The L-space conjecture (25rit030)

The Banff International Research Station will host the “The L-space conjecture” workshop in Banff from February 14 - 23, 2025.

A major task for mathematicians is to understand the types of spaces that we live in well enough to accurately describe all their possible shapes. To do this, mathematicians convert geometric spaces and geometric problems into the language of algebra, where calculations can be made. For instance, to each 3-dimensional space we can associate an algebraic object called a ``group" and in many cases, spaces with the same group are actually the same. Thus understanding the different possible 3-dimensional spaces is determined by the algebraic problem of understanding the different possible groups which can arise. One of the principal goals of our research proposal is to study the possibility of translating geometric properties of 3-dimensional spaces into algebra. The geometric property in question is the ability to cut a space into disjoint surfaces which piece together locally like a deck of cards. Not all 3-dimensional spaces can be cut up like this and it is expected that only those whose groups can be ordered in an algebraically coherent way do. This is what we intend to investigate. It also seems that the existence of such a splitting of the space is equivalent to a certain analytic condition reflecting higher dimensional geometry. This is quite surprising as there is no compelling heuristic which explains why these three conditions should be connected. On the other hand, there is no known space for which they differ and many infinite families for which they are known to be the same. We will investigate whether the analytic condition is the same as the geometric one.

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. The research station is located at The Banff Centre in Alberta and is supported by Canada’s Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta’s Advanced Education and Technology, and Mexico’s Consejo Nacional de Ciencia y Tecnología (CONACYT).