arXiv subjects

Jordan Payette

Publications and source records attributed to Jordan Payette.

6 recordsLinked to original sources

arXiv2024

J-tamed inflation via tame to compatible deformations

We give a complete and self-contained exposition of the $J$-tame inflation lemma: Given any tame almost complex structure $J$ on a symplectic $4$-manifold $(M,\omega)$, and given any compact, embedded, $J$-holomorphic submanifold $Z$, it is always possible to construct a deformation of symplectic forms $\omega_t$ in classes $[\omega_t]=[\omega]+t\mathrm{PD}{Z}$, for $0\leq t$ less than an upper bound $0<T$ that only depends on the self-intersection $Z\cdot Z$. The original proofs of this fact make the unwarranted assumption that one can find a family of normal planes along $Z$ that is both $J$ invariant and $\omega$-orthogonal to $TZ$ -- which amounts, in effect, to assuming the compatibility of $J$ and $\omega$ along $Z$. We explain how the original constructions can be adapted to avoid this assumption when $Z$ has nonpositive self-intersection, and we discuss the difficulties with this line of argument in general to establish the full inflation when $Z$ has positive self-intersection. We overcome this problem by proving a `preparation lemma', which states that prior to inflation, one can isotope $\omega$ within its cohomology class to a new form that still tames $J$ and which is compatible with $J$ along the submanifold $Z$. This preparation lemma can be regarded as an infinitesimal version of the "tamed-to-compatible" conjecture of S. K. Donaldson along an almost-complex submanifold $Z$.

math.SG↗

arXiv2022

Graphes dans les surfaces et ergodicit\'e topologique

The simplest way to make a dynamical system out of a finite connected graph $G$ is to give it a polarization, that is to say a cyclic ordering of the edges incident to a vertex, for each vertex. The phase space $\mathcal{P}(G)$ then consists of all pairs $(v,e)$ where $v$ is a vertex and $e$ is an edge incident to $v$. Such an initial condition gives a position and a momentum. The data $(v,e)$ is of course equivalent to an edge endowed with an orientation $e_{\mathcal O}$. With the polarization, each initial data leads to a leftward walk defined by turning left at each vertex, or making a rebound if there is no other edge. A leftward walk is called complete if it goes through all edges of $G$, not necessarily in both directions. As usual, we define the valence of a vertex as the number of edges incident to it, and we define the valence of a graph as the average of the valences of its vertices. In this article, we prove that if a graph which is embedded in a closed oriented surface of genus $g$ admits a complete leftward walk, then its valence is at most $1 + \sqrt{6g+1}$. We prove furthermore that this result is sharp for infinitely many genera $g$, and that it is asymptotically optimal as $g \to + \infty$. This leads to obstructions for the embeddability of graphs on a surface in a way which admits a complete leftward walk. Since checking that a polarized graph admits a complete leftward walk or not is done in time $4N$, where $N$ is the cardinality of the edges, this obstruction is particularly efficient in terms of computability. This problem has its origins in interesting consequences for what we will call here the topological ergodicity of conservative systems, especially Hamiltonian systems $H$ in two dimensions where the existence of a complete leftward walk corresponds to a topologically ergodic orbit of the system, i.e. an orbit of $H$ visiting all the topology of the surface.

math.CO↗

arXiv2022

Coarse nodal count and topological persistence

Courant's theorem implies that the number of nodal domains of a Laplace eigenfunction is controlled by the corresponding eigenvalue. Over the years, there have been various attempts to find an appropriate generalization of this statement in different directions. We propose a new take on this problem using ideas from topological data analysis. We show that if one counts the nodal domains in a coarse way, basically ignoring small oscillations, Courant's theorem extends to linear combinations of eigenfunctions, to their products, to other operators, and to higher topological invariants of nodal sets. We also obtain a coarse version of the B\'ezout estimate for common zeros of linear combinations of eigenfunctions. We show that our results are essentially sharp and that the coarse count is necessary, since these extensions fail in general for the standard count. Our approach combines multiscale polynomial approximation in Sobolev spaces with new results in the theory of persistence modules and barcodes.

math.SP↗

arXiv2019

Optimal unions of scaled copies of domains and Pólya's conjecture

Given a bounded Euclidean domain $Ω$, we consider the sequence of optimisers of the $k^{\rm th}$ Laplacian eigenvalue within the family consisting of all possible disjoint unions of scaled copies of $Ω$ with fixed total volume. We show that this sequence encodes information yielding conditions for $Ω$ to satisfy Pólya's conjecture with either Dirichlet or Neumann boundary conditions. This is an extension of a result by Colbois and El Soufi which applies only to the case where the family of domains consists of all bounded domains. Furthermore, we fully classify the different possible behaviours for such sequences, depending on whether Pólya's conjecture holds for a given specific domain or not. This approach allows us to recover a stronger version of Pólya's original results for tiling domains satisfying some dynamical billiard conditions, and a strenghtening of Urakawa's bound in terms of packing density.

math.SP↗

arXiv2019

Continuous Covers on Symplectic Manifolds

In this article, we first introduce the notion of a {\it continuous cover} of a manifold parametrised by any compact manifold endowed with a mass 1 volume-form. We prove that any such cover admits a partition of unity where the usual sum is replaced by integrals. We then generalize Polterovich's notion of Poisson non-commutativity to such a context in order to get a richer definition of non-commutativity and to be in a position where one can compare various invariants of symplectic manifolds, for instance the relation between critical values of phase transitions of symplectic balls and eventual critical values of the Poisson non-commutativity. Our first main theorem states that our generalisation of Poisson non-commutativity depends only on real one-parameter spaces since intuitively the Hilbert curve in any high dimensional parameter space fills out the entire manifold and preserves the measure. Our second main theorem states that the Poisson non-commutativity is a (not necessarily strictly) decreasing function of the size of the symplectic balls used to cover continuously any given symplectic manifold. This function has other nice properties as well that do not prevent it from undergoing singularities similar to phase transitions.

math.SG↗

arXiv2018

The Poisson bracket invariant on surfaces

We study the Poisson bracket invariant, which measures the level of Poisson noncommutativity of a smooth partition of unity, on closed symplectic surfaces. Motivated by a general conjecture of Polterovich and building on preliminary work of Buhovsky--Tanny, we prove that for any smooth partition of unity subordinate to an open cover by discs of area at most $c$, and under some localization condition on the cover when the surface is a sphere, then the product of the Poisson bracket invariant with $c$ is bounded from below by a universal constant. Similar results were obtained recently by Buhovsky--Logunov--Tanny for open covers consisting of displaceable sets on all closed surfaces, and their approach was extended by Shi--Lu to open covers by nondisplaceable discs. We investigate the sharpness of all these results.

math.SG↗