U.S.-France Planning Visit: Cancellation Problems

该奖项将资助PI大卫·费什和他的新墨西哥州州立大学研究生的数学研究规划访问，与A·费什领导的仿射空间自同构工作组合作，在仿射消去问题的框架下制定攻击几个问题的策略。Dubouloz在法国第戎戎的勃艮第大学。复数域上的仿射代数集可以实现为复数域上某个有限维的欧几里得空间的子集，由多项式的有限集合的消失定义。代数集合称为簇，如果它不是两个真代数子集的并。给定一个簇，我们可以在它上面构造一个柱面，也是一个簇，通过把它的笛卡尔积与一条复直线相乘。仿射消去问题询问两个簇上的圆柱体的同构是否意味着簇本身的同构。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，否则是开放的。使用代数方法，反例之间的消除类似，但不同的品种，欧几里德空间已被开发。 类似的例子，但性质更接近于欧几里德空间，抵制代数分析。 然而，Festival和Dubouloz博士使用几何方法证明这些不是反例。这表明收敛的基本代数几何性质的品种，影响的取消性质，并保证进一步探索。 在规划访问期间，研究人员将制定一项有两种方法的战略。 一种方法是通过有限群在某些簇上的作用构造的例子。 另一种方法将是通过调查的代数结构的功能领域的品种，其气缸产量仿射空间。这项研究对我们理解欧氏空间的品种和有趣的品种的自同构群有影响。这项研究合并了代数方法的PI与更多的几何方法Dubouloz。 PI和第戎戎工作组之间的合作将丰富NMSU研究生的专业机会。NMSU是联邦指定的西班牙裔服务机构，PI协调在少数民族社区招募数学博士课程。该项目将影响他的教学，研究和论文监督。更直接的是，两名研究生的论文工作，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。 此外，学生将与欧洲同事建立数学联系，这将是他们未来职业生涯的巨大财富。

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.