# 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:

Séminaire Géométrie et Topologie (weekly à Sorbonne)

Symplectix (monthly at Institut Henri Poincaré in Paris)

4/10 seminar on Steenrod operations in quantum cohomology (Zoom-able with varying live location)

# Research articles

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.

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.

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.

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.

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 these slides.

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 and software

### Slides supplementing my articles

Exotic codim=2 contact submanifolds in high dimensions from a Nantes seminar. Gives an simplified, alternative proof of the existence of exotic contact unknots from ``Stabilization of divisors...'' using double branched covers.

CH, GW, and the geography of tight convex hypersurfaces from the Symplectic Zoominar

Minicourse on the Algebraic Giroux criterion (AGC) at the MCM Geometry Summer School

Lecture 1: Background and statement of the AGC

Lecture 2: Algebra and computations using relative Gromov-Witten

Lecture 3: Bao-Honda's definition of (Kuranishi) contact homology

Lecture 4: Sketch of the proof of the AGC

### Computational mathematics

mpci software package. Computes Chern numbers of smooth complete intersections in products of complex projective spaces.

legendrian_links software package. Computes holomorphic disks invariants of Legendrians in R3.

### Notes from expository lectures

Bao-Honda's perturbation scheme for contact homology (from the Sorbonne Kuranishi seminar, 2024)

Orientations for J-disks with spin Lagrangian boundary (from the Uppsala Floer Homotopy Seminar, 2023)

Model Reeb dynamics and holomorphic cylinders (from Obstruction bundle gluing Zoominar, 2020)

# Recent and upcoming talks

Informal geometry seminar, University of Vienna, Austria, Oct. 8, 2024.

Topology seminar, Morningside Center of Mathematics, Chinese Academy of Sciences, Beijing, Sept. 25, 2024.

Symplectic Geometry Seminar, Universität Heidelberg, June 26, 2024.

Geometry and topology seminar, University of Southern California, Apr. 22, 2024.

High dimensional contact topology, American Institute of Mathematics, Apr. 18, 2024.

Kuranishi exchange seminar (expository), Sorbonne Université, Apr. 3, 2024.

Geometry and topology seminar (Zoom), University of Iowa, Mar. 5, 2024.

Séminaire de Topologie, Géométrie et Algèbre, Université de Nantes, Feb. 29, 2024.

Lyon-Grenoble Symplectic Geometry workshop (3 lecture mini-course), Grenoble, Nov. 30-31, 2023.

Symplectix séminaire de topologie symplectique, Paris, Nov. 10 2023 (slides).

Symplectic Zoominar, CRM-Montréal, Princeton/IAS, Tel Aviv, and Paris, Oct. 20, 2023 (slides).

Geometry Summer School (4 lecture mini-course), Morningside Center for Mathematics, Sep. 2023.

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

Topology seminar, University of Glasgow, Oct. 2022.

Humboldt-Universität zu Berlin, June 2022.

Geometry & Topology Seminar, Uppsala University, Mar. 2022.

Floer homotopy reading seminar, Feb. 2022 (expository).

Geometry & Topology, University of Minnesota, Oct. 2021.

Uppsala-Nantes workshop on Lagrangian cobordism, Oct. 2021.

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

Geometry & Topology Seminar, Uppsala University, Mar. 2021.

Topology Seminar (Zoom), UCLA, Jan. 2021.

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

AlgGeomDiffTop Seminar, Rényi Institute, June 2020.

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