Sangjin Lee

Research interest:

My research interests lie in symplectic topology, triangulated category theory, and dynamics. The question motivating my research is "Can we generalize ideas/results/phenomena in the lowest-dimensional symplectic manifolds, i.e., surfaces, to higher-dimensional symplectic manifolds?"

Below, I listed my previous papers/priprints. Each item has a simple description of the article. If you click the arrow icon on the right, you can see what I am particularly interested in recently.

In topology, the answer to such generalization questions is often "No," as lower-dimensional phenomena do not necessarily extend to higher dimensions. However, when a property is compatible with an additional structure—such as a symplectic structure—that is not inherently dimension-sensitive, a higher-dimensional analogue may exist.

Currently, I am particularly interested in the space of stability conditions on a Fukaya category, which is expected to serve as a higher dimensional generalization of the Teichmüller spaces of surfaces. Good references provide further details on this connection, including this. In particular, I believe that we can describe the space of stability conditions on the compact Fukaya category of a plumbing space, building on the results of my tenth listed paper below. After getting the description, I would like to study how similar it is to the Teichmüller spaces of surfaces.

Publications & preprints:

10. Wrapped Fukaya category of plumbings. Joint work with Dogancan Karabas, to appear in Journal of Symplectic Geometry.

Abstract: In this paper, we formulate the wrapped Fukaya category of plumbing spaces. Plumbing spaces are examples of Weinstein manifolds, which appear naturally in various scenarios; Milnor fibers with simple singularities are plumbing spaces, plumbing spaces have simple Lagrangian skeletons, etc. Thus, it is natural to ask about the Fukaya category of plumbing spaces.

In the paper, we formulated the Fukaya category of plumbing spaces through a "local-to-global" approach. The key ingredient is computing the wrapped Fukaya category of "plumbing sectors" that serve as a local model for describing Lagrangian skeletons with self-transversally-intersecting points.

As a corollary of the main theorem, we showed that the Ginzburg dg category of any graded quiver (without potential) is equivalent to the Fukaya category of a specific plumbing space. In other words, we found a geometric model for the Ginzburg dg categories.

9. A computational approach to the Homotopy Theory of DG categories. Joint work with Dogancan Karabas, Preprint, 2024.

Abstract: This paper constitutes the second installment of a series that began with Paper 3. Here, Karabas and I provide explicit descriptions of a cylinder functor and a homotopy colimit functor. The primary objective in constructing these functors is to furnish a computational tool within the realm of dg categories. Consequently, by commencing with "simple" dg categories—particularly semifree dg categories—we can systematically compute their cylinder objects and homotopy colimits in a functorial manner. An application to symplectic topology is also described in the paper, together with a toy example.

8. Pseudo-Anosov autoequivalences arising from Symplectic topology and their hyperbolic actions on stability conditions. Joint work with Hanwool Bae, Preprint, 2023.

Abstract: The notion of pseudo-Anosov autoequivalence is defined by Fan-Filip-Haiden-Katzarkv-Liu, but there are few known examples of it. In the article, we introduce a construction of pseudo-Anosov autoequivalences on N-Calabi-Yau categories associated with trees, motivated by symplectic topology. More precisely, in my thesis, I constructed symplectic automorphisms called "Penner type," which have simple asymptotic behavior. We expected that the Penner type symplectic automorphisms induce pseudo-Anosov autoequivalences because of their simple asymptotic behavior, and the expectation turned out to be true in this article. Moreover, we studied the actions which Penner type autoequivalences induce on the space of stability conditions. The induced actions are hyperbolic under a condition.

7. Barcode entropy for Reeb flows on contact manifolds with Liouville fillings. Joint work with Elijah Fender, Beomjun Sohn, Online ready in Communications in Contemporary Mathematics, doi: https://doi.org/10.1142/S0219199725500440

Abstract: We present a study on the dynamical properties of Reeb flows, focusing on the definition of a new invariant within the following context: a contact manifold equipped with a non-degenerate contact one-form and a Liouville filling. By employing the theory of Persistence modules to the symplectic homology of the filling and the corresponding action functional, we establish an entropy-type invariant that we call barcode entropy. We demonstrate that the barcode entropy remains unchanged for a given contact manifold and contact one-form, regardless of the choice of filling. Furthermore, we establish that the barcode entropy provides a lower bound for the topological entropy of the Reeb flow under consideration.

6. A comparison of categorical and topological entropies on Weinstein manifolds. Joint work with Hanwool Bae, Nagoya Mathematical Journal. Published online 2025:1-30. doi:10.1017/nmj.2025.3

Abstract: We extend our previous work (see item 5) to all compactly supported exact symplectic automorphism on a Weinstein manifold. More precisely, we showed that the categorical entropy of such symplectic automorphisms bounds the topological entropy of it from below.

5. On categorical entropy from the viewpoint of Symplectic topology. Joint work with Hanwool Bae, Dongwook Choa, Wonbo Jeong, and Dogancan Karabas, to appear in Mathematical Research Letters.

Abstract: After Dimitrov-Haiden-Katzarkov-Kontsevich defined the categorical entropy, a question arises: "what is the connection between categorical and topological entropies?". We answer the question in a symplectic setting. More precisely, we fix a type of symplectic automorphism. Then, for a symplectic automorphism of that type, we show that the categorical entropy of it (on the Fukaya category) is smaller than the topological entropy. Also, we answer a question in category theory, which concerns a relation between the localization functor and the categorical entropy.

4. Exotic families of symplectic manifolds with Milnor fibers of ADE types. Joint work with Dongwook Choa and Dogancan Karabas, Math. Z. 307 (2024), no. 4, Paper No. 74, MR 4772662.

Abstract: My previous paper, 'Lefschetz fibrations on cotangent bundles and some plumbings,' constructed diffeomorphic families of Weinstein manifolds. In this paper, we prove that the diffeomorphic familes are exotic, i.e., their members are not symplectomorphic to each other.

3. Homotopy Colimits of Semifree DG Categories and Fukaya Categories of Cotangent Bundles of Lens Spaces. Joint work with Dogancan Karabas, Preprint, 2021.

Abstract: We constructed a cylinder object for a semifree dg category. By using this, we can formulate a homotopy colimit formula which helps us to apply the result of Ganatra-Pardon-Shende. As an example of the fomula, we computed the wrapped Fuakay category of cotangent bundles of lens spaces.

2. Lefschetz fibrations on cotangent bundles and some plumbings. To appear in Selecta Mathematica.

Abstract: In this paper, I gave an algorithm for constructing Lefschetz fibrations on cotangent bundles of manifolds and some plumbing spaces. The main idea of the algorithm is to use Weinstein handle decompositions. As an application of the results, we can obtain infinitely many diffeomorphic families of Weinstein manifolds.

1. Towards a higher-dimensional construction of stable/unstable Lagrangian laminations. Algebraic & Geometric Topology 24-2 (2024), 655--716. DOI 10.2140/agt.2024.24.655

Abstract: I constructed an invariant Lagrangian branched submanifold and an invariant Lagrangian lamination as higher-dimensional generalizations of a train track and a geodesic lamination in the theory of surface.

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