As of September 2017, I'm an NSF postdoctoral research fellow in the Columbia University mathematics department. During the 2016-2017 academic year I was also a Pure Mathematics Instructor at MIT. Before that, from 2011-2016 I was a doctoral student in mathematics at Stanford University working with Yasha Eliashberg.

I work in symplectic geometry, a branch of mathematics lying somewhere at the intersection of smooth topology, differential geometry, algebraic geometry, and mathematical physics. I like to study the complicated algebraic gadgets arising in Floer theory and symplectic field theory and apply them to classification problems in symplectic geometry or try to say new things about Hamiltonian dynamical systems. These invariants also have some rather intruiging connections to things like string theory and algebraic geometry, which means there are lots of surprises in store.

**Email:**kyler@math.columbia.edu or kylersiegel@gmail.com.**Office:**room 609, 2990 Broadway, New York, NY 10027.

*Higher symplectic capacities and the stabilized ellipsoid into polydisk problem*, with Dan Cristofaro-Gardiner, available here.*Counting curves with local tangency constraints*, with Dusa McDuff, arXiv:1906.02394.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.Python code.*Higher symplectic capacities*, 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*, arXiv:1609.08545. Submitted.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, 22(4), 2367-2401 (2018), 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 C*, with Stefan Nemirovski, Inventiones mathematicae, Vol. 203, No. 1 (2016), arxiv:1410.4652^{2}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.

*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 H^{3}, 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.

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

#### Fall 2018: Ordinary differential equations.

- Kylerec workshop in symplectic topology. 2016 version.
- Columbia Symplectic Geometry, Gauge Theory, and Categorification Seminar.