I am a member of the Algebra and Geometry research group at Uppsala University, working on contact and symplectic topology. I received my PhD under the supervision of Ko Honda.
Reach me via my personal email at firstname.lastname@example.org or my university email at email@example.com (corrected in the obvious fashion).
We define a generalization of the Chekanov-Eliashberg algebra, CE, which we call the planar diagram algebra, PDA. It is a non-commutative differential graded algebra with a special filtration. The PDA differential counts holomorphic disks with multiple positive punctures using a combinatorial framework inspired by string topology. Computational software is available here. For a quick overview, see these slides.
How do you count index 1 J-curves on the Lagrangian cylinder over a Legendrian link in R3 when the domain is non-simply connected? This is an obstruction bundle problem which does not always admit a combinatorial solution. We show that such curves can be "perturbed away" using a Legendrian isotopy, implying that any (full) SFT invariant of Legendrian links can be computed using only combinatorial disks.
Combinatorial Reeb dynamics on punctured contact 3-manifolds (accepted at Geometry & Topology)
We develop tools for studying Reeb dynamics on contact 3-manifolds determined by surgery diagrams and for counting holomorphic planes in surgery cobordisms. These tools are used to provide the first examples of closed, tight contact manifolds with vanishing contact homology. For an overview focused on applications, see the slides below.
We describe factorizations of fibered Dehn twists along the boundary of Milnor fibers of Fermat singularities in terms of Dehn twists along Lagrangian spheres. In low dimensions, these factorizations generalize the classical chain relation.
Liouville hypersurfaces and connect sum cobordisms (Journal of Symplectic Geometry, 2021)
A symplectic cobordism construction is described which generalizes Weinstein handle attachment. Applications include proofs of the existence of "fillability" and "non-vanishing contact homology" monoids in the symplectomorphism groups of Liouville domains as well as proofs of the symplectic non-triviality of squares of Dehn twists on cotangent bundles of 2- and 6-dimensional spheres (which are smoothly trivial).
Contact surgery and supporting open books (Algebraic & Geometric Topology, 2013)
Algorithms are described for switching between open book and contact surgery descriptions of a given contact 3-manifold. We use these algorithms to show that every contact 3-manifold can be described by contact surgery along a Legendrian link in 3-space.
Notes, slides, and software
Recent and upcoming talks
Relative RSFT via planar diagrams -- Humboldt-Universität zu Berlin, June 2022.
Relative RSFT via planar diagrams -- Geometry & Topology Seminar, March 2022.
Introduction to Floer homology -- Floer homotopy reading seminar, February 2022.
An algebraic generalization of Giroux's criterion -- Geometry & Topology, University of Minnesota, October 2021.
An algebraic generalization of Giroux's criterion -- Uppsala-Nantes workshop on Lagrangian cobordism, October 2021.
Simplified SFT moduli spaces for Legendrian links -- Geometry & Topology Seminar, Uppsala University, March 2021.
A closed, tight contact 3-manifold with vanishing contact homology -- Topology Seminar, UCLA, January 2021.
A closed, tight contact 3-manifold with vanishing contact homology -- AlgGeomDiffTop Seminar, Rényi Institute, June 2020.
A closed, tight contact 3-manifold with vanishing contact homology -- Virtual Geometry & Topology Seminar, Uppsala University, June 2020.