Periodic Reporting for period 2 - ABCD (Around Braids, Categories and Distances)
Período documentado: 2023-09-01 hasta 2024-08-31
My project aims at proving the usefulness of categorical methods for questions in geometric group theory. Groups are central objects in mathematics: integers form a group under addition, symmetries of an object form a group,... The idea of looking at a group as a geometric object goes back to Gromov in the late 80's, and has had remarkable consequences. If looked at from far away, groups turn out to have behavior that fall in different classes, which in turn govern some global features of the groups. Amongst groups, braid groups have received a large amount of attention, notably because of their appearance in several research contexts (topology, algebra, quantum physics,...). However, despite all this attention, we know surprisingly little about them (and their generalization), in particular from the geometric point of view.
Categories on the other hand are algebraic structures that were introduced in the middle of the twentieth century. They somehow reflect the idea that often, what one cares about is not so much the objects themselves, but rather the ways to relate them -- what's called morphisms between them. These ideas have helped shape and organize a good part of modern algebra. Lately, these concepts met ideas from quantum physics, where researchers were trying to upgrade quantum field theories in dimension 3 (2 in space, 1 in time) to dimension 4 (3+1). A mathematical reformulation suggested to look for groups acting on categories rather than on classical spaces. In particular, Khovanov and Seidel introduced in 2000 an action of the braid groups on a certain category that is quite simple and has interesting features.
The goal of my project is to use this latter Khovanov-Seidel tool to better understand the geometry of braid groups and their extensions (called Artin-Tits groups). Classical approaches often rely on the identification of braids with the symmetries of a space of curves. It appears that Khovanov-Seidel's model is consistent with this approach whenever this makes sense, but can be defined in a much broader generality, which is very promising.
Two questions have been identified as main goals: studying the faithfulness of the so-called Burau representation, and studying the Haagerup property. The first one has been open for almost a century and is one of the most intriguing questions about braids. Proving it to be false would have surprising consequences in the theory of knots, and it being right would be a key step towards other algebraic conjectures. It should also be mentioned that the Burau representation is some kind of shadow of Khovanov-Seidel's construction, which justifies our approach.
The Haagerup property is something very geometric, describing the way braids can, or cannot, act on some large spaces (Hilbert spaces). It is surprising how little we know about this question: an answer is only known in the case of the 3-strand braid group. Finding a way to approach this question from category theory would be quite striking, provide new tools for geometers, but also suggest relevant geometries in the context of category theory that would be of great interest.
Categories on the other hand are algebraic structures that were introduced in the middle of the twentieth century. They somehow reflect the idea that often, what one cares about is not so much the objects themselves, but rather the ways to relate them -- what's called morphisms between them. These ideas have helped shape and organize a good part of modern algebra. Lately, these concepts met ideas from quantum physics, where researchers were trying to upgrade quantum field theories in dimension 3 (2 in space, 1 in time) to dimension 4 (3+1). A mathematical reformulation suggested to look for groups acting on categories rather than on classical spaces. In particular, Khovanov and Seidel introduced in 2000 an action of the braid groups on a certain category that is quite simple and has interesting features.
The goal of my project is to use this latter Khovanov-Seidel tool to better understand the geometry of braid groups and their extensions (called Artin-Tits groups). Classical approaches often rely on the identification of braids with the symmetries of a space of curves. It appears that Khovanov-Seidel's model is consistent with this approach whenever this makes sense, but can be defined in a much broader generality, which is very promising.
Two questions have been identified as main goals: studying the faithfulness of the so-called Burau representation, and studying the Haagerup property. The first one has been open for almost a century and is one of the most intriguing questions about braids. Proving it to be false would have surprising consequences in the theory of knots, and it being right would be a key step towards other algebraic conjectures. It should also be mentioned that the Burau representation is some kind of shadow of Khovanov-Seidel's construction, which justifies our approach.
The Haagerup property is something very geometric, describing the way braids can, or cannot, act on some large spaces (Hilbert spaces). It is surprising how little we know about this question: an answer is only known in the case of the 3-strand braid group. Finding a way to approach this question from category theory would be quite striking, provide new tools for geometers, but also suggest relevant geometries in the context of category theory that would be of great interest.
In the direction of the faithfulness question for the Burau representation, results have been obtained both at the categorical and classical level.
First, we have organized structures coming from stabilitiy conditions by means of automata. These stability conditions are relatively recent tools that are imported from algebraic geometry, and that should help us understand the geometry of the Artin-Tits groups when acting on the Khovanov-Seidel categories. The strategy and first results have been presented in lecture series at CIRM, Marseille, that come with lecture notes.
Second, in joint work with A. Bapat (ANU, Canberra), we have considered an extension of the Burau representation outside of type A, and we have studied its faithfulness. The main surprise to our eyes is that many cases are still open, but we still could settle some of them, which is a very nice outcome. This relies on an extension of geometric results by Bigelow to cases where there is no geometry anymore. We have used two strategies, both of which have been implemented and used for computer calculations. The first one replaces curves in a disk by so-called spherical objects in Khovanov-Seidel's category. We ran an explicit computer search and we found a counterexample to faithfulness in affine type A3. The second one extends a recent stochastic strategy by Gibson, Williamson and Yacobi, that we adapted to the type D context. There we found counterexamples to faithfulness over finite rings in type D4, but not over Z. These results have been presented in a preprint.
Towards the second main goal of proving that classes of Artin-Tits groups have the Haagerup property, we have explored the combinatorics of the moduli space of stability conditions. Unfortunately, our main result is negative, since we proved that the natural set of walls coming from the tesselation of the space of stability conditions in type A3 cannot be used for the Haglund-Paulin criterion. Indeed, the walls cannot tell apart Brunnian braids. The last part of the project in the Haagerup direction has been devoted to look for ways to overcome this issue. Several options have been identified, that will be investigated as follow-ups to this project.
First, we have organized structures coming from stabilitiy conditions by means of automata. These stability conditions are relatively recent tools that are imported from algebraic geometry, and that should help us understand the geometry of the Artin-Tits groups when acting on the Khovanov-Seidel categories. The strategy and first results have been presented in lecture series at CIRM, Marseille, that come with lecture notes.
Second, in joint work with A. Bapat (ANU, Canberra), we have considered an extension of the Burau representation outside of type A, and we have studied its faithfulness. The main surprise to our eyes is that many cases are still open, but we still could settle some of them, which is a very nice outcome. This relies on an extension of geometric results by Bigelow to cases where there is no geometry anymore. We have used two strategies, both of which have been implemented and used for computer calculations. The first one replaces curves in a disk by so-called spherical objects in Khovanov-Seidel's category. We ran an explicit computer search and we found a counterexample to faithfulness in affine type A3. The second one extends a recent stochastic strategy by Gibson, Williamson and Yacobi, that we adapted to the type D context. There we found counterexamples to faithfulness over finite rings in type D4, but not over Z. These results have been presented in a preprint.
Towards the second main goal of proving that classes of Artin-Tits groups have the Haagerup property, we have explored the combinatorics of the moduli space of stability conditions. Unfortunately, our main result is negative, since we proved that the natural set of walls coming from the tesselation of the space of stability conditions in type A3 cannot be used for the Haglund-Paulin criterion. Indeed, the walls cannot tell apart Brunnian braids. The last part of the project in the Haagerup direction has been devoted to look for ways to overcome this issue. Several options have been identified, that will be investigated as follow-ups to this project.
The ABCD project has had several concrete outcomes.
Lecture notes have been written as a companion to the lecture series I gave at the CIRM international conference "Current trends in representation theory,cluster algebras and geometry". These lecture notes are available as a preprint on arXiv, and have been submitted for publication.
A paper aiming at popularize geometric methods in knot and braid theory has been written, and is about to appear in the Bulletin of the Australian Mathematical Society.
Finally, a paper presenting the results obtained with A. Bapat on the lack of faithfulness of extensions of the Burau representation has appeared as a preprint and has been submitted for publication.
Other partial results, relating to the use of automata or to the Haagerup property, will need further investigation before they can appear in a research paper. I am hopeful that these partial results will be instrumental in my work in the next few years.
Lecture notes have been written as a companion to the lecture series I gave at the CIRM international conference "Current trends in representation theory,cluster algebras and geometry". These lecture notes are available as a preprint on arXiv, and have been submitted for publication.
A paper aiming at popularize geometric methods in knot and braid theory has been written, and is about to appear in the Bulletin of the Australian Mathematical Society.
Finally, a paper presenting the results obtained with A. Bapat on the lack of faithfulness of extensions of the Burau representation has appeared as a preprint and has been submitted for publication.
Other partial results, relating to the use of automata or to the Haagerup property, will need further investigation before they can appear in a research paper. I am hopeful that these partial results will be instrumental in my work in the next few years.