Milner 317

Wednesdays, 12:30-1:30 PM

Texas A&M University

**Wednesday, September 27
Milner 317, 12:30 PM**

**Title:** On the
distribution of integral well-rounded lattices in dimension two

**Abstract:** A lattice is
called well-rounded if its minimal vectors span the corresponding
Eucildean space. Well-rounded lattices are very important objects in
lattice theory in connection with packing and covering problems, as
well as the famous conjecture of Minkowski, Frobenius problem, etc. In
this talk we completely describe well-rounded full-rank sublattices of
Z^2, as well as their determinant and minima sets. We show that the
determinant set has positive density, deriving an explicit lower bound
for it, while the minima set has density 0. We will also discuss
formulas for the number of such lattices with a fixed determinant and
with a fixed minimum. Our results extend automatically to well-rounded
sublattices of any lattice AZ^2, where A is an element of the real
orthogonal group O_2(R).