# Russell Avdek

I am a mathematician working in the Geometry and Physics research group at Uppsala University and will join the Laboratoire de Mathématiques d'Orsay later this year. I usually study holomorphic curves and topological aspects of contact and symplectic manifolds. Here is my CV.

Reach me via my personal email at first_name.last_name@gmail.com or my university email at first_name.last_name@math.uu.se (corrected in the obvious fashion).

# Research articles

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.

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.

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).

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

Geometry Seminar, Université Libre de Brussels, Feb. 2023.

Topology seminar, University of Glasgow, October 2022.

Humboldt-Universität zu Berlin, June 2022.

Geometry & Topology Seminar, March 2022.

Floer homotopy reading seminar, February 2022 (expository).

Geometry & Topology, University of Minnesota, October 2021.

Uppsala-Nantes workshop on Lagrangian cobordism, October 2021.

Advances In Symplectic Topology, Institut Henri Poincaré, May 2021 (video).

Geometry & Topology Seminar, Uppsala University, March 2021.

Topology Seminar, UCLA, January 2021.

Obstruction Bundle Gluing Zoominar, October 2020 (expository, video).

AlgGeomDiffTop Seminar, Rényi Institute, June 2020.

Virtual Geometry & Topology Seminar, Uppsala University, June 2020.