PIMS-BIRS TeamUp: The Geometry of Log Unit Lattices (25frg504)

The Banff International Research Station will host the PIMS-BIRS TeamUp: "The Geometry of Log Unit Lattices" workshop at UBCO Kelowna from May 25 - June 8, 2025.

The exploration of lattices straddles the interface between mathematics and computer science. Lattices associated to rings of integers in number fields - which are generalizations of the integers and rational numbers (i.e. fractions) to higher dimensions - have crucial applications in modern information protection and have consequently undergone extensive investigation in the past few decades. For example, they feature prominently in secure cloud computing and represent one of the foremost research thrusts in designing cryptographic systems that are resistant to attacks by quantum computers.

Much more elusive are the so-called log unit lattices of number fields. They are to date only well understood in a few special cases, and attempts at deploying them in cryptography have met with mixed success. Our team of six scholars and PhD students from academic institutions in Calgary, Edmonton and Toronto will leverage its combined expertise in number theory, statistics and computing, along with prior research into lattices, to tackle difficult open problems on log unit lattices. Our aim is to investigate their shapes and geometric features, and carefully analyze how commonly (or infrequently) log unit lattices with particular desirable structural properties occur overall. Our research will significantly advance the overall understanding of log unit lattices and expand the general body of knowledge on these objects.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. BIRS is supported by Canada’s Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), and Alberta’s Advanced Education and Technology.