Curvature estimate for a graphical surface, which is an non-parametric surface, is a natural consideration to analyze the surface. However, a graph with no geometric condition may have curvature as high as desired at some point on a bounded domain, although the height is generally bounded on the domain. We can find a result related to the estimate for the minimum value of the mean curvature curvature of a space-like graph. In this talk, we consider a Lorentzian product space based on a Riemannian manifold and provide Heinz-type estimates for the space-like graph of a function defined on a closed geodesic ball in the Riemannian manifold.
In 1976, Cappell and Shaneson constructed an infinite family of smooth homotopy 4-spheres, called Cappell-Shaneson homotopy 4-spheres. Cappell-Shaneson homotopy 4-spheres are the most notable potential counterexamples to the smooth 4-dimensional Poincare conjecture and they are related to other important conjectures including Gluck, Schoenflies, the slice-ribbon conjectures. In this talk, I would like to give a survey on Cappell-Shaneson homotopy 4-spheres.
In this talk, we will introduce specific smoothing of topologically embedded surfaces in 4-manifolds. The main idea of the smoothing is motivated from the 4-dimensional light bulb theorem introduced in 2020 by D. Gabai. First, we review Gabai's work and other follow-up results. After then, we discuss our recent progress on this smoothing of topological embeddings, called light bulb smoothing and introduce some applications of the result.
Ulrich bundles are vector bundles which admit completely linear resolutions. They can be characterized in several different ways, and thus it is not strange that they naturally appear in various problems in algebra and geometry. In particular, Ulrich bundles over hypersurfaces can be understood via linear determinantal representations and matrix factorizations. In this short talk, I will survey what are known on Ulrich bundles over hypersurfaces, and introduce a few recent results on smooth cubic fourfolds.
A polygon space is the moduli space of vectors in the 3-dimensional Euclidean space, which has been studied in various fields of mathematics. In this talk, I will discuss an SYZ mirror symmetry of bending systems of polygon spaces and its cluster structure. This work is based on joint work with Siu-Cheong Lau and Xiao Zheng.
A quandle is a non-empty set with a binary operation with conditions derived from Reidemeister moves. A quandle can be generalized to a biquandle, which is a non-empty set with two binary operations. When we define invariants of (surface-)links by using quandles or biquandles, we use orientations of (surface-)links. In this reason, it is difficult to define invariants for non-orientable surface-links by using (bi)quandles. In 2009, Kamada and Oshiro constructed symmetric quandles, a quandle with a map called a good involution. We can obtain invariants of unoriented (surface-)links by using symmetric quandles. In this talk, we define symmetric biquandles analogously to the symmetric quandles. We introduce symmetric biquandle coloring invariants of unoriented surface-links. This is a joint work with Sang Youl Lee.
We give a classification of Gorenstein Fano bi-equivariant compactifications of semisimple complex Lie groups with rank two, and determine which of them are equivariant K-stable and admit (singular) Kahler-Einstein metrics. As a consequence, we obtain several explicit examples of K-stable Fano varieties admitting (singular) Kahler-Einstein metrics. We also compute the greatest Ricci lower bounds, equivalently the delta invariants for K-unstable varieties. This gives us three new examples on which each solution of the Kahler-Ricci flow is of type II. This is joint work with Jae-Hyouk Lee and Sungmin Yoo.
In this talk, I will focus on free boundary minimal hypersurfaces in a ball and their rigidity results. First, we show that any minimal hypersurface with free boundary in a closed geodesic ball in a round open hemisphere $\Bbb S^{n+1}_+$ which is Killing-graphical is a geodesic disk. Second, we will talk about an analogous result for free boundary maximal hypersurfaces in a region bounded by a de Sitter space in the Lorentz-Minkowski space. More precisely, any smooth, compact free boundary maximal hypersurface in a de Sitter ball is the spacelike coordinate planar disk passing through the center of the de Sitter space. Notably, there is no more condition such as the Killing-graphical. Therefore free boundary maximal hypersurfaces in the de Sitter space are substantially rigid.
A map between topological spaces is said to be proper if the pre-images of compact subsets are compact. If the spaces are bounded domains in Euclidean spaces and the map extends continuously to the boundary, the properness of the maps is equivalent to the boundary being mapped to the boundary. Hence if the domains have special boundary structures, the map is expected to have a certain rigidity. In this talk I will introduce some rigidity results on proper holomorphic maps between bounded domains in complex Euclidean spaces.
We show that the second homology of the configuration spaces of a planar graph is generated under the operations of embedding, disjoint union, and edge stabilization by three atomic graphs: the cycle graph with one edge, the star graph with three edges, and the theta graph with four edges. We give an example of a non-planar graph for which this statement is false.
Since C. S. Peirce emphasized the importance of self-distributivity in algebraic structures in 1880, these structures have been actively studied by a number of scholars, including Mayer, Toyoda, Bruck, Galkin, etc. While homology theories of associative structures, such as groups and rings, have been extensively studied, starting with the work of Eileenberg and Hochschild, it has not been long since the study of homology theories of non-associative distributed structures began to be active. In this talk, we discuss what we are currently studying about homology theories of non-associative distributed structures.
In this talk, we introduce new notions of symmetric operators such as semi-symmetric shape operator and structure Jacobi operator in complex hyperbolic two-plane Grassmannians. Next we prove that there does not exist a Hopf real hypersurface in complex hyperbolic two-plane Grassmannians with such notions.
For a nondegenerate projective curve C in P^r of degree d and arithmetic genus g, G. Castelnuovo gave an upper bound \pi_0 (d,r) on genus g. Moreover, he proved that if d \geq 2r + 1 then a curve C which attains the possibly maximal genus always lies on a surface of minimal degree. Castelnuovo's theorem has been studied by G. Halphen, G Fano and recently by Eisenbud-Harris and I. Petrakiev. In this talk, we will briefly report on a recent progress improving the classification theory of curves of maximal and near the maximal genus.
The goal of this paper is to investigate the geometry of warped product Legendrian submanifolds in a Sasakian space form without boundary from the extrinsic point of view, subject to the following conditions: topology and geometry of a base manifold, geometric analysis of both a warping function and Ricci curvature of the base manifold. We establish sharp estimates of the relationship between the second fundamental form and the warping function, and also provide some trivial results for warped product Legendrian submanifolds by using the Ricci curvature along the gradient of the warping function. Taking the clue from the Bochner formula and the second-order ordinary differential equation, we find the characterization for topology of the base space of warped product Legendrian submanifolds via the first non-zero eigenvalue of the warping function, and prove that the base space is isometric to the Euclidean space R^p or the Euclidean sphere S^p under some extrinsic conditions.
Let $F_1(LG(2,4))$ be the space of lines in the Lagrangian Grassmannian $LG(2,4)$. Then $F_1(LG(2,4))$ is isomorphic to $\mathbb{P}^3$ which classifies the vertex point of a line in $Gr(2,4)$. Also the universal family of lines is nothing but the pair of linear subspaces $(V_1, V_2)$, $V_1<V_2<V_1^{\bot}$. Under this correspondence, one can describe the moduli space of twisted cubics in $LG(2,4)$ as a projective bundle over $Gr(2,4)$. Here, the role of $Gr(2,4)$ is the choice of the quadric surface and a ruling in $LG(2,4)$. In this talk, starting from the definition of rational curves, we sketch the proof of these facts.
In 2008, Lomonaco and Kauffman introduced a knot mosaic system to define a quantum knot system. In 2014, Kuriya and Shehab proved Lomonaco-Kauffman conjecture which means that knot mosaic type is a complete invariant of tame knots. The mosaic number of a knot K is the smallest integer n for which K can be represented on an n × n mosaic board. In this talk, we define a mosaic system for surface-links using marked graph diagrams and discuss the mosaic number of marked graph mosaics. This is joint work with Sam Nelson.
In this talk, we discuss the first nonzero eigenvalue $\lambda_{p,1}$ of the $p$-Laplacian on free boundary hypersurfaces in the unit ball evolving along the inverse mean curvature flow. We show that $\lambda_{p,1}$ is monotone decreasing along the flow. Using the convergence of free boundary disks in the unit ball, we give a lower bound of $\lambda_{p,1}$ of a free boundary disk type hypersurface in the unit ball. This is joint work with Pak Tung Ho.
In its original formulation, which motivated Hilbert's 15th problem, Schubert calculus consists in the study of the cohomology of grassmannians by an interplay of geometric, algebraic and combinatorial tools. Since then the scope of Schubert calculus has expanded in two orthogonal directions. On the one hand more general geometric objects have been considered, including for instance flag varieties, homogeneous varieties and bundles associated to them. On the other hand it was observed that the general framework could be reproduced for a wide variety of functors like K-theory, cobordism and their equivariant versions. In this talk I will give a brief overview of some recent developments in the context of oriented cohomology theories (e.g algebraic cobordism) and Bredon equivariant cohomology.