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Julian Chaidez

Publications and source records attributed to Julian Chaidez.

At least 19 recordsLinked to original sources
arXiv2026

Convex hypersurfaces and robust heterodimensional dynamics

We prove that any closed orientable hypersurface in a contact manifold of dimension five or greater is isotopic to a robustly non-convex hypersurface via an arbitrarily $C^0$-small isotopy. This strengthens a recent result of the first author and yields a strong counterpart to the groundbreaking density theorem of Honda-Huang and Giroux. This is proven by combining a new convexity obstruction via heteroclinics and recent advances in robust heterodimensional dynamics due to Li-Turaev to produce a robust deconvexifying plug, which is a local and robust convexity obstruction.

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arXiv2025

Pseudo-Anosov Reeb flows and contact structures

We introduce the notion of a pseudo-Anosov contact structure, which admits a type of singular contact form with pseudo-Anosov Reeb flow. We prove that contact homology detects the free homotopy classes of closed orbits of any pseudo-Anosov Reeb flow and that any pseudo-Anosov contact structure is universally tight and torsion free. Many applications are given, including new cases of the Finiteness Conjecture for transitive pseudo-Anosov flows. Our proofs use a flavor of contact homology graded by a free homotopy class of loops, defined for any contact manifold. We establish several properties of this type of contact homology that may be of independent interest.

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arXiv2025

On dynamical invariants of coadjoint orbits of 3D volume-preserving diffeomorphisms

The helicity, or asymptotic linking number, is a functional of exact volume-preserving vector fields on 3-manifolds, invariant under volume-preserving transformations. It is known to exhibit remarkable uniqueness properties: many invariant functionals reduce to functions of helicity. We examine how severely this uniqueness can fail. On integral homology spheres with the $C^{1}$-topology, the failure is extreme: for every $C^{1}$-open set of nonvanishing exact fields of fixed helicity, some other global dynamical invariant is continuous and non-constant in that set; the Ruelle invariant if some field is non-Anosov, and topological entropy otherwise. In particular, on the three-sphere, the Ruelle invariant is everywhere independent of helicity. This implies, in a very strong sense, a negative answer to a question of Arnold and Khesin on the somewhere density of coadjoint orbits of 3D volume-preserving diffeomorphisms, when considered for the $C^1$-topology. On arbitrary three-manifolds, we also answer the question in the negative using local rather than global invariants.

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arXiv2025

On Zoll Contact 5-Spheres

We prove that any contact form on the standard contact 5-sphere with Zoll Reeb flow is strictly contactomorphic to a scaling of the standard Zoll contact form.

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arXiv2024

Contact homology and linearization without dga homotopies

This article clarifies the status of linearized contact homology given the foundations of the contact dg-algebra established by Pardon. In particular, we prove that the set of isomorphism classes of linearized contact homologies of a closed contact manifold is a contact invariant.

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arXiv2024

Robustly non-convex hypersurfaces in contact manifolds

We construct the first examples of hypersurfaces in any contact manifold of dimension 5 and larger that cannot be $C^2$-approximated by convex hypersurfaces, contrasting sharply with the foundational results of Giroux in dimension $3$ and Honda-Huang in the $C^0$ case. The main technical step is the first construction of a dynamical blender in the contact setting.

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arXiv2023

Elementary SFT Spectral Gaps And The Strong Closing Property

We formulate elementary SFT spectral invariants of a large class of symplectic cobordisms and stable Hamiltonian manifolds, in any dimension. We give criteria for the strong closing property using these invariants, and verify these criteria for Hofer near periodic systems. This extends the class of symplectic dynamical systems in any dimension that satisfy the strong closing property.

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arXiv2023

Newton-Okoukov bodies and symplectic embeddings into non-toric rational surfaces

We develop new methods of both constructing and obstructing symplectic embeddings into non-toric rational surfaces using the theory of Newton-Okoukov bodies. Applications include sharp embedding results for concave toric domains into non-toric rational surfaces, and new cases of non-existence for infinite staircases in the non-toric setting.

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arXiv2023

Combed Trisection Diagrams and Non-Semisimple 4-Manifold Invariants

Given a triple $H$ of (possibly non-semisimple) Hopf algebras equipped with pairings satisfying a set of properties, we describe a construction of an associated smooth, scalar invariant $\tau_H(X,\pi)$ of a simply connected, compact, oriented $4$-manifold $X$ and an open book $\pi$ on its boundary. This invariant generalizes an earlier semisimple version and is calculated using a trisection diagram $T$ for $X$ and a certain type of combing of the trisection surface. We explain a general calculation of this invariant for a family of exotic 4-manifolds with boundary called Stein nuclei, introduced by Yasui. After investigating many low-dimensional Hopf algebras up to dimension 11, we have not been able to find non-semisimple Hopf triples that satisfy the criteria for our invariant. Nonetheless, appropriate Hopf triples may exist outside the scope of our explorations.

math.QA↗
arXiv2023

Legendrian embedded contact homology

We give a construction of embedded contact homology (ECH) for a contact $3$-manifold $Y$ with convex sutured boundary and a pair of Legendrians $Λ_+$ and $Λ_-$ contained in $\partial Y$ satisfying an exactness condition. The chain complex is generated by certain configurations of closed Reeb orbits of $Y$ and Reeb chords of $Λ_+$ to $Λ_-$. The main ingredients include: a general Legendrian adjunction formula for curves in $\mathbb{R} \times Y$ with boundary on $\mathbb{R} \times Λ$; a relative writhe bound for curves in contact $3$-manifolds asymptotic to Reeb chords; and a Legendrian ECH index with an accompanying ECH index inequality. The (action filtered) Legendrian ECH of any pair $(Y,Λ)$ of a closed contact $3$-manifold $Y$ and a Legendrian link $Λ$ can also be defined using this machinery after passing to a sutured link complement. This work builds on ideas present in Colin-Ghiggini-Honda's proof of the equivalence of Heegaard-Floer homology and ECH. The independence of our construction of choices of almost complex structure and contact form should require a new flavor of monopole Floer homology. It is beyond the scope of this paper.

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arXiv2022

Contact homology and higher dimensional closing lemmas

We develop methods for studying the smooth closing lemma for Reeb flows in any dimension using contact homology. As an application, we prove a conjecture of Irie, stating that the strong closing lemma holds for Reeb flows on ellipsoids. Our methods also apply to other Reeb flows, and we illustrate this for a class of examples introduced by Albers-Geiges-Zehmisch.

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arXiv2022

The Ruelle Invariant And Convexity In Higher Dimensions

We construct the Ruelle invariant of a volume preserving flow and a symplectic cocycle in any dimension and prove several properties. In the special case of the linearized Reeb flow on the boundary of a convex domain $X$ in $\mathbb{R}^{2n}$, we prove that the Ruelle invariant $\text{Ru}(X)$, the period of the systole $c(X)$ and the volume $\text{vol}{X}$ satisfy \[\text{Ru}(X) \cdot c(X) \le C(n) \cdot \text{vol}{X}\] Here $C(n) > 0$ is an explicit constant dependent on $n$. As an application, we construct dynamically convex contact forms on $S^{2n-1}$ that are not convex, disproving the equivalence of convexity and dynamical convexity in every dimension.

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arXiv2022

3d Convex Contact Forms And The Ruelle Invariant

Let $X \subset \mathbb{R}^4$ be a convex domain with smooth boundary $Y$. We use a relation between the extrinsic curvature of $Y$ and the Ruelle invariant $\text{Ru}(Y)$ of the natural Reeb flow on $Y$ to prove that there exist constants $C > c > 0$ independent of $Y$ such that \[c < \frac{\text{Ru}(Y)^2}{\text{vol}(X)} \cdot \text{sys}(Y) < C\] Here $\text{sys}(Y)$ is the systolic ratio, i.e. the square of the minimal period of a closed Reeb orbit of $Y$ divided by twice the volume of $X$. We then construct dynamically convex contact forms on $S^3$ that violate this bound using methods of Abbondandolo-Bramham-Hryniewicz-Salomão. These are the first examples of dynamically convex contact $3$-spheres that are not strictly contactomorphic to a convex boundary $Y$.

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arXiv2021

Computing Reeb dynamics on 4d convex polytopes

We study the combinatorial Reeb flow on the boundary of a four-dimensional convex polytope. We establish a correspondence between "combinatorial Reeb orbits" for a polytope, and ordinary Reeb orbits for a smoothing of the polytope, respecting action and Conley-Zehnder index. One can then use a computer to find all combinatorial Reeb orbits up to a given action and Conley-Zehnder index. We present some results of experiments testing Viterbo's conjecture and related conjectures. In particular, we have found some new examples of polytopes with systolic ratio $1$.

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arXiv2021

4-Manifold Invariants From Hopf Algebras

The Kuperberg invariant is a topological invariant of closed 3-manifolds based on finite-dimensional Hopf algebras. In this paper, we initiate the program of constructing 4-manifold invariants in the spirit of Kuperberg's 3-manifold invariant. We utilize a structure called a Hopf triplet, which consists of three Hopf algebras and a bilinear form on each pair subject to certain compatibility conditions. In our construction, we present 4-manifolds by their trisection diagrams, a four-dimensional analog of Heegaard diagrams. The main result is that every Hopf triplet yields a diffeomorphism invariant of closed 4-manifolds. In special cases, our invariant reduces to Crane-Yetter invariants and generalized dichromatic invariants, and conjecturally Kashaev's invariant. As a starting point, we assume that the Hopf algebras involved in the Hopf triplets are semisimple. We speculate that relaxing semisimplicity will lead to even richer invariants.

math.QA↗
arXiv2021

Lattice Formulas For Rational SFT Capacities

We initiate the study of the rational SFT capacities of Siegel using tools in toric algebraic geometry. In particular, we derive new (often sharp) bounds for the RSFT capacities of a strongly convex toric domain in dimension $4$. These bounds admit descriptions in terms of both lattice optimization and (toric) algebraic geometry. Applications include (a) an extremely simple lattice formula for for many RSFT capacities of a large class of convex toric domains, (b) new computations of the Gromov width of a class of product symplectic manifolds and (c) an asymptotics law for the RSFT capacities of all strongly convex toric domains.

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arXiv2021

ECH embedding obstructions for rational surfaces

Let $(Y,A)$ be a smooth rational surface or a possibly singular toric surface with ample divisor $A$. We show that a family of ECH-based, algebro-geometric invariants $c^{\text{alg}}_k(Y,A)$ proposed by Wormleighton obstruct symplectic embeddings into $Y$. Precisely, if $(X,ω_X)$ is a $4$-dimensional star-shaped domain and $ω_Y$ is a symplectic form Poincaré dual to $[A]$ then \[(X,ω_X)\text{ embeds into }(Y,ω_Y)\text{ symplectically } \implies c^{\text{ECH}}_k(X,ω_X) \le c^{\text{alg}}_k(Y,A)\] We give three applications to toric embedding problems: (1) these obstructions are sharp for embeddings of concave toric domains into toric surfaces; (2) the Gromov width and several generalizations are monotonic with respect to inclusion of moment polygons of smooth (and many singular) toric surfaces; and (3) the Gromov width of such a toric surface is bounded by the lattice width of its moment polygon, addressing a conjecture of Averkov--Hofscheier--Nill.

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