2008-2009

Lenny Fukshansky

**Monday, September 22
Millikan 213
(Pomona College), 3:00 PM**

**Title: **On distribution
of well-rounded sublattices of Z^2

Abstract: We study the distribution
of well-rounded sublattices of Z^2 by means of investigating the
structure of the set C of its similarity classes. We prove that C has
structure of an infinitely generated non-commutative monoid, and define
the notion of minima and determinant weight for each similarity class
in C. We show that these similarity classes are in bijective
correspondence with certain ideals in Gaussian integers, and construct
an explicit parametrization of lattices in each such similarity class
by elements in the corresponding ideal. We use this parametrization to
investigate some basic analytic properties of zeta function of
well-rounded sublattices of Z^2, including its order of the pole,
growth of coefficients, and some related features.