PIMS-UBC Rising Stars Lecture, Department Colloquium: Maggie Miller

Abstract: Often, interesting knotting vanishes when allowed one extra dimension, e.g. knotted circles in 3-space all become isotopic when included into 4-space. Hughes, Kim and I recently found a new counterexample to this principle: for g>1, there exists a pair of 3-dimensional genus-g solids in the 4-sphere with the same boundary, and that are homeomorphic relative to their boundary, but do not become isotopic rel boundary even when their interiors are pushed into the 5-dimensional ball. This proves a conjecture of Budney and Gabai for g>1 in a very strong sense.

In this talk, I’ll describe some motivation from 3-dimensional topology and useful/weird facts about higher-dimensional knots (e.g. knotted surfaces in 4-manifolds), show how to construct interesting codimension-2 knotting in dimensions 4 and 5 (joint with Mark Hughes and Seungwon Kim), and talk about related open problems.