Some explicit solutions to the Riemann-Hilbert problem

Hilbert-Schmidt groups as infinite-dimensional Lie groups and their Riemannian geometry

We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional groups, is calculated. We show that for some of these groups the Ricci curvature is $-\infty$.

The dimension of the Hilbert Space of Geometic quantization of vortices on a Riemann surface

In this article we calculate the dimension of the Hilbert space of Kahler quantization of the moduli space of vortices on a Riemann surface. This dimension is given by the holomorphic Euler characteristic of the quantum line bundle.

The higher Riemann-Hilbert correspondence and principal $2$-bundles

In this note we show that the holonomies for higher local systems provided by the higher Riemann-Hilbert correspondence coincide with those associated to principal $2$-bundles where both formalisms meet.

On critical normal sections for two-dimensional immersions in R^n and a Riemann-Hilbert problem

For orthonormal normal sections of two-dimensional immersions in R^4 we define torsion coefficients and a functional for the total torsion. We discuss normal sections which are critical for this functional. In particular, a global estimate for the torsion coefficients of a critical normal section in terms of the curvature of the normal bundle is provided.

Riemann-Hilbert 映射的导数

Derivative of the Riemann-Hilbert map

本文研究闭 Riemann 面上全纯联络的 Riemann-Hilbert 映射，计算其导数并刻画单射点轨迹，从而统一恢复多个已有结果。