Kyler Siegel's homepage

About me.

I'm an assistant professor in the mathematics department at the University of Southern California, and also a visitor at Seoul National University until the end of the calendar year. Previously, I was a Ritt Assisant Professor in the mathematics department at Columbia University, and an NSF postdoctoral research fellow and pure mathematics instructor at Columbia during 2017-2019 and at MIT during 2016-2017. During Fall 2021 I was a member at the Institute for Advanced Study. I gratefully acknowledge the National Science FoundationMy work is currently supported by the NSF Division of Mathematical Sciences grants DMS-2506639, DMS-2533874, and DMS-2105578.
and the Simons FoundationMy work is currently supported by a Simons Foundation Travel Support for Mathematicians award.
for supporting my work.

Research interests.

My background is mostly in symplectic geometry, a branch of mathematics lying somewhere at the intersection of smooth topology, differential geometry, algebraic geometry, and mathematical physics, with some input from string theory. I like to study the complicated algebraic structures arising from pseudoholomorphic curve invariants such as Floer theory and symplectic field theory, which are defined by counting maps from Riemann surfaces into a target space subject to a certain elliptic PDE. At the moment I am particularly interested in interactions between quantitative symplectic geometry (e.g. Hamiltonian flows in Euclidean space) and complex algebraic geometry (e.g. singular algebraic curves).

Contact.

• Email: kyler.siegel@usc.edu or kylersiegel@gmail.com.

Mathematics papers.^{1 Note: there may sometimes be minor differences between the published and arXiv versions, with the former usually slightly more up to date.}

Sesquicuspidal curves, scattering diagrams, and symplectic nonsqueezing, with Dusa McDuff, arXiv:2412.00561. Informal overview.

Abstract. Abstract:

We construct a large new family of rational algebraic curves in the complex projective plane with a $(p,q)$ cusp singularity. More precisely, we classify all such pairs $(p,q)$ for curves which are rigid (in a suitable sense), finding a phase transition from discrete to continuous as the ratio $p/q$ crosses the fourth power of the golden ratio. In particular, for many values of $(p,q)$, our curves solve the minimal degree problem for plane curves with a $(p,q)$ cusp singularity. Our technique relies on (i) explicit bijections between curves in log Calabi-Yau surfaces and curves in nontoric blowups of toric surfaces, (ii) the tropical vertex group and its connections with relative Gromov--Witten invariants, and (iii) recent positivity results for scattering diagrams. As our main application, we completely solve the stabilized symplectic embedding problem for four-dimensional ellipsoids into the four-dimensional round ball. The answer is neatly encoded in a single piecewise smooth function which transitions from an infinite Fibonacci staircase to an explicit rational function. Many of our results also extend to other target spaces, e.g. del Pezzo surfaces and more general rational surfaces.

Singular algebraic curves and infinite symplectic staircases, with Dusa McDuff, arXiv:2404.14702.

Abstract. Abstract:

We show that the infinite staircases which arise in the ellipsoid embedding functions of rigid del Pezzo surfaces 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 $\mathbb{Q}$-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.

On symplectic packing problems in higher dimensions, with Yuan Yao, Mathematische Annalen, to appear (or arXiv:2312.13224).

Abstract. Abstract:

Let $B^{2n}(R)$ denote the closed $2n$-dimensional symplectic ball of area $R$, and let $\Sigma_g(L)$ be a closed symplectic surface of genus $g$ and area $L$. We prove that there is a symplectic embedding $\bigsqcup\limits_{i=1}^k B^4(R_i) \times \Sigma_g (L) \hookrightarrow \operatorname{int}(B^4(R))\times \Sigma_g (L)$ if and only if there exists a symplectic embedding $\bigsqcup\limits_{i=1}^k B^4(R_i) \hookrightarrow \operatorname{int}(B^4(R))$. This lies in contrast with the standard higher dimensional ball packing problem $\bigsqcup\limits_{i=1}^k B^{2n}(R_i) \hookrightarrow \operatorname{int}(B^{2n}(R))$ 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 curves of genus zero.

Ellipsoidal superpotentials and singular curve counts, with Dusa McDuff, arXiv:2308.07542.

Abstract. Abstract:

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.

A tree formula for the ellipsoidal superpotential of the complex projective plane, Proceedings of Gökova Geometry-Topology Conference, to appear (or arXiv:2404.14707).

Abstract. Abstract:

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.

Ellipsoidal superpotentials and stationary descendants, with Grigory Mikhalkin, arXiv:2307.13252.

Abstract. Abstract:

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.

Symplectic capacities, unperturbed curves, and convex toric domains, with Dusa McDuff, Geometry & Topology, to appear (or arXiv:2111.00515).

Abstract. Abstract:

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.

On the embedding complexity of Liouville manifolds, with Sheel Ganatra, Journal of Differential Geometry, to appear (or arxiv 2012.04627).

Abstract. Abstract:

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 $\mathcal{L}_\infty$ 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.

Computing higher symplectic capacities I, International Mathematics Research Notices (2022) (or arXiv:1911.06466). Python code (updated 3/27/20).

Abstract. Abstract:

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 we prove that the resulting structure coefficients count punctured pseudoholomorphic curves in cobordisms between ellipsoids. As sample applications, we produce new previously inaccessible obstructions for the stabilized ellipsoid problem and for stabilized polydisk embeddings, and we give new counts of curves with tangency constraints.

Counting curves with local tangency constraints, with Dusa McDuff, Journal of Topology (2021) (or arXiv:1906.02394). Python code.

Abstract. Abstract:

We construct invariants of any semipositive symplectic manifold which count rational curves satisfying multibranched tangency constraints to a local divisor. We also construct analogous invariants counting punctured curves with negative ends on a small skinny ellipsoid, and we prove that these counts coincide at least in dimension four. We then give a formula describing how tangency constraints arise as point constraints are pushed together, and we use this to recursively compute all invariants in dimension four in terms of Gromov-Witten invariants of blowups.

Higher symplectic capacities and the stabilized ellipsoid problem for integral ellipsoids, with Dan Cristofaro-Gardiner and Richard Hind, Journal of Fixed Point Theory and Applications (2022) (or arxiv:2102.07895).

Abstract. Abstract:

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 is 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.

Higher symplectic capacities, Algebraic & Geometric Topology, to appear (or arXiv:1902.01490).

Abstract. Abstract:

We construct a new family of symplectic capacities indexed by certain symmetric polynomials, defined using rational symplectic field theory. We prove various structural properties of the capacities and discuss the connections with the equivariant L-infinity structure on symplectic cohomology and curve counts with tangency conditions. We also give some preliminary computations in basic examples and show that they give new state of the art symplectic embedding obstructions.

Squared Dehn twists and deformed symplectic invariants, Journal of Symplectic Geometry (2021) (or arXiv:1609.08545).

Abstract. Abstract:

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}.

Subflexible symplectic manifolds, with Emmy Murphy, Geometry & Topology (2018) (or arXiv:1510.01867).

Abstract. Abstract:

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.

Rationally convex domains and singular Lagrangian surfaces in $\mathbb{C}^2$, with Stefan Nemirovski, Inventiones mathematicae (2016) (or arxiv:1410.4652).

Abstract. Abstract:

We give a complete characterization of those disk bundles over surfaces which embed as rationally convex strictly pseudoconvex domains in 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.

Survey articles.

Symplectic field theory: an overview, with Richard Hind, Celebratio Mathematica, to appear (or arXiv:2410.19936).

Abstract. Abstract:

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

Some older papers.

My PhD thesis is available here. Note: this is essentially subsumed by the above three papers, which are also slightly more up-to-date.

A Geometric proof of a faithful linear-categorial surface mapping class group action (Columbia senior thesis), arXiv:1108.367v1.

Abstract. Abstract:

We give completely combinatorial proofs of the main results of [3] using polygons. Namely, we prove that the mapping class group of a surface with boundary acts faithfully on a finitely-generated linear category. Along the way we prove some foundational results regarding the relevant objects from bordered Heegaard Floer homology.

Stick index of knots and links in the cubic lattice, with Colin Adams, Michelle Chu, Thomas Crawford, Stephanie Jensen, and Liyang Zhang, Journal of Knot Theory and its Ramifications, Vol. 21, No. 5 (2012), arXiv:1205.5256.

Abstract. Abstract:

The cubic lattice stick index of a knot type is the least number of sticks necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p, p + 1)-torus knots, and show how composing and taking satellites can be used to obtain the cubic lattice stick index for a relatively large infinite class of knots. We present several bounds relating cubic lattice stick index to other known invariants. Finally, we define and consider lattice torsion index.

Applying Poincare's polyhedron theorem to groups of hyperbolic isometries, preprint (Columbia junior thesis).

Abstract. Abstract:

We present a computer algorithm which confirms that an approximately discete subgroup of PSL(2,C) is in fact discrete. The algorithm proceeds by constructing the Dirichlet domain of the subgroup in H³ , and then checks that the hypotheses of Poincares Theorem for Fundametal Polyhedra are satisfied. In order to check that the hypotheses are exactly satisfied, we rely on group theoretical properties resulting from certain geometric conditions of the Dirichlet domain. We begin in Section 1 with a review of relevant background material. In Section 2 we provide a formal statement of the problem, including the restrictions we impose upon the subgroup of PSL(2,C). In Section 3 we introduce the geometric conditions which must apply for our algorithm to hold, and we prove that these are generically satisfied. We then show in Section 4 that these conditions are in fact sufficient to verify the hypotheses of Poincare's Theorem. In Section 5 we look at the field containing our matrix entries.

Other projects.

Symplectic embedding machine. Symplectic embedding machine:

In these videos, we use computer optimization techniques to find a Hamiltonian flow which squeezes the four-dimensional ellipsoid E(1,5) into the four-ball of smallest possible area. Using pseudoholomorphic curve invariants, one can prove that the smallest possible ball has area 5/2. The numerical embedding found here is close to the theoretical optimal. The left video depicts (in the moment map projection) the final flow as a function of training time. The center video depicts the best Hamiltonian flow found in the moment map projection, while the right video is a 3D projection. See here for some code (in Tensorflow).

For an in depth study of this approach to symplectic embeddings, see the work of my summer 2019 REU students Alex Gajewski, Eli Goldin, Jakwanul Safin, Navtej Singh and Junhui Zhang (coadvised by Simon Rezchikov, with special thanks to Thibault Langlais).

Numerical computations of symplectic capacities. Code and report available here (based on a workshop hosted by Tel Aviv University and ICERM).

A Novel Unsupervised Clustering Algorithm for Binning DNA Fragments in Metagenomics, with Kristen Altenburger, Yusing Hon, Jessey Lin, and Chenglong Yu, Current Bioinformatics, Vol. 10, No. 2 (2015).

Abstract. Abstract:

As metagenomic projects gather large quantities of genomic data from novel microbial species, there is a great need for improved computational tools to sort and understand such data. An important part is the ``binning problem'', namely to determine the number of species present in a sample and to sort DNA fragments by species of origin. We present PuzzleCluster, a new clustering algorithm for unsupervised binning in metagenomics. Besides implementing a new clustering approach, PuzzleCluster introduces several other new features, such as using word agreement information for increased clustering accuracy, and estimating clustering parameters by fitting data with the expectation maximization algorithm. PuzzleCluster uses no prior assumptions about the genetic makeup or number of species present. Our tests show that PuzzleCluster frequently outperforms the best existing unsupervised binning programs.

Teaching.

• Spring 2025: Complex analysis (USC Math 520)

• Spring 2025: Introduction to theory of complex variables (USC Math 475)

• Fall 2024: Almost toric methods in symplectic and algebraic geometry (USC Math 635)

• Some past courses: Differential geometry, Geometry and transformations, Calculus I, Quantitative aspects of Morse and Floer theory, Differential geometry, Fundamental concepts of analysis, Ordinary differential equations, Introduction to Modern Algebra I.

Notes.

• Some old notes (no longer under warrantee)

• Quantitative aspects of Morse and Floer theory (part of the lecture notes for USC Math 635 Spring 2022)

Conferences and seminars.

Southern California Symplectothon Symplectothon:

• Symplectothon 2024 on global Kuranishi charts (notes, syllabus)
• Symplectothon 2022 on Floer theory in symmetric products (syllabus)

Western hemisphere virtual symplectic seminar (dormant)

Kylerec workshop in symplectic topology. 2016 version.

USC Geometry and Topology

artificial intelligence and mathematics artificial intelligence and mathematics:

A discussion on the role of artificial intelligence in mathematics took place in the USC mathematics department on March 14th, 2025. The slides are available here in case they might be useful for facilitating similar discussions elsewhere.