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Kyler Siegel

Publications and source records attributed to Kyler Siegel.

18 recordsLinked to original sources
arXiv2025

Lectures on stabilized ellipsoid embeddings

These notes are based on a five-part minicourse on stabilized symplectic embeddings given in Les Marécottes, Switzerland during a September 2025 workshop. Our main goal is to explain the recent resolution of the (restricted) stabilized ellipsoid embedding problem by D. McDuff and the author. Along the way we also introduce various other ideas which shed light on the context and hint at possible generalizations. Some of the concepts covered include sesquicuspidal curves, symplectic inflation, multidirectional tangency constraints, well-placed curves, cluster transformations, Looijenga pairs, toric models, scattering diagrams, and the tropical vertex theorem.

math.SG↗
arXiv2025

Singular algebraic curves and infinite symplectic staircases

We show that the infinite staircases which arise in the ellipsoid embedding functions of rigid del Pezzo surfaces (with their monotone symplectic forms) can be entirely explained in terms of rational sesquicuspidal symplectic curves. Moreover, we show that these curves can all be realized algebraically, giving various new families of algebraic curves with one cusp singularity. Our main techniques are (i) a generalized Orevkov twist, and (ii) the interplay between algebraic $\Q$-Gorenstein smoothings and symplectic almost toric fibrations. Along the way we develop various methods for constructing singular algebraic (and hence symplectic) curves which may be of independent interest.

math.SG↗
arXiv2025

Sesquicuspidal curves, scattering diagrams, and symplectic nonsqueezing

We solve the stabilized symplectic embedding problem for four-dimensional ellipsoids into the four-dimensional round ball. The answer is neatly encoded by a piecewise smooth function which exhibits a phase transition from an infinite Fibonacci staircase to an explicit rational function related to symplectic folding. Our approach is based on a bridge between quantitative symplectic geometry and singular algebraic curve theory, and a general framework for approaching both topics using scattering diagrams. In particular, we construct a large new family of rational algebraic curves in the complex projective plane with a (p,q) cusp singularity, many of which solve the classical minimal degree problem for plane curves with a prescribed cusp. A key role is played by the tropical vertex group of Gross--Pandharipande--Siebert and ideas from mirror symmetry for log Calabi--Yau surfaces. Many of our results also extend to other target spaces, e.g. del Pezzo surfaces and more general rational surfaces.

math.AG↗
arXiv2024

Symplectic field theory: an overview

We summarize some of the main ideas and results around symplectic field theory, from its early inception up to recent and ongoing developments.

math.SG↗
arXiv2024

On the embedding complexity of Liouville manifolds

We define a family of symplectic invariants which obstruct exact symplectic embeddings between Liouville manifolds, using the general formalism of linearized contact homology and its L-infinity structure. As our primary application, we investigate embeddings between normal crossing divisor complements in complex projective space, giving a complete characterization in many cases. Our main embedding results are deduced explicitly from pseudoholomorphic curves, without appealing to Hamiltonian or virtual perturbations.

math.SG↗
arXiv2024

A tree formula for the ellipsoidal superpotential of the complex projective plane

The ellipsoidal superpotential of the complex projective plane can be interpreted as a count of rigid rational plane curves of a given degree with one prescribed cusp singularity. In this note we present a closed formula for these counts as a sum over trees with certain explicit weights. This is a step towards understanding the combinatorial underpinnings of the ellipsoidal superpotential and its mysterious nonvanishing and nondecreasing properties.

math.SG↗
arXiv2024

Higher symplectic capacities

We construct new families of symplectic capacities indexed by certain symmetric polynomials, defined using rational symplectic field theory. In particular, we introduce a sequence of capacities based on an L-infinity structure on linearized contact homology and rational curve counts with local tangency constraints. We prove various structural properties of these capacities and give some preliminary computations which show that they give state of the art symplectic embedding obstructions in basic examples.

math.SG↗
arXiv2024

Symplectic capacities, unperturbed curves, and convex toric domains

We use explicit pseudoholomorphic curve techniques (without virtual perturbations) to define a sequence of symplectic capacities analogous to those defined recently by the second named author using symplectic field theory. We then compute these capacities for all four-dimensional convex toric domains. This gives various new obstructions to stabilized symplectic embedding problems which are sometimes sharp.

math.SG↗
arXiv2023

On Symplectic Packing Problems in Higher Dimensions

Let $B^{2n}(R)$ denote the closed $2n$-dimensional symplectic ball of area $R$, and let $Σ_g(L)$ be a closed symplectic surface of genus $g$ and area $L$. We prove that there is a symplectic embedding $\bigsqcup_{i=1}^k B^4(R_i) \times Σ_g (L) \overset{s}\hookrightarrow \operatorname{int}(B^4(R))\times Σ_g (L)$ if and only if there exists a symplectic embedding $\bigsqcup_{i=1}^k B^4(R_i) \overset{s}\hookrightarrow \mathrm{int}(B^4(R))$. This lies in contrast with the standard higher dimensional ball packing problem $\bigsqcup\limits_{i=1}^k B^{2 n}(R_i)\overset{s}\hookrightarrow \mathrm{int}(B^{2n}(R))$ for $n >2$, which we conjecture (based on index behavior for pseudoholomorphic curves) is controlled entirely by Gromov's two ball theorem and volume considerations. We also deduce analogous results for stabilized embeddings of concave toric domains into convex domains, and we establish a stabilized version of Gromov's two ball theorem which holds in any dimension. Our main tools are: (i) the symplectic blowup construction along symplectic submanifolds, (ii) an h-principle for symplectic surfaces in high dimensional symplectic manifolds, and (iii) a stabilization result for pseudoholomorphic holomorphic curves of genus zero.

math.SG↗
arXiv2023

Ellipsoidal superpotentials and singular curve counts

Given a closed symplectic manifold, we construct invariants which count (a) closed rational pseudoholomorphic curves with prescribed cusp singularities and (b) punctured rational pseudoholomorphic curves with ellipsoidal negative ends. We prove an explicit equivalence between these two frameworks, which in particular gives a new geometric interpretation of various counts in symplectic field theory. We show that these invariants encode important information about singular symplectic curves and stable symplectic embedding obstructions. We also prove a correspondence theorem between rigid unicuspidal curves and perfect exceptional classes, which we illustrate by classifying rigid unicuspidal (symplectic or algebraic) curves in the first Hirzebruch surface.

math.SG↗
arXiv2023

Ellipsoidal superpotentials and stationary descendants

We compute stationary gravitational descendants in symplectic ellipsoids of any dimension, and use these to derive a number of new recursive formula for punctured curve counts in symplectic manifolds with ellipsoidal ends. Along the way we develop a framework in which punctured curve counts can be explicitly computed using the standard complex structure on affine space. Finally, we initiate the study of "infinitesimal symplectic cobordisms", which serve as elementary building blocks for symplectic cobordisms between ellipsoids.

math.SG↗
arXiv2022

Higher Symplectic Capacities and the Stabilized Embedding Problem for Integral Ellipsoids

The third named author has been developing a theory of "higher" symplectic capacities. These capacities are invariant under taking products, and so are well-suited for studying the stabilized embedding problem. The aim of this note is to apply this theory, assuming its expected properties, to solve the stabilized embedding problem for integral ellipsoids, when the eccentricity of the domain has the opposite parity of the eccentricity of the target and the target is not a ball. For the other parity, the embedding we construct is definitely not always optimal; also, in the ball case, our methods recover previous results of McDuff, and of the second named author and Kerman. There is a similar story, with no condition on the eccentricity of the target, when the target is a polydisc: a special case of this implies a conjecture of the first named author, Frenkel, and Schlenk concerning the rescaled polydisc limit function. Some related aspects of the stabilized embedding problem and some open questions are also discussed.

math.SG↗
arXiv2021

Counting curves with local tangency constraints

We construct invariants for any closed semipositive symplectic manifold which count rational curves satisfying tangency constraints to a local divisor. More generally, we introduce invariants involving multibranched local tangency constraints. We give a formula describing how these invariants arise as point constraints are pushed together in dimension four, and we use this to recursively compute all of these invariants in terms of Gromov--Witten invariants of blowups. As a key tool, we study analogous invariants which count punctured curves with negative ends on a small skinny ellipsoid.

math.SG↗
arXiv2021

Squared Dehn twists and deformed symplectic invariants

We establish an infinitesimal version of fragility for squared Dehn twists around even dimensional Lagrangian spheres. The precise formulation involves twisting the Fukaya category by a closed two-form or bulk deforming it by a half-dimensional cycle. As our main application, we compute the twisted and bulk deformed symplectic cohomology of the subflexible Weinstein manifolds constructed in \cite{murphysiegel}.

math.SG↗
arXiv2021

Computing higher symplectic capacities I

We present recursive formulas which compute the recently defined "higher symplectic capacities" for all convex toric domains. In the special case of four-dimensional ellipsoids, we apply homological perturbation theory to the associated filtered L-infinity algebras and prove that the resulting structure coefficients count punctured pseudoholomorphic curves in cobordisms between ellipsoids. As sample applications, we produce new previously inaccessible obstructions for stabilized embeddings of ellipsoids and polydisks, and we give new counts of curves with tangency constraints.

math.SG↗
arXiv2017

Subflexible symplectic manifolds

We introduce a class of Weinstein domains which are sublevel sets of flexible Weinstein manifolds but are not themselves flexible. These manifolds exhibit rather subtle behavior with respect to both holomorphic curve invariants and symplectic flexibility. We construct a large class of examples and prove that every flexible Weinstein manifold can be Weinstein homotoped to have a nonflexible sublevel set.

math.SG↗
arXiv2015

Rationally Convex Domains and Singular Lagrangian Surfaces in $\mathbb{C}^2$

We give a complete characterization of those disk bundles over surfaces which embed as rationally convex strictly pseudoconvex domains in $\mathbb{C}^2$. We recall some classical obstructions and prove some deeper ones related to symplectic and contact topology. We explain the close connection to Lagrangian surfaces with isolated singularities and develop techniques for constructing such surfaces. Our proof also gives a complete characterization of Lagrangian surfaces with open Whitney umbrellas, answering a question first posed by Givental in 1986.

math.CV↗