Dongha Lee

Talks

2025. 10. 2. KAIST Math Graduate Student Seminar, Daejeon, South Korea.

Title: The Renormalization of Volume and Chern-Simons Invariant for Hyperbolic 3-ManifoldsAbstract: In this talk, we extend the classical theory of minimal surfaces studied in Euclidean and Riemannian geometry to a more general framework in Weitzenböck and Riemann-Cartan geometry, which incorporates torsion. After providing a gentle introduction to minimal surface theory, we present theorems that generalize classical results concerning the holomorphic nature of the Hopf differential, the conformality of the Gauss map, and the minimality of surfaces.

2025. 2. 6. The 20th East Asian Conference on Geometric Topology, Tokyo, Japan.

Title: The Renormalization of Volume and the Chern-Simons Invariant for Hyperbolic 3-ManifoldsAbstract: For hyperbolic 3-manifolds, many interesting results support a deep relationship between volume and the Chern-Simons invariant. In this talk, we consider noncompact hyperbolic 3-manifolds having infinite volume, specifically convex-cocompact hyperbolic 3-manifolds. For these manifolds, there is a well-defined invariant called the renormalized volume which replaces classical volume. We will renormalize the Chern-Simons invariant and discover a close relationship with the renormalized volume.

2024. 5. 21. KIAS Geometry and Topology Group Seminar, Seoul, South Korea.

Title: Chern-Simons Theory for 3-ManifoldsAbstract: We review the Chern-Simons theory for 3-manifolds. The talk will start with a gentle introduction to principal bundles and their connections, and reach the definition and properties of the Chern-Simons 3-form and invariant. We will also discuss holonomy representation and the case of hyperbolic 3-manifolds.Title: The Renormalization of Volume and the Chern-Simons Invariant for Hyperbolic 3-ManifoldsAbstract: For hyperbolic 3-manifolds, many interesting results support a deep relationship between volume and the Chern-Simons invariant. In this talk, we consider noncompact hyperbolic 3-manifolds having infinite volume. For these manifolds, there is a well-defined invariant called the renormalized volume which replaces classical volume. We will renormalize the Chern-Simons invariant and discover a close relationship with the renormalized volume.

2024. 4. 20. 2024 KMS Spring Meeting, Daejeon, South Korea.

Title: The renormalization of volume and Chern-Simons invariant for hyperbolic 3-manifoldsAbstract: For hyperbolic 3-manifolds, there are interesting results supporting a deep relationship between hyperbolic volume and the Chern-Simons invariant. For noncompact hyperbolic 3-manifolds having infinite volume, specifically convex-cocompact hyperbolic 3-manifolds, there is a well-defined invariant called the renormalized volume which replaces classical volume. In this talk, we renormalize the Chern-Simons invariant for convex-cocompact hyperbolic 3-manifolds and discover a close relationship with the renormalized hyperbolic volume. It reveals a connection to Weitzenböck geometry and introduces a complex-valued quantity related to mean curvature.

2024. 3. 28. KAIST Math Graduate Student Seminar, Daejeon, South Korea.

Title: The Renormalization of Volume and Chern-Simons Invariant for Hyperbolic 3-ManifoldsAbstract: For hyperbolic manifolds, many interesting results support a deep relationship between hyperbolic volume and the Chern-Simons invariant. In this talk, we consider noncompact hyperbolic 3-manifolds having infinite volume. For these manifolds, there is a well-defined invariant called the renormalized volume which replaces classical volume. The talk will start from a gentle introduction to hyperbolic geometry and reach the renormalization of the Chern-Simons invariant, which has a close relationship with the renormalized hyperbolic volume.

2022. 11. 2. PK2 Topology Workshop, Busan, South Korea.

Title: The Renormalization of Volume and Chern-Simons Invariant for Hyperbolic ManifoldsAbstract: In this talk, we consider geometric invariants on noncompact hyperbolic 3-manifolds. The talk consists of three parts: the renormalization of volume for hyperbolic 3-manifolds having infinite volume, a brief introduction to Chern-Simons invariant, and its renormalization for hyperbolic 3-manifolds.

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