arXiv subjects

Morgan Weiler

Publications and source records attributed to Morgan Weiler.

12 recordsLinked to original sources

arXiv2025

Persistent Homology for Labeled Datasets: Gromov-Hausdorff Stability and Generalized Landscapes

Techniques from metric geometry have become fundamental tools in modern mathematical data science, providing principled methods for comparing datasets modeled as finite metric spaces. Two of the central tools in this area are the Gromov-Hausdorff distance and persistent homology, both of which yield isometry-invariant notions of distance between datasets. However, these frameworks do not account for categorical labels, which are intrinsic to many real-world datasets, such as labeled images, pre-clustered data, and semantically segmented shapes. In this paper, we introduce a general framework for labeled metric spaces and develop new notions of Gromov-Hausdorff distance and persistent homology which are adapted to this setting. Our main result shows that our persistent homology construction is stable with respect to our novel notion of Gromov-Hausdorff distance, extending a classic result in topological data analysis. To facilitate computation, we also introduce a labeled version of persistence landscapes and show that the landscape map is Lipschitz.

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arXiv2024

Anchored symplectic embeddings

Given two four-dimensional symplectic manifolds, together with knots in their boundaries, we define an ``anchored symplectic embedding'' to be a symplectic embedding, together with a two-dimensional symplectic cobordism between the knots (in the four-dimensional cobordism determined by the embedding). We use techniques from embedded contact homology to determine quantitative critera for when anchored symplectic embeddings exist, for many examples of toric domains. In particular we find examples where ordinarily symplectic embeddings exist, but they cannot be upgraded to anchored symplectic embeddings unless one enlarges the target domain.

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arXiv2023

Torus knotted Reeb dynamics and the Calabi invariant

We establish the existence of a secondary Reeb orbit set with quantitative action and linking bounds for any contact form on the standard tight three-sphere admitting the standard transverse positive $T(p,q)$ torus knot as an elliptic Reeb orbit with a canonically determined rotation number. This can be interpreted through an ergodic lens for Reeb flows transverse to a surface of section. Our results also allow us to deduce an upper bound on the mean action of periodic orbits of naturally associated classes of area preserving diffeomorphisms of the associated Seifert surfaces of genus $(p-1)(q-1)/2$ in terms of the Calabi invariant, without the need for genericity or Hamiltonian hypotheses. Our proofs utilize knot filtered embedded contact homology, which was first introduced and computed by Hutchings for the standard transverse unknot in the irrational ellipsoids and further developed in our previous work. We continue our development of nontoric methods for embedded contact homology and establish the knot filtration on the embedded contact homology chain complex of the standard tight three-sphere with respect to positive $T(p,q)$ torus knots, where there are nonvanishing differentials. We also obtain obstructions to the existence of exact symplectic cobordisms between positive transverse torus knots.

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arXiv2023

A classification of infinite staircases for Hirzebruch surfaces

The ellipsoid embedding function of a symplectic manifold gives the smallest amount by which the symplectic form must be scaled in order for a standard ellipsoid of the given eccentricity to embed symplectically into the manifold. It was first computed for the standard four-ball (or equivalently, the complex projective plane) by McDuff and Schlenk, and found to contain the unexpected structure of an "infinite staircase," that is, an infinite sequence of nonsmooth points arranged in a piecewise linear stair-step pattern. Later work of Usher and Cristofaro-Gardiner--Holm--Mandini--Pires suggested that while four-dimensional symplectic toric manifolds with infinite staircases are plentiful, they are highly non-generic. This paper concludes the systematic study of one-point blowups of the complex projective plane, building on previous work of Bertozzi-Holm-Maw-McDuff-Mwakyoma-Pires-Weiler, Magill-McDuff, Magill-McDuff-Weiler, and Magill on these Hirzebruch surfaces. We prove a conjecture of Cristofaro-Gardiner--Holm--Mandini--Pires for this family: that if the blowup is of rational weight and the embedding function has an infinite staircase then that weight must be $1/3$. We show also that the function for this manifold does not have a descending staircase. Furthermore, we give a sufficient and necessary condition for the existence of an infinite staircase in this family which boils down to solving a quadratic equation and computing the function at one specific value. Many of our intermediate results also apply to the case of the polydisk (or equivalently, the symplectic product of two spheres).

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arXiv2023

Torus knot filtered embedded contact homology of the tight contact 3-sphere

Knot filtered embedded contact homology was first introduced by Hutchings in 2015; it has been computed for the standard transverse unknot in irrational ellipsoids by Hutchings and for the Hopf link in lens spaces L(n,n-1) via a quotient by Weiler. While toric constructions can be used to understand the ECH chain complexes of many contact forms adapted to open books with binding the unknot and Hopf link, they do not readily adapt to general torus knots and links. In this paper, we generalize the definition and invariance of knot filtered embedded contact homology to allow for degenerate knots with rational rotation numbers. We then develop new methods for understanding the embedded contact homology chain complex of positive torus knotted fibrations of the standard tight contact 3-sphere in terms of their presentation as open books and as Seifert fibered spaces. We provide Morse-Bott methods, using a doubly filtered complex and the energy filtered perturbed Seiberg-Witten theory developed by Hutchings and Taubes, and use them to compute the T(2,q) knot filtered embedded contact homology, for q odd and positive. In the sequel we complete the computation for positive T(p,q) knots (where there is a nonvanishing differential) and use our results to deduce quantitative existence results for torus knotted Reeb dynamics on the tight 3-sphere and the mean action of area preserving diffeomorphisms of once punctured surfaces of arbitrary genus arising as Seifert surfaces of positive torus knots.

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arXiv2022

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 an open dense set of symplectic forms whose ellipsoid embedding functions are completely described by finitely many obstructions, while there is simultaneously a Cantor set of symplectic forms for which an infinite number of obstructions are needed. In the latter case, we say that the embedding function has 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. Our work computes the function on infinitely many intervals and thereby indicates a method of proof for a conjecture of Usher.

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arXiv2022

Staircase Patterns in Hirzebruch Surfaces

The ellipsoidal capacity function of a symplectic four manifold $X$ measures how much the form on $X$ must be dilated in order for it to admit an embedded ellipsoid of eccentricity $z$. In most cases there are just finitely many obstructions to such an embedding besides the volume. If there are infinitely many obstructions, $X$ is said to have a staircase. This paper gives an almost complete description of the staircases in the ellipsoidal capacity functions of the symplectic Hirzebruch surfaces $H_b$ formed by blowing up the projective plane with weight $b$. We describe an interweaving, recursively defined, family of obstructions to symplectic embeddings of ellipsoids that show there is an open dense set of shape parameters $b$ that are blocked, i.e. have no staircase, and an uncountable number of other values of $b$ that do admit staircases. The remaining $b$-values form a countable sequence of special rational numbers that are closely related to the symmetries discussed in Magill--McDuff (arXiv:2106.09143). We show that none of them admit ascending staircases. Conjecturally, none admit descending staircases. Finally, we show that, as long as $b$ is not one of these special rational values, any staircase in $H_b$ has irrational accumulation point. A crucial ingredient of our proofs is the new, more indirect approach to using almost toric fibrations in the analysis of staircases by Magill (arXiv:2204.12460). In particular, the structure of the relevant mutations of the set of almost toric fibrations on $H_b$ is echoed in the structure of the set of blocked $b$-intervals.

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arXiv2020

Infinite staircases for Hirzebruch surfaces

We consider the embedding capacity functions $c_{H_b}(z)$ for symplectic embeddings of ellipsoids of eccentricity $z$ into the family of nontrivial rational Hirzebruch surfaces $H_b$ with symplectic form parametrized by $b\in [0,1)$. This function was known to have an infinite staircase in the monotone cases ($b= 0$ and $ b= 1/3$). It is also known that for each $b$ there is at most one value of $z$ that can be the accumulation point of such a staircase. In this manuscript, we identify three sequences of open, disjoint, blocked $b$-intervals, consisting of $b$-parameters where the embedding capacity function for $H_b$ does not contain an infinite staircase. There is one sequence in each of the intervals $(0,1/5)$, $(1/5,1/3)$, and $(1/3,1)$. We then establish six sequences of associated infinite staircases, one occurring at each endpoint of the blocked $b$-intervals. The staircase numerics are variants of those in the Fibonacci staircase for the projective plane (the case $b=0$). We also show that there is no staircase at the point $b=1/5$, even though this value is not blocked. The focus of this paper is to develop techniques, both graphical and numeric, that allow identification of potential staircases, and then to understand the obstructions well enough to prove that the purported staircases really do have the required properties. A subsequent paper will explore in more depth the set of $b$ that admit infinite staircases.

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arXiv2020

Symplectic embeddings of four-dimensional polydisks into half integer ellipsoids

We obtain new sharp obstructions to symplectic embeddings of four-dimensional polydisks $P(a,1)$ into four-dimensional ellipsoids $E(bc,c)$ when $1\le a< 2$ and $b$ is a half-integer. When $1 \leq a < 2-O(b^{-1})$ we demonstrate that $P(a,1)$ symplectically embeds into $E(bc,c)$ if and only if $a+b\le bc$. Our results show that inclusion is optimal and extend the result by Hutchings \cite{H} when $b$ is an integer. Our proof is based on a combinatorial criterion developed by Hutchings \cite{H} to obstruct symplectic embeddings.

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arXiv2020

Embedded contact homology of prequantization bundles

The 2011 PhD thesis of Farris demonstrated that the ECH of a prequantization bundle over a Riemann surface is isomorphic as a Z/2Z-graded group to the exterior algebra of the homology of its base. We extend this result by computing the Z-grading on the chain complex, permitting a finer understanding of this isomorphism and a stability result for ECH. We fill in a number of technical details, including the Morse-Bott direct limit argument and the classification of certain J-holomorphic buildings. The former requires the isomorphism between filtered Seiberg-Witten Floer cohomology and filtered ECH as established by Hutchings-Taubes. The latter requires the work on higher asymptotics of pseudoholomorphic curves by Cristofaro-Gardiner--Hutchings--Zhang to obtain the writhe bounds necessary to appeal to an intersection theory argument of Hutchings-Nelson.

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arXiv2018

Mean action of periodic orbits of area-preserving annulus diffeomorphisms

An area-preserving diffeomorphism of an annulus has an "action function" which measures how the diffeomorphism distorts curves. The average value of the action function over the annulus is known as the Calabi invariant of the diffeomorphism, while the average value of the action function over a periodic orbit of the diffeomorphism is the mean action of the orbit. If an area-preserving annulus diffeomorphism is a rotation near the boundary of the annulus, and if its Calabi invariant is less than the maximum boundary value of the action function, then we show that the infimum of the mean action over all periodic orbits of the diffeomorphism is less than or equal to its Calabi invariant.

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arXiv2012

Pattern Avoidance in Poset Permutations

We extend the concept of pattern avoidance in permutations on a totally ordered set to pattern avoidance in permutations on partially ordered sets. The number of permutations on $P$ that avoid the pattern $\pi$ is denoted $Av_P(\pi)$. We extend a proof of Simion and Schmidt to show that $Av_P(132) \leq Av_P(123)$ for any poset $P$, and we exactly classify the posets for which equality holds.

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