Characters in Low-Dimensional Topology

University Of Texas At Austin, Austin TX

Investigators

Abstract

This award supports participation by U.S.-based junior researchers in the conference "Characters in Low-Dimensional Topology" held June 2-6, 2018 in Montreal, Canada at the University of Quebec at Montreal, hosted by the Centre de Recherches Mathematiques. The conference provides an international forum for the dissemination and discussion of recent important advances in low-dimensional topology and geometry. This is a large and active area of research in pure mathematics, whose basic goal is to understand the structure of mathematical objects known as manifolds, which arise naturally in modeling many scientific phenomena. Manifolds are "spaces" whose small-scale structure depends only on the number of degrees of freedom, called the dimension: for example, on a small scale a 1-dimensional manifold looks just like a line, a 2-dimensional manifold looks just like a flat plane, a 3-dimensional manifold looks like a region of the space in which we live, and so on. Although all manifolds of a given dimension are the same on a small scale, their large-scale structures can be very different and quite complicated; many mathematical techniques have been developed to investigate this. "Low-dimensional" means dimensions 3 and 4, which are of special interest: we live in a manifold that has three dimensions if we consider only the spatial universe, but four dimensions if time is included as well. The conference brings together experts and emerging researchers to report on recent results and explore future directions. This award enables young mathematicians from the U.S., graduate students and postdoctoral researchers, to participate. A particular effort will be made to support diversity by encouraging the attendance of women and other groups that are traditionally underrepresented in mathematics. The conference focuses on the many different geometric and topological techniques that have recently been developed to study 3-dimensional manifolds. It is now known that 3-manifolds possess rigid geometric structures, and recent related developments have brought new insights not only to low-dimensional topology but also to neighboring fields. Meanwhile, new homology theories, having their origins both in quantum physics and in gauge theory, have led to dramatic progress on old questions in knot theory and 3-and 4-dimensional topology. Despite these advances on several fronts, relatively little is known about the interplay between analysis, geometry, and algebraic invariants in dimension 3. The conference highlights ongoing research focused on finding direct connections between these several aspects of 3-dimensional topology. The meeting brings together experts and early-career mathematicians working on the new invariants of 3-manifolds, on geometric group theory applied to 3-manifolds, and on more classical geometric and topological techniques. The website for the conference is cirget.uqam.ca/boyerfest//en This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.