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.
