BIMSA >
Research seminar in Discrete Mathematics
Research seminar in Discrete Mathematics
On lattice extensions
On lattice extensions
Speaker
Lenny Fukshansky
Time
Tuesday, October 20, 2026 5:05 PM - 6:15 PM
Venue
Online
Online
Zoom 787 662 9899
(BIMSA)
Abstract
A classical theorem of Ces\`aro (1884) asserts, loosely speaking, that the probability of $k \geq 2$ randomly picked integers being coprime is $1/\zeta(k)$. This theorem has been rediscovered several times during the 20-th century and generalized in various ways. One such generalization (G. Maze et al., 2011) is a computation of probability that a collection of $k < n$ points in $\zed^n$ is extendable to a basis. Then, given such an extendable collection, one can ask in how many ways it can be extended by vectors with bounded norm as this bound tends to infinity? We start by answering this question, which leads us into a more general topic of lattice extensions.
A lattice $\Lambda$ is said to be an extension of a sublattice $L$ of smaller rank if $L$ is equal to the intersection of $\Lambda$ with the subspace spanned by $L$. The idea of lattice extensions is implicit in several constructions of reduction theory, such as Minkowski and HKZ reduced bases, the study of stability and canonical flag of a lattice, etc. Our goal is to initiate a more systematic study of the geometry of lattice extensions. To this end, we prove the existence of a small-determinant extension of a given lattice and then look at successive minima and covering radius. We investigate extensions (within an ambient lattice) preserving the successive minima of the given lattice, as well as extensions preserving the covering radius. This is joint work with Maxwell Forst.
A lattice $\Lambda$ is said to be an extension of a sublattice $L$ of smaller rank if $L$ is equal to the intersection of $\Lambda$ with the subspace spanned by $L$. The idea of lattice extensions is implicit in several constructions of reduction theory, such as Minkowski and HKZ reduced bases, the study of stability and canonical flag of a lattice, etc. Our goal is to initiate a more systematic study of the geometry of lattice extensions. To this end, we prove the existence of a small-determinant extension of a given lattice and then look at successive minima and covering radius. We investigate extensions (within an ambient lattice) preserving the successive minima of the given lattice, as well as extensions preserving the covering radius. This is joint work with Maxwell Forst.
Speaker Intro
Lenny Fukshansky is a Professor of Mathematics at Claremont McKenna College, where he has been since 2007. Prior to that, he held a postdoctoral appointment at Texas A&M University and shorter visiting positions at IHES and Max Planck Institute in Bonn. He received his Ph.D. in mathematics in 2004 from UT Austin under the supervision of Jeffrey Vaaler. His main research interests are in number theory, discrete geometry, and the theory of Euclidean lattices.