Research

RESEARCH INTERESTS AND PROJECTS

(Click to expand)

I am interested in various aspects of symplectic and contact topology and geometry. The topics are listed so that those topics listed first are most often on my mind on a given day (as of March 2023).

1. Locally conformal symplectic (LCS) geometry: One may consider a sort of symplectic geometry for which there is an ambiguity of local scale.

2. Legendrian contact homology: I am especially interested in Legendrian contact homology in higher dimensions.

3. Contact handlebody theory / convex hypersurface theory: I am interested in understanding how rigid and flexible phenomena seen in 3-dimensional contact manifold can be extended to higher dimensions using these tools.

4. Quantitative non-squeezing:

PUBLICATIONS AND PREPRINTS

Preprints/submitted works are in orange Published/accepted works are in green

(6) Products of locally conformal symplectic manifolds

Co-author: Baptiste Chantraine

Preprint: arXiv (2024)

Abstract: Given two locally conformal symplectic (LCS) structures on manifolds M_1 and M_2, we construct a natural R^+ torsor of locally conformal symplectic structures on a certain covering space M_1 ⊞ M_2 of M_1 x M_2. As the smooth construction of M_1 ⊞ M_2 is natural from the perspective of flat line bundles, we use this language to phrase the LCS theory. This construction shares many properties with, and in a sense generalizes, the standard symplectic product. Notably, for a Hamiltonian isotopy f_t of an LCS manifold M, there is an associated family of Lagrangian embeddings F_t of M into M⊞M, in which certain fixed points of f_t are in bijection with intersection points of F_0 (the diagonal) and F_1. Using a Lagrangian intersection result of the first author and E. Murphy, we may conclude that if f_t is a C^0-small Hamiltonian isotopy, then the number of fixed points of f_1 is bounded below by the rank of the Novikov theory associated to the Lee class of the LCS structure on M. Finally, we end the paper by constructing the suspension of a Lagrangian submanifold along a Hamiltonian isotopy in the LCS theory, again generalizing the symplectic setting. 

(5) Obstructions for exact submanifolds with symplectic applications

Preprint: arXiv (2023)

Abstract: Suppose X^N is a closed oriented manifold, α in H*(X;R) is a cohomology class, and Z in H_{N-k}(X) is an integral homology class. We ask the following question: is there an oriented embedded submanifold Y^{N-k} in X with homology class Z such that the restriction of α to Y is zero in H*(Y;R)? In this article, we provide a family of computable obstructions to the existence of such `exact' submanifolds in a given homology class which arise from studying formal deformations of the de Rham complex. In the final section, we apply these obstructions to prove that the following symplectic manifolds admit no non-separating exact (a fortiori contact-type) hypersurfaces: Kähler manifolds, symplectically uniruled manifolds, and the Kodaira--Thurston manifold.




(4) Differential graded algebras for trivalent plane graphs and their representations

Accepted: Quantum Topology

Slides from lecture at Columbia Geometry and Topology Seminar (April 2022)

Abstract: To any trivalent plane graph embedded in the sphere, Casals and Murphy associate a differential graded algebra (dg-algebra), in which the underlying graded algebra is free associative over a commutative ring. Our first result is a generalization of the Casals–Murphy dg-algebra to non-commutative coefficients. In generalizing, we prove various functoriality properties which did not appear in the commutative setting, notably including changing the chosen face at infinity of the graph. Our second result is to prove that rank r representations of this dg-algebra, over a field F, correspond to colorings of the faces of the graph by elements of the Grassmannian Gr(r,2r;F) so that bordering faces are transverse, up to the natural action of PGL(2r,F). Underlying the combinatorics, the dg-algebra is a computation of the fully non-commutative Legendrian contact dg-algebra for Legendrian satellites of Legendrian 2-weaves, though we do not prove as such in this paper. The graph coloring problem verifies that for Legendrian 2-weaves, rank r representations of the Legendrian contact dg-algebra correspond to constructible sheaves of microlocal rank r. This is the first such verification of this conjecture for an infinite family of Legendrian surfaces. 

(3) On certain quantifications of Gromov's non-squeezing theorem

Coauthors: Antoine Song, Umut Varolgunes, Jonathan Zhu

Appendix: Joé  Brendel

Accepted: Geometry and Topology, see accepted version on arXiv

Lecture at virtual `Low-dimensional Topology and Symplectic Geometry Weekend' workshop (May 2021)

Marked-up slides from the talk

Abstract: Let R>1 and let B be the Euclidean 4-ball of radius R with a closed subset E removed. Suppose that B embeds symplectically into the unit cylinder D^2 x R^2. By Gromov's non-squeezing theorem, E must be non-empty. We prove that the Minkowski dimension of E is at least 2, and that this result is optimal for R<=sqrt(2). In an appendix by Joé Brendel, it is shown that the lower bound is optimal for R < sqrt(3). We also discuss the minimum volume of E in the case that the symplectic embedding extends, with bounded Lipschitz constant, to the entire ball. 

 







(2) Getting a handle on contact manifolds

Preprint: arXiv (2019)

Thesis version: More background and details, as well as a bit more speculation.

Public lecture for the layperson

Abstract: We develop the details of a surgery theory for contact manifolds of arbitrary dimension via convex structures, extending the 3-dimensional theory developed by Giroux. The theory is analogous to that of Weinstein manifolds in symplectic geometry, with the key difference that the vector field does not necessarily have positive divergence everywhere. The surgery theory for contact manifolds contains the surgery theory for Weinstein manifolds via a sutured model for attaching critical points of low index. Using this sutured model, we show that the existence of convex structures on closed contact manifolds is guaranteed, a result equivalent to the existence of supporting Weinstein open book decompositions.

(1) Appendix to "Differential algebra of cubic planar graphs" (by Roger Casals and Emmy Murphy)

Published: Advances in Mathematics (2018)

Summary: In the body of the paper, Casals and Murphy construct a differential graded algebra (dg-algebra) over any field for any trivalent planar graph with auxiliary choices, and prove that different auxiliary choices yield equivalent (dg-isomorphic) dg-algebras. Hence, any trivalent planar graph has a well-defined set of augmentations. We prove that this is in bijection with the set of face colorings of the graph by elements of the projective line (over the specified field). Supposing the Casals-Murphy dg-algebra is the Legendrian Contact Homology of a certain Legendrian surface specified by the trivalent planar graph, this verifies that augmentations are sheaves for such Legendrian surfaces.