Generic complexity in computational topology: Breaking through the bottlenecks (2011-2013)

Three-dimensional spaces play a key role in many fields of science, from genetics to cosmology and electrodynamics. Practical algorithms to study these spaces are elusive, and many existing algorithms are too slow for real applications. By addressing the key bottleneck of normal surfaces using the new concept of generic complexity, we will capture the theoretical limitations of current algorithms, overcome these limitations with new effective algorithms, and deploy these to yield new experimental and theoretical results for applications such as the homeomorphism problem and the famous virtual fibration conjecture. The results will have a major impact in the burgeoning fields of computational topology and generic complexity.