U.S.-France Planning Visit: Cancellation Problems

美法计划访问：取消问题

• 批准号: 0936691
• 负责人: David Finston
• 金额: $ 2万
• 依托单位: New Mexico State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0936691&HistoricalAwards=false
• 关键词: U.S.; France; Planning; Visit; Cancellation

This award will fund mathematics research planning visits for PI David Finston and his New Mexico State University graduate students to develop strategies to attack several problems under the framework of the affine cancellation problem in collaboration with the working group on automorphisms of affine space led by A. Dubouloz at Université de Bourgogne in Dijon, France. An affine algebraic set over the field of complex numbers can be realized as a subset of a Euclidean space of some finite dimension over the complex field, defined by the vanishing of a finite collection of polynomials. The algebraic set is called a variety if it is not the union of two proper algebraic subsets. Given a variety, one can construct a cylinder over it, also a variety, by taking its Cartesian product with a complex line. The affine cancellation problem asks whether an isomorphism of cylinders over two varieties implies an isomorphism of the varieties themselves. The answer is no in general, although the most intriguing case, where one of the varieties is a complex Euclidean space, is known to have a positive solution in low dimensions and is open otherwise. Using algebraic methods, counterexamples to cancellation among varieties resembling, but distinct from, Euclidean spaces have been developed. Similar examples, but with properties even closer to those of Euclidean spaces, resist algebraic analysis. However, Drs. Finston and Dubouloz have used geometric methods to show that these are not counterexamples. This suggests a convergence on essential algebrao-geometric properties of varieties that influence the cancellation property and warrant further exploration. During the planning visits, the researchers will develop a strategy with two approaches. One approach will be through examples constructed as quotients by actions of finite groups on certain varieties. Another approach will be through investigation of the algebraic structure of function fields of varieties whose cylinders yield affine spaces. The research has implications for our understanding of Euclidean spaces as varieties and of automorphism groups of interesting varieties.The research merges algebraic methods of the PI with the more geometric approach of Dubouloz. Collaborations between the PI and the working group in Dijon will enrich professional opportunities for NMSU graduate students. NMSU is a federally designated Hispanic Serving Institution, and the PI coordinates recruiting in the minority community for the doctoral program in mathematics. The project will influence his teaching, research, and thesis supervision. More immediately, the thesis work of the two graduate students, both members of underrepresented minority groups, will be enriched by their exposure to more geometry. In addition, the students will develop mathematical connections with European colleagues that will be great assets in their future careers.

该奖项将资助PI David Finston和他的新墨西哥州立大学研究生访问数学研究计划，与法国第戎的勃艮第大学A. Dubouloz领导的仿射空间自同构工作组合作，在仿射抵消问题的框架下制定解决几个问题的策略。复数域上的仿射代数集可以实现为复域上有限维欧几里得空间的子集，由有限多项式集合的消失来定义。如果代数集合不是两个固有代数子集的并集，则称为变体。给定一个变量，我们可以在它上面构造一个圆柱体，同样是一个变量，通过取它与复直线的笛卡尔积。仿射对消问题问的是两个变量上柱体的同构是否意味着两个变量本身的同构。一般来说，答案是否定的，尽管最有趣的情况是，其中一个变种是一个复杂的欧几里得空间，已知在低维有一个正解，在其他地方是开放的。使用代数方法，反例之间的抵消类似，但不同于欧几里得空间已开发。类似的例子，但性质更接近欧几里得空间，抵制代数分析。然而,Drs。Finston和Dubouloz使用几何方法来证明这些不是反例。这表明了影响消去性质的基本代数几何性质的收敛性，值得进一步探索。在计划访问期间，研究人员将制定一项包含两种方法的策略。一种方法是通过有限群对某些变量的作用构造成商的例子。另一种方法是研究柱面产生仿射空间的变元的函数场的代数结构。该研究对我们理解欧几里得空间的变体和有趣变体的自同构群具有重要意义。该研究将PI的代数方法与更几何的Dubouloz方法相结合。PI与第戎工作组之间的合作将丰富新密西根州立大学研究生的专业机会。新密歇根州立大学是联邦政府指定的西班牙裔服务机构，PI负责协调少数族裔社区数学博士课程的招聘工作。该项目将影响其教学、研究和论文指导。更直接的是，两位研究生的论文工作，都是代表性不足的少数群体的成员，将通过接触更多的几何来丰富他们的论文工作。此外，学生将与欧洲同事建立数学联系，这对他们未来的职业生涯将是巨大的财富。