Matrix Problems in Hilbert Spaces

Hilbert series and degree bounds for matrix (semi-)invariants

We study the ring R(n,m) of invariants for the left-right action of SL_n \times SL_n on m-tuples of n by n complex matrices. We show that R(3,m) is generated by invariants of degree less equal 309 for all m. Then, we use a combinatorial description of the invariants to show that R(n,m) cannot be generated by invariants of degree < n^2 for large m. We also compute the Hilbert series for several cases.

Chern-Simons矩阵模型的希尔伯特空间、形变双电流代数作用及共形极限

On the Hilbert Space of the Chern-Simons Matrix Model, Deformed Double Current Algebra Action, and the Conformal Limit

本文从几何量子化角度研究Chern-Simons矩阵模型的希尔伯特空间，证明其与箭图簇截面空间同构，构造DDCA不可约作用，并定义共形极限，证明其可积模结构，解决Dorey-Tong-Turner猜想。

Irregular Riemann-Hilbert correspondence, Alekseev-Meinrenken dynamical r-matrices and Drinfeld twists

In 2004, Enriquez-Etingof-Marshall suggested a new approach to the Ginzburg-Weinstein linearization theorem for a quasitriangular Lie bialgebra $(\g,r)$. This approach is based on solving a system of PDEs for a gauge transformation between the classical r-matrix $r$ and the Alekseev-Meinrenken dynamical r-matrix. They proved that the semiclassical limit of an admissible Drinfeld twist gives rise to a solution of the PDEs. In this paper, we explain that preferred gauge transformations can be constructed as connection maps for a certain irregular Riemann-Hilbert problem (provided $r$ is the stan…