Mathematics

The support norm sn(ξ) of a contact structure ξ is the minimum of the negative Euler characteristics of the pages of the open books supporting ξ . In this paper we prove additivity of the support norm for tight contact structures.

Let π:(M,ω)→B be a (non-singular) Lagrangian torus fibration on a compact, complete base B with prequantum line bundle (L, ∇ L )→(M,ω) . For a positive integer N and a compatible almost complex structure J on (M,ω) invariant along the fiber of π , let D be the associated Spin c Dirac operator with coefficients in L ⊗N . Then, we give an orthogonal family { ϑ ~ b } b∈ B BS of sections of L ⊗N indexed by the Bohr-Sommerfeld points B BS , and show that each ϑ ~ b converges to a delta-function section supported on the corresponding Bohr-Sommerfeld fiber π −1 (b) and the L 2 -norm of D ϑ ~ b converges to 0 by the adiabatic(-type) limit. Moreover, if J is integrable, we also give an orthogonal basis { ϑ b } b∈ B BS of the space of holomorphic sections of L ⊗N indexed by B BS , and show that each ϑ b converges to a delta-function section supported on the corresponding Bohr-Sommerfeld fiber π −1 (b) by the adiabatic(-type) limit. We also explain the relation of ϑ b with Jacobi's theta functions when (M,ω) is T 2n .

We construct examples in any odd dimension of contact manifolds with finite and non-zero algebraic torsion (in the sense of Latschev-Wendl), which are therefore tight and do not admit strong symplectic fillings. We prove that Giroux torsion implies algebraic 1 -torsion in any odd dimension, which proves a conjecture by Massot-Niederkrueger-Wendl. These results are part of the author's PhD thesis.

In this article we explore compactifications of cluster varieties of finite type in complex dimension two. Cluster varieties can be viewed as the spec of a ring generated by theta functions and a compactification of such varieties can be given by a grading on that ring, which can be described by positive polytopes [17]. In the examples we exploit, the cluster variety can be interpreted as the complement of certain divisors in del Pezzo surfaces. In the symplectic viewpoint, they can be described via almost toric fibrations over $\R^2$ (after completion). Once identifying them as almost toric manifolds, one can symplectically view them inside other del Pezzo surfaces. So we can identify other symplectic compactifications of the same cluster variety, which we expect should also correspond to different algebraic compactifications. Both viewpoints are presented here and several compactifications have their corresponding polytopes compared. The finiteness of the cluster mutations are explored to provide cycles in the graph describing monotone Lagrangian tori in del Pezzo surfaces connected via almost toric mutation [34].

Motivated rational homotopy theory, we consider a class of Liouville manifolds which we call "algebraically hyperbolic". These Liouville manifolds are characterized by the property that (the pre-triangulated closure of) their wrapped Fukaya category has exponential algebraic growth, in a sense which we define. On the one hand, we prove that the class of algebraically hyperbolic Liouville manifolds is closed under various natural geometric operations, which allows one to construct many examples. On the other hand, under certain non-degeneracy hypotheses, we prove that Reeb flows on the ideal boundary of an algebraically hyperbolic Liouville manifold are always chaotic. This leads to various concrete applications, including new constructions of Weinstein manifolds which are not algebraic varieties and of contact manifolds whose Reeb flows have positive topological entropy. In dimension 3, the latter property implies a strong form of the Weinstein conjecture. Finally, we consider algebraically hyperbolic Weinstein manifolds from the perspective of their skeleton. In particular, we describe concrete criteria at the level of the skeleton which ensure that a Weinstein manifold is algebraically hyperbolic. This provides some connection between stratified-topological properties of skeleta of Weinstein manifolds and Reeb dynamics on their ideal boundary.

This paper is concerned with the "almost existence" phenomenon for periodic orbits of Hamiltonian dynamical systems. In particular, we recover this result in both some standard and some novel cases via feral curves and an adiabatic degeneration.

In this article we study the Arnold conjecture in settings where objects under consideration are no longer smooth but only continuous. The example of a Hamiltonian homeomorphism, on any closed symplectic manifold of dimension greater than 2, having only one fixed point shows that the conjecture does not admit a direct generalization to continuous settings. However, it appears that the following Arnold-type principle continues to hold in continuous settings: Suppose that X is a non-smooth object for which one can define spectral invariants. If the number of spectral invariants associated to X is smaller than the number predicted by the (homological) Arnold conjecture, then the set of fixed/intersection points of X is homologically non-trivial, hence it is infinite. We recently proved that the above principle holds for Hamiltonian homeomorphisms of closed and aspherical symplectic manifolds. In this article, we verify this principle in two new settings: C 0 Lagrangians in cotangent bundles and Hausdorff limits of Legendrians in 1-jet bundles which are isotopic to the zero section. An unexpected consequence of the result on Legendrians is that the classical Arnold conjecture does hold for Hausdorff limits of Legendrians in 1-jet bundles. Dedicated to Claude Viterbo on the occasion of his 60th birthday.

How does one measure the failure of Hochschild homology to commute with colimits? Here I relate this question to a major open problem about dynamics in contact manifolds -- the assertion that Reeb orbits exist and are detected by symplectic homology. More precisely, I show that for polarizably Weinstein fillable contact manifolds, said property is equivalent to the failure of Hochschild homology to commute with certain colimits of representation categories of tree quivers. So as to be intelligible to algebraists, I try to include or black-box as much of the geometric background as possible.

This paper deals with the problem of generalising Gromov's non squeezing theorem to an infinite dimensional Hilbert phase space setting. By following the lines of the proof by Hofer and Zehnder of finite dimensional non-squeezing, we recover an infinite dimensional non-squeezing result by Kuksin for symplectic diffeomorphisms which are non-linear compact perturbations of a symplectic linear map. We also show that the infinite dimensional non-squeezing problem, in full generality, can be reformulated as the problem of finding a suitable Palais-Smale sequence for a distinguished Hamiltonian action functional.

This work is concerned with Bielawski's hyperkähler slices in the cotangent bundles of homogeneous affine varieties. One can associate such a slice to the data of a complex semisimple Lie group G , a reductive subgroup H⊆G , and a Slodowy slice S⊆g:=Lie(G) , defining it to be the hyperkähler quotient of T ∗ (G/H)×(G×S) by a maximal compact subgroup of G . This hyperkähler slice is empty in some of the most elementary cases (e.g. when S is regular and (G,H)=( SL n+1 , GL n ) , n≥3 ), prompting us to seek necessary and sufficient conditions for non-emptiness. We give a spherical-geometric characterization of the non-empty hyperkähler slices that arise when S= S reg is a regular Slodowy slice, proving that non-emptiness is equivalent to the so-called a -regularity of (G,H) . This a -regularity condition is formulated in several equivalent ways, one being a concrete condition on the rank and complexity of G/H . We also provide a classification of the a -regular pairs (G,H) in which H is a reductive spherical subgroup. Our arguments make essential use of Knop's results on moment map images and Losev's algorithm for computing Cartan spaces.