Geometry of Numbers and Diophantine Approximation

This mathematical research project is a joint project of the Austrian Science Fund FWF and the Russian Science Foundation RSF. On the Austrian side 5 researchers (in Graz, Vienna, and Leoben) are working in the project, while on the Russian side 10 scientists are involved (in Khabarovsk and Moscow). Geometry of Numbers and Diophantine Approximation are two closely related parts of number theory, which is a branch of mathematics which is concerned with properties of (whole, rational or real) numbers. A typical question from the geometry of numbers asks for the number of points of a multi-dimensional lattice which are contained in a certain given set. Obviously the answer will depend on the fine-meshedness of the lattice, and on the size of the set which should contain the lattice points. What is a precise mathematical formulation of these observations, and how large is the error in an approximation formula? Or, to state a second question, how large does a (convex) test set have to be so that it must necessarily contain at least one lattice point? Diophantine approximation, which is named after Diophantus of Alexandria and looks back on a history spanning several millenia, is concerned with the question how closely real numbers can be approximated by rational numbers (that is, by fractions). For example, the circle constant pi cannot be exactly represented as a fraction (it is irrational), but it can be closely approximated by the fraction 22/7. An even better approximation, which was already known to Chinese scholars 1500 years back, is given by 355/113. How can such approximations be obtained? How close can the approximation be, given a certain restriction on the size of the denominator? How good can the approximation be if pi is replaced by Euler`s constant e, and what does this tell us about the numbers pi and e? These are typical questions of Diophantine approximation. Both disciplines, geometry of numbers as well as Diophantine approximation, have a long and strong history among Austrian mathematicians. Two of the most important Austrian mathematicians of the second half of the twentieth century, Edmund Hlawka and Wolfgang Schmidt, have been working extensively in these fields. In recent years there have been many exciting developments in these areas, such as the development of the so-called parametric geometry of numbers. Because of these new developments, several long-standing open problems have finally come within reach. This joint FWF-RSF research project offers the unique chance to combine the (partially complementary) expertise of the Austrian and the Russian number-theoretic schools, in order to solve some of the most important open problems in these areas of mathematics.

This project supported the intensive cooperation of Russian and Austrian mathematicians. The focus lies on fundamental questions concerning the nature of our numbers, for instance diophantine approximations and the solution of diophantine equations.