Diophantine Approximation and related subjects such as the theory of
Uniform Distribution and Dynamical Systems are one of the modern areas
in contemporary Number Theory, which have seen tremendous development in
the recent decades.
In this project we work on various challenging problems related to
inequalities for Diophantine exponents, badly approximable subspaces of
Euclidean space, problems related to angles of inclination of subspaces,
singularity phenomena, applications to Kronecker sequence, general and
special problems in inhomogeneous approximation, and the behavior of
various types of irrationality measure functions and Diophantine spectra.
We use classical and modern methods of Geometry of Numbers, Metric
Number Theory and Dynamical Systems. One of the important new approaches
is to combine the analysis of the geometry of best approximations with
relatively new original constructions from Parametric Geometry of
Numbers. We also apply methods of Metric Number Theory and extract new
geometric approaches and methods related to application of theory of
Dynamical Systems. This will lead to new and unexpected results in many
of the proposed problems.