Russell Avdek

I am a mathematician working in the areas of contact and symplectic topology. I received my PhD under the supervision of Ko Honda in 2013. In 2021 I joined the Algebra and Geometry research group at Uppsala University.

I can be reached 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

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.

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.

Slides

Recent seminar talks

  • An algebraic Giroux criterion (video) -- Advances In Symplectic Topology, Institut Henri Poincaré, May 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.

  • Model Reeb dynamics and holomorphic cylinders (video)-- Obstruction Bundle Gluing Zoominar, October 2020.

  • 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.