The news of Ian Agols claimed proof of the Virtually Haken/fibered conjectures is very exciting and I look forward to reading Danny Calegari’s summaries of Ian’s talks. But at the same time, I remember the time after Perelmann announced his proof of the Poincare conjecture and the dire predictions of the end of 3-manifold topology, and I wonder what the aftermath will be this time.
The most visible aspect of this subject has always been its open problems – The Poincare conjecture at first, then Thurston’s suite of problems, including the VHC. Many (most? all?) fields are guided by major open problems, but 3-dimensional topology stands out for borrowing most of its most powerful techniques from outside – algebra, geometry, dynamical systems, gauge theory, etc. These connections have invigorated and popularized the field, but they have also created a situation in which when a major problem falls, so does the field’s public image.
Every field goes through cycles in which an innovation changes the landscape, the new ideas are explored, refined and consolidated, then the field quiets down while waiting for the next innovation. I hope and expect that the proof of the VHC will be the beginning of a phase of exploration and consolidation, rather than the beginning of dormancy.
In the long term, of course, it’s much harder to say. There are still plenty of open questions in 3-dimensional topology, though none as visible as Thurston’s conjectures. I’ve mentioned WYSIWYG geometry/topology in past posts, though this has not yet coalesced into a single major open problem worthy of the attention that Thurston’s conjectures received. There are plenty of other directions as well that I am unqualified to write about.
But it’s interesting that the proofs of the major conjectures that drove topology research for 100 years come at the same time as real world applications of topology are suddenly appearing. You may have read about knot theory being used to understand the dynamics of DNA, but there are also problems emerging related to understanding the structure of large data sets. These days, a science experiment may produce results consisting of hundreds or thousands of points in a vector space with hundreds or thousands of dimensions. Organizing this data into a form that can be understood by a human requires a great deal of topological understanding. On the other hand, most of the techniques that have been developed so far are topologically rather naive. While there may be no direct connection to 3-dimensional topology, I would claim that the intuition and ideas that come from dealing with with abstract 3-dimensional spaces can be extremely valuable for understanding these problems. (I will give more specific evidence for this in upcoming posts.)
I believe that applying ideas from low dimensional topology to problems in data analysis has the potential to not only create more outside interest in the field, but also to enrich the field by suggesting new directions to explore. In my next few posts, I plan to describe some of the types of problems that arise in understanding large data sets, describe the ways that ideas from different areas of topology have already been used to attack these problems, and discuss/speculate on how ideas from low dimensional topology could be used in the future.