Russell Avdek

I'm a mathematician who studies algebraic and topological aspects of symplectic manifolds and their friends, working at Institut de Mathématiques de Jussieu at Sorbonne Université as a CNRS chargé de recherche.

I completed my PhD at USC with Ko Honda in 2013, worked for a few years in industry as a software/machine learning engineer, and did postdocs at Uppsala University and Laboratoire de Mathématiques d'Orsay as a MathInGreaterParis fellow before my current poste. Here is my CV and my GitHub.

Reach me via my personal email at first_name.last_name@gmail.com or my university email at first_name@imj-prg.fr, corrected in the obvious fashion.

Current seminars

I'm co-organizing the following seminars in fall 2024:

Research articles

Bourgeois' contact manifolds are tight (with Zhengyi Zhou)

We prove that Bourgeois' contact structures on  MxT2 determined by the supporting open books of a contact manifold M are always tight. The proof is based on a contact homology computation leveraging holomorphic foliations and Kuranishi structures.

Symplectic mapping class relations from pencil pairs (submitted)

We describe symplectic mapping class relations between products of positive Dehn twists along Lagrangian spheres in Weinstein 4-manifolds, all of which are affine varieties. The relations are obtained by applying classification results for Fano 3-folds and polarized K3 surfaces of small genus to a general methodology -- finding pencil pairs. See these slides for an overview.

Stabilization of divisors in high-dimensional contact manifolds (submitted)

A stabilization operation is defined for codimension 2 contact submanifolds in contact manifolds of dimension at least 5. The definition is such that (1) a given contact manifold is overtwisted iff its standard transverse unknot is stabilized and (2) transverse stabilization preserves the formal contact isotopy class and intrinsic contact structure of a link. We prove that many such contact submanifolds are non-simple. Slides and videos are below.

An algebraic generalization of Giroux's criterion (submitted)

We compute the contact homologies of neighborhoods of convex hypersurfaces of any dimension. The result is expressed in terms of homotopy classes of augmentations of the dividing set, or alternatively bilinearized homology theories which we define for free, commutative DGAs over the rationals. See the MCM slides below for an overview with lots of background. Some applications (to appear in an upcoming article) are reviewed here.

A filtered generalization of the Chekanov-Eliashberg algebra (to appear in Quantum Topology)

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.

Simplified SFT moduli spaces for Legendrian links (Journal of Symplectic Geometry, 2023)

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 (Geometry & Topology, 2023)

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 these slides.

Symplectic mapping class group relations generalizing the chain relation (with Bahar Acu, International Journal of Mathematics, 2016)

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 reprove Ding-Geiges' ``contact Lickorish-Wallace theorem'': Every contact 3-manifold can be described by contact surgery along a Legendrian link in 3-space.

Notes and software

Slides supplementing my articles

Computational mathematics

Notes from expository lectures

Recent and upcoming talks