Geodesible contact structures
Geometry and Topology 12 (2008) 1729-1776
In this paper, we study and almost completely classify contact structures on closed 3– manifolds which are totally geodesic for some Riemannian metric. Due to previously known results, this amounts to classifying contact structures on Seifert manifolds which are transverse to the fibers. Actually, we obtain the complete classification of contact structures with negative (maximal) twisting number (which includes the transverse ones) on Seifert manifolds whose base is not a sphere, as well as partial results in the spherical case.
Infinitely many universally tight torsion free contact structures with vanishing Ozsváth-Szabó invariants
Mathematische Annalen 353 (2012), n° 4, 1351-1376
Non-vanishing of the Ozsváth-Szabó contact invariant is a powerful way to prove tightness of contact structures but this invariant is known to vanish in the presence of Giroux torsion. In this note, we construct, on infinitely many manifolds, infinitely many isotopy classes of universally tight torsion free contact structures whose Ozsváth-Szabó invariant neverthe- less vanishes. Along the way, we prove a conjecture of K Honda, W Kazez and G Matić about their contact topological quantum field theory.
Tightness in contact metric 3-manifolds
Inventiones Mathematicae 188 (2012), n° 3, 621-657
This paper begins the study of relations between Riemannian geometry and global properties of contact structures on 3-manifolds. In particular we prove an analog of the sphere theorem from Riemannian geometry in the setting of contact geometry. Specifically, if a given three dimensional contact manifold (M,ξ) admits a complete compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure ξ is tight; in particular, the contact structure pulled back to the universal cover is the standard contact structure on S³. We also describe geometric conditions in dimension three for ξ to be universally tight in the nonpositive curvature setting.
Weak and strong fillability of higher dimensional contact manifold
Inventiones Mathematicae 192 (2013) n°3, 287-373
For contact manifolds in dimension three, the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent. We extend these facts to higher dimensions: in particular, we define a natural generalization of weak fillings and prove that it is indeed weaker (at least in dimension five), while also being obstructed by all known manifestations of "overtwistedness". We also find the first examples of contact manifolds in all dimensions that are not symplectically fillable but also cannot be called overtwisted in any reasonable sense. These depend on a higher dimensional analogue of Giroux torsion, which we define via the existence in all dimensions of exact symplectic manifolds with disconnected contact boundary.
Quantitative Darboux theorems in contact geometry
Transactions of the AMS 368 (2016) n°11, 7845-7881
This paper begins the study of relations between Riemannian geometry and contact topology on (2n+1)-manifolds and continues this study on 3-manifolds. Specifically we provide a lower bound for the radius of a geodesic ball in a contact (2n+1)-manifold (M,ξ) that can be embedded in the standard contact structure on ℝ2n+1, that is on the size of a Darboux ball. The bound is established with respect to a Riemannian metric compatible with an associated contact form α for ξ. In dimension three, it further leads us to an estimate of the size for a standard neighborhood of a closed Reeb orbit. The main tools are classical comparison theorems in Riemannian geometry. In the same context, we also use holomorphic curves techniques to provide a lower bound for the radius of a PS-tight ball.
Examples of non-trivial contact mapping classes in all dimensions
avec Klaus Niederkrüger
International Mathematics Research Notices (IMRN) 2016 (15), 4784-4806
We give examples of contactomorphisms in every dimension that are smoothly isotopic to the identity but that are not contact isotopic to the identity. In fact, we prove the stronger statement that they are not even symplectically pseudo-isotopic to the identity. We also give examples of pairs of contactomorphisms which are smoothly conjugate to each other but not by contactomorphisms.
On the contact mapping class group of Legendrian circle bundles
avec Emmanuel Giroux
Compositio Mathematica 2017 153(2), 294-312
In this paper, we determine the group of contact transformations modulo contact isotopies for Legendrian circle bundles over closed surfaces of nonpositive Euler characteristic. These results extend and correct those presented by the first author in a former work. The main ingredient we use is connectedness of certain spaces of embeddings of surfaces into contact 3-manifolds. In the third section, this connectedness question is studied in more details with a number of (hopefully instructive) examples.
Contactomorphism groups and Legendrian flexibility
avec Sylvain Courte
We explain a connection between the algebraic and geometric properties of groups of contact transformations, open book decompositions, and flexible Legendrian embeddings. The main result is that, if a closed contact manifold (V,ξ) has a supporting open book whose pages are flexible Weinstein manifolds, then both the connected component of identity in its automorphism group and its universal cover are uniformly simple groups: for every non-trivial element g, every other element is a product of at most 128(dimV+1) conjugates of g±1. In particular any conjugation invariant norm on these groups is bounded.
Formalising perfectoid spaces
Certified Programs and Proofs 2020, 299-312
Perfectoid spaces are sophisticated objects in arithmetic geometry introduced by Peter Scholze in 2012. We formalised enough definitions and theorems in topology, algebra and geometry to define perfectoid spaces in the Lean theorem prover. This experiment confirms that a proof assistant can handle complexity in that direction, which is rather different from formalising a long proof about simple objects. It also confirms that mathematicians with no computer science training can become proficient users of a proof assistant in a relatively short period of time. Finally, we observe that formalising a piece of mathematics that is a trending topic boosts the visibility of proof assistants amongst pure mathematicians.