일시 : 2026년 7월 3일 (금) 오후 2시~ 4시
장소 : 아산이학관 622호
발표자 1: 송교범 (럿거스 대학교 수학과 대학원 재학 중)
Title: Revised Demailly’s Affineness Criterion and Algebraization of Entire Grauert Tubes
Abstract: An entire Grauert tube is the tangent bundle $TM$ of a Riemannian manifold $(M,g)$ endowed with a Stein manifold structure satisfying the homogeneous complex Monge–Amp\`ere equation (HCMA). In 1982, Burns conjectured that every entire Grauert tube is an affine variety. This conjecture suggests a deep connection between Stein and affine geometry, and would also have significant implications for the algebraic rigidity of nonnegatively curved manifolds. In this talk, we provide a partial answer to Burns conjecture: the complement of a codimension-one subset of an entire Grauert tube is affine. This result is obtained by establishing a generalized version of Demailly’s criterion for affineness of Stein manifolds, which may be of independent interest.
발표자 2: 허진 (브라운 대학교 수학과 대학원 대학 중)
제목: Nested Isotopic Deformations Preserving the Surface-area-to-volume Ratio
초록: The surface-area-to-volume ratio (SA/V) is a basic geometric quantity that appears naturally in the study of growing shapes. Motivated by recent observations of microorganisms whose growth maintains a nearly constant SA/V ratio, we study continuous deformations of embeddings that preserve this quantity.
In this talk, we define a generalized SA/V ratio for embeddings using Hausdorff measures and introduce SA/V-preserving isotopic deformations, or SIDs: continuous one-parameter families of topological embeddings along which this ratio remains constant. While trivial examples of SIDs are easy to construct, producing nested SIDs is more subtle. We initiate the study of such examples by constructing several embedded n-spheres admitting nested SIDs and discussing their geometric properties.