Ideal triangle groups, dented tori, and numerical analysis

On exact triangles consisting of stable vector bundles on tori

In this paper, we consider the exact triangles consisting of stable vector bundles on one-dimensional complex tori, and give a geometric interpretation of them in terms of the corresponding Fukaya category via the homological mirror symmetry.

Twistor triangles in the period domain of complex tori

We study the geometry of the (generalized) twistor triangles $\triangle J_1J_2J_3$ in the period domain of compact complex tori of complex dimension $2n$ by the means of the representation theory of the algebras (of real dimension 8) generated by the complex structures $J_1,J_2,J_3$. Considering the period domain as the homogeneous space for $G=GL_{4n}(\mathbb{R})$, we introduce on it a $G$-invariant pseudometric and define pseudometric invariants, helping us to distinguish triangles from a reasonable class up to $G$-equivalence.

Geometric interpretation for exact triangles consisting of projectively flat bundles on higher dimensional complex tori

Let $(X^n, \check{X}^n)$ be a mirror pair of an $n$-dimensional complex torus $X^n$ and its mirror partner $\check{X}^n$. Then, a simple projectively flat bundle $E(L,\mathcal{L})\rightarrow X^n$ is constructed from each affine Lagrangian submanifold $L$ in $\check{X}^n$ with a unitary local system $\mathcal{L} \rightarrow L$. In this paper, we first interpret these simple projectively flat bundles $E(L,\mathcal{L})$ in the language of factors of automorphy. Furthermore, we give a geometric interpretation for exact triangles consisting of three simple projectively flat bundles $E(L,\mathcal{L}…

具有有限单值群的 Lamé 方程的特征刻画与枚举

Characterization and enumeration on Lamé equations with finite monodromy

本文完整刻画了平坦环面上具有有限单值群的 Lamé 方程，结合 dessin d'enfants 与球面环面几何证明了解的存在性，并对非半整数情形给出显式计数公式。