부산대학교  기하학, 위상수학  세미나
(PNU Geometry and Topology seminar)

Title. Duality in Rabinowitz Fukaya category

Abstract: In this talk, I will first briefly introduce Rabinowitz Floer homology of a pair of Lagrangian submanifolds in a Liouville domain. Then I will explain that it fits into a long exact sequence, which is a Floer theoretic analogue of the long exact sequence associated with the pair $(M, \partial M)$ for a compact oriented manifold $M$ with boundary. I will finally discuss that it also admits a Floer theoretic analogue of Poincaré duality under a certain condition. This is based on a joint work with Wonbo Jeong and Jongmyeong Kim.

Title. Formal exponential maps and the Atiyah class of dg manifolds

Abstract: Exponential maps arise naturally in the contexts of Lie theory and smooth manifolds. The infinite jets of these classical exponential maps are related to Poincaré --Birkhoff--Witt isomorphism and the complete symbols of differential operators. We will investigate the question on how to extend these maps to dg manifolds. As an application, we will show there is an L-infinity structure on the space of vector fields in connection with the Atiyah class of a dg manifold. In particular, for the dg manifold arising from a foliation, we induce an L-infinity structure on the deRham complex associated with the foliation. As a special case, it is related to Kapranov’s L-infinity structure on the Dolbeault complex of a Kähler manifold. This is a joint work with Mathieu Stiénon and Ping Xu.

Title: Rigidity for 3-dimensional non-convex polyhedra.

Abstract: In this talk, we will first briefly survey the rigidity problems of 3-dimensional polyhedra with respect to their edge lengths, dihedral angles, and facial angles. We will then show that a 3-dimensional hyperbolic, spherical, and Euclidean polyhedron can be uniquely determined by its dihedral angles and edge lengths, regardless of whether it is non-convex or self-intersecting, under three plausible conditions: (1) the polyhedron is composed solely of convex faces, (2) there are no partially flat vertices, and (3) any triple of vertices is not collinear. Additionally, we will provide various counterexamples that arise when our conditions are violated and pose further questions and conjectures. There is little prerequisite knowledge required for this talk, except for combinatorics related to planar graphs and 2-dimensional spherical geometry.

Title: Classification of singular del Pezzo surfaces 

Abstract: In Kähler geometry, classifying Fano varieties with a Kähler-Einstein metric is an important problem. Although proving the existence of a Kähler-Einstein metric on a Fano variety is challenging, recent progress has provided strong tools to attack this problem. In this talk, we will explain how to prove the existence of Kähler-Einstein metrics on quasi-smooth del Pezzo hypersurfaces with higher index.

Title: Geometric Inequalities on Capillary Hypersurfaces

Abstract: In this talk, I will present some recent results on geometric inequalities on capillary hypersurfaces. I will also discuss some related rigidity results, which were obtained in collaboration with Prof. J. Pyo and with Prof H.Li and Prof. Y. Hu.

Title: Birational geometry of generalized Hessenberg varieties and the generalized Shareshian-Wachs conjecture

Abstract: Hessenberg varieties are subvarieties of flag varieties with interesting properties in both algebro-geometric and combinatorial perspectives. The Shareshian-Wachs conjecture connects their cohomology with the chromatic quasi-symmetric functions of the associated graphs, which are refinements of the chromatic polynomials. In this talk, we introduce generalized Hessenberg varieties and study their birational geometry via blowups. As a result, natural maps from Hessenberg varieties to projective spaces or the permutohedral varieties are decomposed into explicit blowups and projective bundle maps. As a byproduct, we also provide an elementary proof of the Shareshian-Wachs conjecture and its natural generalization. This is joint work with Prof. Young-Hoon Kiem.

Title: Unique toric structure on a Fano manifold

Abstract: A symplectic manifold is called toric if it admits an effective Hamiltonian action of a compact torus whose dimension is half the dimension of M. Delzant proved that every compact symplectic toric manifold M is Kahler and so M is a smooth projective toric variety. In this talk, I will talk about the following conjecture posed by Dusa McDuff: If M is Fano, then M admits at most one toric structure. If time permits, I will explain some positive results related to the conjecture. This is joint work with Eunjeong Lee, Mikiya Masuda, and Seonjeong Park

Title: Slowly converging geometric equations 

Abstract: In geometric variational problems such as minimal surface, mean curvature flow, Harmonic map and its heat flow, challenges often reduce down to questions on the asymptotic behavior near singularity and infinity. In this talk, we discuss the rate and direction of convergence for slowly converging solutions. Previously, they were constructed under so called the Adams-Simon positivity condition on the limit. We conversely prove that every slowly converging solution necessarily satisfies such a condition and the condition dictates possible dynamics. The result can be placed as a generalization of Thom's gradient conjecture. This is a joint work with Pei-Ken Hung at Minnesota.

Title: Geometry of Hermitian matrices and their determinants 

Abstract: Recall that a rank of the matrix A is the size of a maximal nonvanishing minor of A. This simple observation induces the rank stratification of matrices, in particular, the set of m×n matrices of rank at most k becomes an algebraic variety for each k. On the other hand, it is well-known that any finite dimensional division algebra over the real numbers R is isomorphic to R, C the complex number field, H the quaternions, or O the octonions (Hurwitz 1898, Kervaire/Milnor 1958). In this talk, we will first consider 3×3 Hermitian matrices over R, C, H, and O. Their rank stratifications provide determinantal descriptions of the Severi varieties. An interesting observation is that the gradient map of each of Hermitian determinantal cubics is a Cremona transformation. If time permits, I will talk about the relationship between these Cremona transformations and Hessian matrix factorizations.