arXiv subjects

Nicki Magill

Publications and source records attributed to Nicki Magill.

7 recordsLinked to original sources

arXiv2026

Quadrilateral mutations and symplectic embeddings

We study the relationship between almost toric base diagrams, perfect exceptional classes, and optimal ellipsoid embeddings for $H_b=\mathbb{CP}^2_1 \# \overline{\mathbb{CP}\!}\,{}^2_b$ and $P_b=S^2_1\times S^2_b$. Starting from a quadrilateral almost toric base diagram with one Delzant corner, we encode the three non-Delzant corners by a recursive triple. We show that every quadrilateral obtained via a well-defined sequence of mutations from the initial diagrams is encoded by a recursive triple in the same way. Moreover, geometric mutation of these diagrams corresponds to algebraic mutation of the associated triples. These algebraic mutations are the recursive operations used to generate the $(p,q)$-perfect classes for $H$. We apply this dictionary to realize every $(p,q)$-perfect class for $H$ by an explicit sequence of almost toric mutations for suitable values of $b$. We also prove the analogous realization result for triples of quasi-perfect classes for $P$, showing that these classes are in fact $(p,q)$-perfect. Finally, we apply these results to ellipsoid embedding problems, including visible embeddings, visible obstructions, and ATF-visible staircases.

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arXiv2025

Curvy points, the perimeter, and the complexity of convex toric domains

We study the related notions of curvature and perimeter for toric boundaries and their implications for symplectic packing problems in dimension 4; a natural setting for this is a generalized version of convex toric domain which we also study, where there are no conditions on the moment polytope at all aside from convexity. We show that the subleading asymptotics of the ECH and elementary ECH capacities recover the perimeter of such domains in their liminf, without any genericity required, and hence the perimeter is an obstruction to a full filling. As an application, we give the first examples of the failure of packing stability by open subsets of compact manifolds with smooth boundary or with no boundary at all; this has implications for long-term super-recurrence. We also show that a single smooth point of positive curvature on the toric boundary obstructs the existence of an infinite staircase, and we build on this to completely classify smooth (generalized) convex toric domains which have an infinite staircase. We also extend a number of theorems to generalized convex toric domains, in particular the "concave to convex", embedding theorem and the "accumulation point theorem". A curvy point forces "infinite complexity"; we raise the question of whether an infinitely complex domain can ever have an infinite staircase and we give examples with infinite staircases and arbitrarily high finite complexity.

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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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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

Unobstructed embeddings in Hirzebruch surfaces

This paper continues the study of the ellipsoid embedding function of symplectic Hirzebruch surfaces parametrized by $b \in (0,1)$, the size of the symplectic blow-up. Cristofaro-Gardiner, et al. (arxiv: 2004.13062) found that if the embedding function for a Hirzebruch surface has an infinite staircase, then the function is equal to the volume curve at the accumulation point of the staircase. Here, we use almost toric fibrations to construct full-fillings at the accumulation points for an infinite family of recursively defined irrational $b$-values implying these $b$ are potential staircase values. The $b$-values are defined via a family of obstructive classes defined in Magill-McDuff-Weiler (arxiv:2203.06453). There is a correspondence between the recursive, interwoven structure of the obstructive classes and the sequence of possible mutations in the almost toric fibrations. This result is used in Magill-McDuff-Weiler (arxiv:2203.06453) to show that these classes are exceptional and that these $b$-values do have infinite staircases.

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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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arXiv2021

Staircase symmetries in Hirzebruch surfaces

This paper continues the investigation of staircases in the family of Hirzebruch surfaces formed by blowing up the projective plane with weight b, that was started in Bertozzi, Holm et al. in arXiv:2010.08567. We explain the symmetries underlying the structure of the set of b that admit staircases, and show how the properties of these symmetries arise from a governing Diophantine equation. We also greatly simplify the techniques needed to show that a family of steps does form a staircase by using arithmetic properties of the accumulation function. There should be analogous results about both staircases and mutations for the other rational toric domains considered, for example, by Cristofaro-Gardiner et al. in arXiv:2004.07829 and by Casals--Vianna in arXiv:2004.13232.

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