参加会议，安排了介绍性演讲。会议将为与会者提供一个平台，开始探索与重点主题相关的研究方面，预计他们将在几年内为这些领域的研究进展做出贡献。更多信息可在会议网站上找到：http://www2.gsu.edu/~matfxe/gsu-usc/foca.htmlThe会议将重点讨论积极特征方法的最新发展。近年来，利用Frobenius同态的相关概念，对正特征方法的研究取得了巨大的发展。这包括特征p的紧闭理论，通过紧闭理论导出的奇点，以及代数几何中特征0（通过约简到特征p）的相应奇点。会议的亮点，Frobenius算子和Cartier代数，也涉及Frobenius同态在非常基本的方式。它们都与紧密闭包理论，两族几何，f模，f稳定性，测试理想，跳跃数，反规范覆盖等有着复杂的联系。最近的研究还包括定义不变的“Frobenius复杂度”，以衡量环Frobenius算子是否有限生成。本次会议将为与会者提供一个接触主题、交流思想、建立合作并为这些研究领域做出贡献的机会。

This award supports participation in the conference Recent Developments in Positive Characteristic Methods in Commutative Algebra, held during March 13-15, 2015, at Georgia State University, Atlanta, Georgia. The motivation behind the conference is to bring together researchers, including postdoctoral researchers and graduate students, in commutative algebra and related areas, in order to exchange ideas and discuss recent developments in positive characteristic methods. Top researchers in the field as well as postdocs and graduate students will be invited to speak at the conference. To make the conference accessible to all participants, introductory talks are scheduled. The conference will provide a platform for the participants to start exploring research aspects related to the highlighted topics, and it is anticipated that they will contribute research progress in these areas within a few years. More information can be found on the conference web site: http://www2.gsu.edu/~matfxe/gsu-usc/foca.htmlThe conference will focus on recent developments in positive characteristic methods. Research on positive characteristic methods has seen tremendous developments in recent years via using concepts related to the Frobenius homomorphisms. This includes the tight closure theory in characteristic p, singularities derived via tight closure theory, as well as the corresponding singularities in characteristic zero (via reduction to characteristic p) in algebraic geometry. The highlights of the conference, Frobenius operators and Cartier algebras, also involve the Frobenius homomorphisms in very fundamental ways. Both of them have intricate connections with the tight closure theory, birational geometry, F-modules, F-stability, test ideals, jumping numbers, anti-canonical covers, etcetera. Very recent research also includes the invariant "Frobenius complexity" being defined in order to measure whether a ring Frobenius operators is finitely generated. This conference will provide an opportunity for the participants to get exposed to the topics, exchange ideas, forge collaborations, and contribute to these research areas.