Waseda Young Researchers Meeting on Geometry

Abstract: In a warped product with an open interval as its base, we can define a graph hypersurface and consider the mean curvature flow of these hypersurfaces. In this talk, we show that the mean curvature flow preserves the graph structure of hypersurfaces for all time and study its asymptotic behavior under certain conditions on the warping functions. Additionally, we will mention some ongoing studies.

Abstract: The notion of a dually flat structure was introduced by Amari-Nagaoka for studying statistical inference, informatics, machine learning and so on. Prior to Amari’s work, Efron gave pioneering insights into the extrinsic geometry of a one parameter curved exponential family in two-dimensional exponential family. In this talk, we will focus on the dually flat geometry of plane curves and give its characterization from the viewpoint of singularity theory. In particular, new ingredients are the e/m-envelopes or the e/m-caustics, which arise when we replace the distance-squared function in Euclidean geometry with the Bregman divergence in dually flat geometry. Nevertheless, our discussion still makes sense in higher-dimensional cases. Our approach also describes catastrophe phenomena in the root selection problem of maximum likelihood estimation.

Abstract: Koszul-Vinberg manifolds are a generalization of Hessian manifolds and induce left-symmetric algebroid structures on the cotangent bundles. There are many similarities between Koszul-Vinberg and Poisson geometry. Then, as Poisson structures are generalized Jacobi structures, can Koszul-Vinberg structures be generalized a natural structures? In this talk, by generalizing the notion of left-symmetric algebroid, we define a natural generalization of Koszul-Vinberg manifolds and show properties of the structures.

Abstract: As one of the deformation quantizations of K\"{a}hler manifolds, Karabegov proposed a deformation quantization with separation of variables and its construction method for K\"{a}hler manifolds. In particular, for locally symmetric K\"{a}hler manifolds, the construction methods were also proposed by Sako-Suzuki-Umetsu and Hara-Sako, inspired by Karavegov's construction. These methods state that there is the star product with separation of variables on locally symmetric K\"{a}hler manifolds satisfying some recurrence relations. By using this construction method, the explicit star products were obtained \(\mathbb{C}^{N}\), \(\mathbb{C}P^{N}\), \(\mathbb{C}H^{N}\), and arbitrary one- and two-dimensional ones. In general, however, it is not easy to determine the general terms of the recurrence relations given by Hara-Sako. In this talk, we focus on the complex Grassmannian \(G_{2,4}(\mathbb{C})\) and give the explicit formula for its star product from the explicit form of the general terms of the recurrence relations for \(G_{2,4}(\mathbb{C})\). If time permits, we also present an outlook on twistor theory as one of the future directions of this work. This talk is based on the joint work with Akifumi Sako (Tokyo University of Science). 

Abstract: The notion of pseudo-Riemannian symmetric $R$-spaces was introduced by H. Naitoh as a generalization of the notion of symmetric $R$-spaces in 1984. I will show that a maximal antipodal set of the pseudo-Riemannian symmetric $R$-space associated with a semisimple symmetric graded Lie algebra can be obtained as an orbit of a Weyl group.

Abstract: The tt* equations(topological-antitopological fusion equations) were introduced by S. Cecotti and C. Vafa(1991) for describing massive deformations of supersymmetric conformal field theories. In general, the tt*-equation is highly nonlinear and it has been solved in very special cases: the sinh-Gordon equation and the tt*-Toda equation. In mathematics, B. Dubrovin(1993) formulated certain cases of tt*-equation as an isomonodromic deformation of a meromorphic linear differential equation with two poles. Much later, M. Guest, A. Its and C.-S. Lin solved the “Toda type” of tt*-equation from the view point of isomonodromy theory(2015, 2020) and p.d.e. theory(2015). In physics, Cecotti and Vafa(1993) used Dubrovin’s formulation and classified “solvable” tt*-equations by upper unitriangular matrices. In this talk, we explain Dubrovin’s formulation and its isomonodromy aspects. Furthermore, we give a mathematical formulation of classification introduced by Cecotti, Vafa and we construct tt*-equations from the Cartan matrices of ADE. In the language of the TERP structure, C. Hertling, C. Sevenheck(2007) and C. Sabbah(2011) constructed such tt*-equations by using results on the ADE-singularities. We give an alternative proof by using the Vanishing Lemma.

Abstract: We study the closure of a complex subtorus in the complement of toric divisors. If the closure is a complex submanifold in the symplectic toric manifold, then the submanifold inherits the equivariant Hamiltonian subtorus action that comes from the symplectic toric manifold. In this talk, we discuss the moment polytope of the submanifold under the Hamiltonian subtorus action.