Email: ksiegel@math.stanford.edu, kylersiegel@gmail.comOffice: 380-S, 450 Serra Mall, Stanford CA, 94305 |

- I am a fourth year graduate student in mathematics at Stanford University, working with Yakov Eliashberg.

*Rationally convex domains and singular Lagrangian surfaces in C^2,*with Stefan Nemirovski.*Inventiones mathematicae,*(2015).**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.

*The Classification of (n-1)-Connected (2n)-Manifolds*, talk given for Stanford Student Topology Seminar, November 7th, 2014.

*A Symplectic Conspectus*, talk given for Stanford Student Symplectic Seminar (S^2 x S^2), November 7th, 2014.

*Introduction to the h-Principle*, talk given for Further Advances in Symplectic Flexibility, workshop at Asilomar, CA, May 16th, 2014.

*The Ubiquity of ADE Classifications in Nature*, talk given for KIDDIE Colloquium, March 21st, 2014.

*Homological Mirror Symmetry for the Projective Line*, talk given for Fukaya Categories and Picard-Lefshetz Theory Reading Seminar, March 14th, 2014.

*Exterior Differential Systems*, talk given for Cartan Seminar, February 20th, 2014.

*Mirror Symmetry: Introduction to the B Model*, talk given for Stanford Student Symplectic Seminar (S^2 x S^2), January 21st, 2014.

*Donaldson Hypersurfaces*, notes from La LLagonne Summer School, June 17-21, 2013.

*Lagrangian Floer Cohomology for Real Projective Space Inside Complex Projective Space*, talk given for Stanford Student Symplectic Seminar (S^2 x S^2), January 28th, 2013.

*Positive Scalar Curvature and Surgery*, talk given for AJOSSCS, October 23rd, 2012.

*The Space of Negative Scalar Curvature Metrics*, talk given for AJOSSCS, October 30th, 2012.

*Bott Periodicity and Clifford Algebras*, talk given for KIDDIE Colloquium, November 26th, 2012.

*A geometric proof of a faithful linear-categorial surface mapping class group action,*arXiv:1108.3676v1.**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. 10, No. 2 (2015).**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.***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.

Last updated: September 2015