Diophantine methods, lattices, and arithmetic theory of quadratic forms (11w5011)
Wai Kiu Chan (Wesleyan University)
Lenny Fukshansky (Claremont McKenna College)
Rainer Schulze-Pillot (Universitaet des Saarlandes)
Jeffrey Vaaler (University of Texas at Austin)
1. Classical arithmetic theory of quadratic forms and lattices.
2. Diophantine problems and the theory of height functions.
In spite of the close connections of these areas which have been described in the overview section of this proposal and although people in each of these areas frequently meet at various conferences, it is quite rare for them to meet altogether for a joint workshop. This program would provide such a unique opportunity, which we are hoping will result in fruitful mathematical communication and lead to significant future progress. We believe that bringing together people working in these different areas with a unifying connection to the theory of quadratic forms will motivate the application of original methods to existent problems as well as give rise to new research questions. There is a variety of prominent research directions that lie in the intersection of these two areas. Here are just a few of them:
1. Representation problems for quadratic forms and lattices over global fields and rings, in particular finding representatives of small height in orbits of representations under the automorphism group of the form and counting representations of bounded height.
2. Small zeros (with respect to height) of individual quadratic forms and of systems of quadratic forms, originating in the work of Cassels and Siegel, its various generalizations, and related Diophantine problems with the use of heights.
3. Classical Hermite's constant, geometry of numbers, and various generalizations with the use of height functions, explicit reduction theory of definite and indefinite quadratic forms.
Diophantine methods with the use of height functions are usually based on geometry of numbers and ideas from lattice theory. The target of these methods often lies in the realm of quadratic forms theory. This is the motivation for bringing together people working in these different, but related areas.
We believe the time for our proposed workshop is just right. The recent years have seen some remarkable progress in both, the arithmetic of quadratic forms and diophantine methods with height functions, as well as in the intersection of the two fields, with many new players emerging. It is also remarkable that some of the methods used in both fields are starting to look increasingly alike. The proposed workshop would ensure that people are aware of each others' work and promote the exchange of ideas to further the subject along.