Abstract: 

The aim of this mini-course is to provide an introduction to the study of Levi-flat real hypersurfaces through examples, in particular the unit tangent bundles of closed hyperbolic surfaces. While reviewing fundamental concepts and tools in foliation theory and several complex variables, we plan to cover the following topics: the Milnor-Wood inequality and Matsumoto rigidity theorem, Brunella's convexity, and Hopf's ergodicity theorem.

Abstract:  We introduce in this talk a class of ``residue functions'', each of which ``deforms'' holomorphically certain weighted L^2 norm on the ambient complex manifold X to an L^2 norm on the union of certain log-canonical (lc) centres of a given lc pair (X,D). The properties of such residue functions can be encoded into a sequence of analytic adjoint ideal sheaves, which fit into various residue short exact sequences. These sequences are useful in facilitating induction on (co)dimension of lc centres in geometric problems involving lc singularities. Moreover, they contain seemingly more concrete subadjunction information. An argument that uses the residue exact sequence to show that the lc threshold 𝜆 of an effective ℚ-divisor D_2 relative to an lc pair (X,D_1) (assuming that |D_2| does not contain the mlc Z of (X,D_1) ) depends only on D_2|_{Z} will be presented to illustrate this point.

Abstract:  Gromov introduced the notion of Kähler hyperbolic manifolds which in particular generalizes Kähler manifolds of Riemannian sectional curvature bounded from above by a negative constant. Gromov's basic estimate on such manifolds yields a vanishing theorem for harmonic forms and also a lower bound for the bottom of the spectrum of the Laplacian of the given Kähler metric. The bound is determined by a uniform constant and the so-called `Kähler hyperbolicity length' of the metric. In this talk, I shall explain a method to obtain the lower bounds for the Laplacian of the complete Kähler-Einstein metrics on bounded symmetric domains, using the aforementioned estimate. The method in particular provides the optimal lower bound on the complex hyperbolic space (and polydiscs) which is sharper than McKean's estimate (1970). This is joint work with Young-Jun Choi and Kang-Hyurk Lee.

Abstract: In this talk, I will talk about the CR Paneitz operator and some of its properties. I will then talk about some related results in CR geometry, including the CR positive mass theorem and the convergence of the CR Yamabe flow. Finally, I will mention some of the very recent results about the spectrum of the CR Paneitz operator in the non-embeddable case, which is a joint work with Yuya Takeuchi.

Abstract: In this talk, we discuss cohomology isomorphism of symmetric power of cotangent bundle of ball quotient and its toroidal compactification. The Hodge theory for compact complex manifolds gives the canonical isomorphism between Dolbeault cohomologies and the space of harmonic forms. This theory has been extended to L^2-Dolbeault cohomologies for a non-compact Kähler manifold that possesses a global Kähler potential with boundary gradient for the metric.

To extend these results for L^2-Dolbeault cohomologies associated with the symmetric power of cotangent bundle of complex hyperbolic space forms with finite volume, we establish a cohomological isomorphism theorem between an L^2-Dolbeault cohomology and a sheaf cohomology of a certain holomorphic vector bundle for its toroidal compactification.

This is based on a joint work with Aeryeong Seo of Kyungpook National University.

Abstract: This talk concerns the local biholomorphic invariants, namely CR geometry, of real hypersurfaces in complex spaces. Within the hierarchy of k-nondegenerate CR structures (k = 1, 2, …), this local geometry is fully understood only in the 1-nondegenerate (Levi-nondegenerate) case. For k > 1, many fundamental questions remain open, and this talk focuses on recent progress for 2-nondegenerate CR hypersurfaces.

By analyzing general forms of defining equations, we associate to each point of a 2-nondegenerate CR hypersurface a canonical model 2-nondegenerate CR structure. These model structures can themselves be viewed as CR invariants. This talk presents a detailed study of their properties, including descriptions of their moduli spaces, symmetry algebras, homogeneous models, explicit defining equations, and the geometric meaning of several invariants encoded in the fourth jet of their defining equations. Based on joint work with Jan Gregorovič and Martin Kolář, published in Mathematische Annalen 392 (2025), 1615–1663. DOI: 10.1007/s00208-025-03138-1.

Abstract: The classical division problem asks whether a holomorphic function f belongs to the ideal generated by given holomorphic functions g_1,⋯,g_r on a complex manifold X. In this talk, we study a generalized version of this problem: given plurisubharmonic functions 𝛗_1,⋯,𝛗_r, under what conditions does f belong to the sum of the multiplier ideal sheaves Σ_{i=1}^r ℐ(𝛗_i)? We establish an existence theorem with sharp L^2 estimates for this generalized division problem. We also discuss applications to the L^2 extension theorem and to the openness conjecture for multiplier ideal sheaves.

Abstract: Let X=𝔹^n/Γ be a noncompact complex hyperbolic space form of finite volume; and \overline{X}⊇X be the Mumford compactification of X. We consider the construction of symmetric differentials on \overline{X} which vanish on the infinity \overline{X}-X. These symmetric differentials can be viewed as a generalization of the cusps forms on modular curves. It is known that these differentials exist only when the lattice Γ is `sufficiently small'. We introduce a geometric quantity called the canonical radius r of Γ in relation to the `size' of Γ. By the method of L^2-estimates, we show that when r ≥ r^* for some fixed r^*=r^*(n)>0 depending only on the dimension n, the desired symmetric differentials exist.

Abstract: The classical Schwarz-Pick Lemma in one complex variable states that, for a holomorphic function f: 𝔻→𝔻 between unit discs 𝔻⊂ℂ,  

|f'(z)| / (1- |f(z)|^2 ) ⩽ 1/ (1 - |z|^2)

for all z ∈𝔻. The equality holds for all points if and only if f is a biholomorphism. Geometrically, this is equivalent to the statement  f^* g_𝔻 \leq g_𝔻, where g_𝔻 denotes the Poincaré metric (or the Bergman metric) of 𝔻.  This fundamental result has been significantly generalized over the decades by many mathematicians, including Ahlfors, Chern, Lu, Yau, and Royden.  In this presentation, we introduce a new Schwarz Lemma for bounded domains with Bergman metrics. The key ingredient of our proof is the Cauchy-Schwarz inequality from probability theory (or information geometry). We also discuss how this result compares with existing Schwarz lemmas. This is a joint work with H. Seo and S. Yoo.