First meeting, Ben Gurion University, 12.11.18, Building 58 room -101
Bounding the covolume of lattices in products.
The Kazhdan-Margulis theorem ensures that, in a semisimple Lie group
without compact factors, the set of covolumes of all lattices is
bounded away from 0. The goal of this talk, based on joint work with
Adrien Le Boudec, is to discuss the validity of that theorem, and some
of its relatives, in the broader context of lattices in products of
compactly generated simple locally compact groups.
On algorithms in SL(n,Z)
I will introduce and analyze two algorithms in SLn(Z):
In 1983 D. carter and G.Keller proved that in SL(n,O) (where n>2 and O is the
ring of integers of an algebraic number field) there exists an integer
v_n(O)depending only on n and O such that every matrix in SL(n,O) is a
product of at most v_n(O) elementary matrices. We ask the question: How
does this decomposition behave?
Based on the work of S.I.Adian and J.Mennicke for the case of SLn(Z),
we can deduce an upper bound for the entries of those elementary
After the work of R.K. Dennis and L.N. Vaserstein, we can show that
for n>>0, every matrix in SL(n,Z)is a product of six commutators, and
again, get a bound for those matrices.
0.01 % improvement of the Liouville property for discrete harmonic functions on Z^2.
Let u be a harmonic function on the plane. The Liouville theorem
claims that if |u| is bounded on the whole plane, then u is
identically constant. It appears that if u is a harmonic function on
the lattice Z^2, and |u| < 1 on 99,99% of Z^2, then u is a constant
function. Based on a joint work (in progress) with A. Logunov,
Eu. Malinnikova and M. Sodin.
What is the sum of angles of an n-gon?...
Wait, I meant an n-gon in the 3-dimensional space!
You are cordially invited.
The support of BGU Center for advanced studies in mathematics is gratefully acknowledged.