Symplectic cohomology and a conjecture of Viterbo

We identify a new class of closed smooth manifolds for which there exists a uniform bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero section in a unit cotangent disk bundle, settling a well-known conjecture of Viterbo from 2007 as the special case of $T^n.$ This class of manifolds is defined in topological terms involving the Chas-Sullivan algebra and the BV-operator on the homology of the free loop space, contains spheres and is closed under products. We discuss generalizations and various applications.Comment: 20 pages; improvements in the exposition, new titl

On the Geometry of the Moduli Space of Real Binary Octics

The moduli space of smooth real binary octics has five connected components. They parametrize the real binary octics whose defining equations have 0, 1, ..., 4 complex-conjugate pairs of roots respectively. We show that the GIT-stable completion of each of these five components admits the structure of an arithmetic real hyperbolic orbifold. The corresponding monodromy groups are, up to commensurability, discrete hyperbolic reflection groups, and their Vinberg diagrams are computed. We conclude with a simple proof that the moduli space of GIT-stable real binary octics itself cannot be a real hyperbolic orbifold.Comment: 23 page

Validated Computation of Heteroclinic Sets

In this work we develop a method for computing mathematically rigorous enclosures of some one dimensional manifolds of heteroclinic orbits for nonlinear maps. Our method exploits a rigorous curve following argument build on high order Taylor approximation of the local stable/unstable manifolds. The curve following argument is a uniform interval Newton method applied on short line segments. The definition of the heteroclinic sets involve compositions of the map and we use a Lohner-type representation to overcome the accumulation of roundoff errors. Our argument requires precise control over the local unstable and stable manifolds so that we must first obtain validated a-posteriori error bounds on the truncation errors associated with the manifold approximations. We illustrate the utility of our method by proving some computer assisted theorems about heteroclinic invariant sets for a volume preserving map of $\mathbb{R}^3$.Comment: 31 pages, 20 figure

Simplified SFT moduli spaces for Legendrian links

We study moduli spaces $\mathcal{M}$ of holomorphic maps $U$ from Riemann surfaces to $\mathbb{R}^{4}$ with boundaries on the Lagrangian cylinder over a Legendrian link $\Lambda \subset (\mathbb{R}^{3}, \xi_{std})$. We allow our domains, $\Sigma$, to have non-trivial topology in which case $\mathcal{M}$ is the zero locus of an obstruction function $\mathcal{O}$, sending a moduli space of holomorphic maps in $\mathbb{C}$ to $H^{1}(\Sigma)$. In general, $\mathcal{O}^{-1}(0)$ is not combinatorially computable. However after a Legendrian isotopy, $\Lambda$ can be made left-right-simple, implying that any $U$ of index $1$ is a disk with one or two positive punctures for which $\pi_{\mathbb{C}}\circ U$ is an embedding. Moreover, any $U$ of index $2$ is either a disk or an annulus with $\pi_{\mathbb{C}} \circ U$ simply covered and without interior critical points. Therefore any SFT invariant of $\Lambda$ is combinatorially computable using only disks with $\leq 2$ positive punctures.Comment: 42 pages. V3: Minor change

Hodge integrals and degenerate contributions

Degenerate contributions to higher genus Gromov-Witten invariants of Calabi-Yau 3-folds are computed via Hodge integrals. The vanishing of contributions of covers of elliptic curves conjectured by Gopakumar and Vafa is proven. A formula for degree 1 covers for all genus pairs is computed in agreement with M-theoretic calculations of Gopakumar and Vafa. Finally, these results lead to a proof of a formula in the tautological ring of the moduli space of curves previously conjectured by Faber.Comment: 21 pages, LaTeX2e. Expanded discussion of relationships with the M-theoretic calculations of Gopakumar-Vafa. New results for arbitrary 3-folds following a suggestion of Jinzenji and Xion

On ε\varepsilon Approximations of Persistence Diagrams

Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently observed in nature. In this paper a theoretical framework for the algorithmic computation of an arbitrarily good approximation of the persistent homology is developed. We study the filtrations generated by sub-level sets of a function $f : X \to \mathbb{R}$, where $X$ is a CW-complex. In the special case $X = [0,1]^N$, $N \in \mathbb{N}$ we discuss implementation of the proposed algorithms. We also investigate a priori and a posteriori bounds of the approximation error introduced by our method.Comment: 26 pages; changed title; added revision

Four-periodic infinite staircases for four-dimensional polydisks

The ellipsoid embedding function of a symplectic four-manifold measures the amount by which its symplectic form must be scaled in order for it to admit an embedding of an ellipsoid of varying eccentricity. This function generalizes the Gromov width and ball packing numbers. In the one continuous family of symplectic four-manifolds that has been analyzed, one-point blowups of the complex projective plane, there is a full measure set of symplectic forms whose ellipsoid embedding functions are completely described by finitely many obstrutions, while there is simultaneously a Cantor set of symplectic forms for which an infinite number of obstructions are needed. The latter case is called an infinite staircase. In this paper we identify a new infinite staircase when the target is a four-dimensional polydisk, extending a countable family identified by Usher in 2019. We use almost toric fibrations to compute key upper bounds and outline an analogy with the case of the one-point blowup of the complex projective plane. Our work indicates a method of proof for a conjecture of Usher. We further describe the continued fractions of ellipsoid eccentricities which provide key lower bounds on the ellipsoid embedding function, and we explain Python code for efficiently exploring the space of symplectic embeddings.Comment: 55 pages, 15 figure

Contact Homology of Orbit Complements and Implied Existence

For Reeb vector fields on closed 3-manifolds, cylindrical contact homology is used to show that the existence of a set of closed Reeb orbit with certain knotting/linking properties implies the existence of other Reeb orbits with other knotting/linking properties relative to the original set. We work out a few examples on the 3-sphere to illustrate the theory, and describe an application to closed geodesics on $S^2$ (a version of a result due to Angenent).Comment: Section 5 was removed, as well as any assertions in the main theorems which depend on it. Changes to introduction, several typos correcte