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

Title: The cohomology of 4-dimensional toric orbifolds

Abstract: Toric orbifolds are even-dimensional orbifolds equipped with locally standard torus action. Each of them corresponds to a polytopes $P$ and a characteristic function $\lambda$. There is an interesting interplay between their topology and the combinatorial data of $P$ and $\lambda$. In this talk, I will explain how this combinatorial information determines the algebraic structure of their cohomology in the 4-dimensional case.

This is joint work with Xin Fu and Jongbaek Song.

Title. Classification of equivariant Legendrian embeddings of rational homogeneous spaces into nilpotent orbits

Abstract. Nilpotent orbits, that is, the adjoint orbits of nilpotent elements in projectivized semi-simple Lie algebras, are homogeneous varieties admitting invariant contact structures. The most well-known examples are the odd-dimensional projective spaces. In this case, it is classically well known that every equivariant Legendrian embedding of a rational homogeneous space can be realized as one of the so-called subadjoint varieties. In this talk, I will discuss the case of other nilpotent orbits. Namely, I will classify equivariant Legendrian embeddings of rational homogeneous spaces into nilpotent orbits.

Title: Random dynamics on surfaces 

Abstract: In this talk, we discuss random dynamical systems and group actions on surfaces. Random dynamical systems, especially understanding stationary measures, can play an important role for understanding a group action. For instance, when a group action on torus is given by toral automorphisms, using random dynamics, Benoist-Quint classified all orbit closures. We will study group actions on surfaces by diffeomorphisms, using random dynamics including absolute continuity of stationary measures, classification of orbit closure, and exact dimensionality of stationary measures. This talk will be mostly about the ongoing joint work with Aaron Brown, Davi Obata, and Yuping Ruan.

Title: Toric lifting problem for piecewise linear spheres with a few vertices 

Abstract: Given a simplicial complex $K$ with $m$ vertices, toric spaces over $K$ are defined as quotients of the moment-angle complex $\mathcal{Z}_K \subset \left(D^2 \right)^m$ by freely acting subtori $T^r$ of the torus $T^m$ acting on $\mathcal{Z}_K$, equipped with the induced $T^{m-r}$-action. Similarly, real toric spaces with the induced $\mathbb{Z}_2^{m-r}$-action are defined from the real moment-angle complex $\mathbb{R} \mathcal{Z}_K \subset \left(D^1 \right)^m$ with the $\mathbb{Z}_2^m$-action. The involution on $\mathcal{Z}_K$ induced by complex conjugation on $D^2 \subset \mathbb{C}$ allows us to view $\mathbb{R} \mathcal{Z}_K$ as the fixed point set, inheriting the $\mathbb{Z}_2^m$-action. The toric lifting problem asks which real toric spaces with a $\mathbb{Z}_2^{m-r}$-action arise from toric spaces with a $T^{m-r}$-action. In this talk, I will discuss this problem for piecewise linear spheres with a few vertices.

This is a joint work with Suyoung Choi and Mathieu Vall\'{e}e.

Title. The Border Rank of the 4 × 4 Determinant Tensor is Twelve

Abstract. For a given tensor, its tensor rank is defined as the smallest number of decomposable tensors required to express the tensor as the sum of them, and its border rank is defined as the smallest number such that the tensor can be expressed as the limit of tensors of tensor rank at most the number. Both are natural generalizations of matrix rank, but finding these ranks for a given tensor is regarded as a challenging problem, because the methods to determine the matrix rank have not been generalized to the tensor and border ranks yet. Even for the fundamental tensors such as matrix multiplication, determinant, and permanent tensors, their tensor and border ranks remain only partially understood. In this talk, I determine the border rank of the 4 × 4 determinant tensor to be twelve. This is joint work with Jong In Han and Yeongrak Kim.

Title: GIT stability and complex structures on $SU(3)$

Abstract: The flag variety consists of all flags in $\mathbb{C}^3$, and it has a structure of projective space. On the other hand, it can be regarded as the homogeneous space $SU(3)/T$, and $SU(3)$ has a structure of a (non-algebraic) holomorphic torus bundle over a projective variety. In this talk, I will explain variants of these structures from the viewpoint of equivariant symplectic geometry and GIT quotients. This talk is based on joint work with Yoshinori Hashimoto and Hisashi Kasuya.

Title: Hecke Correspondences from Localized Mirror Construction 

Abstract: Nakajima’s quiver varieties provide geometric representations of deformed Kac-Moody algebras, where Hecke correspondences play the role of creation and annihilation operators. From the viewpoint of mirror symmetry, one expects these varieties to arise from the deformation spaces of Lagrangian branes; however, a concrete symplectic model realizing this correspondence has been missing. In this talk, I will describe a Floer-theoretic construction that identifies Nakajima quiver varieties with Maurer-Cartan deformation spaces of framed Lagrangian branes. Building on this identification, I will show that the Hecke correspondences themselves appear as the supports of the Floer cohomology between families of such branes. Finally, I will explain how the localized mirror functor induces a quasi-equivalence realizing a local form of Homological Mirror Symmetry. This is based on joint work in progress with Siu-Cheong Lau.

Title: Geometry of locally symmetric hypersurfaces 

Abstract: A Riemannian manifold is called a (locally) symmetric space if, at each point, the geodesic symmetry is a (local) isometry. Symmetric spaces can be regarded as Riemannian manifolds with parallel curvature tensor, and they admit a decomposition into a Riemannian product of a Euclidean space (possibly trivial) and irreducible symmetric spaces. In this talk, I will review basic definitions and several known results concerning symmetric spaces and hypersurfaces, and then discuss the classification of locally symmetric hypersurfaces in symmetric spaces. This talk is based on ongoing joint work with Yuri Nikolayevsky and Ruy Tojeiro.