1. Team Profile

Our research team is composed of energetic, mid-career researchers with expertise spanning multiple disciplines and research directions. We have made notable contributions to Fourier analysis, symplectic geometry and dynamical systems, representation theory of finite groups, Lie groups, spherical homotopy groups, and quantum information science. Over the past five years, our work has been published in a number of internationally renowned academic journals, including the Journal of Functional Analysis, Communications in Mathematical Physics, Journal of Differential Equations, Monatshefte für Mathematik, Annali di Matematica Pura ed Applicata, Linear Algebra and its Applications, and Quantum Information Processing, among others. During the same period, the team has led multiple provincial- and ministerial-level research projects, including eight projects funded by the National Natural Science Foundation of China (NSFC).

2. Team Members

Guoquan Li, Yuyu Wang, Yuchen Wang, Xingzhong Liu, Zhiguang Hu, Yang Liu, Pan Lian, Bin Chen

3. Representative Publications

Guiqiao Xu, “On the power of standard information for L²-approximation in the average case setting,” Journal of Complexity 59 (2020) 101482.

Linan Zhong, Yuyu Wang, “Detection of a nontrivial product in the stable homotopy groups of spheres,” Algebraic and Geometric Topology 13 (2013) 3009–3029.

Ming Li, Chao Liang, and Xingzhong Liu, “A closing lemma for non-uniformly hyperbolic singular flows,” Communications in Mathematical Physics 405(8) (2024), No. 195.

Pan Lian, “The bounds of the odd dimensional Clifford-Fourier kernels,” Annali di Matematica Pura ed Applicata (1923–) (2021).

Bin Chen, Shao-Ming Fei, “Complementary measurement-induced quantum uncertainty based on metric adjusted skew information,” International Journal of Quantum Information (2021) 2150001.

Yang Liu, Yong Yang, “On Huppert’s ρ-σ conjecture,” Monatshefte für Mathematik (2021).

Zhiguang Hu, Philip B. Zhang, “Determinants and characteristic polynomials of Lie algebras,” Linear Algebra and its Applications 563 (2019.2.15): 426–439.

4. Representative Research Highlights

4.1 Research Theme I: Generalized Fourier Transform Theory

In the past five years, the team has made several advances in the theory of generalized Fourier transforms. In particular, we have explicitly constructed the integral kernel of the Dunkl transform in the dihedral group setting; partially proved the boundedness of the (k,a)-Fourier kernel introduced by Said–Kobayashi–Ørsted; and obtained pointwise estimates for the integral kernel of the Clifford-Fourier transform defined by Brackx–De Schepper–Sommen. Related papers have been published in journals such as the Journal of Functional Analysis, Journal of Fourier Analysis and Applications, Proceedings of the American Mathematical Society, Journal of Physics A: Mathematical and Theoretical, and Journal of Mathematical Physics.

4.2 Research Theme II: Hamiltonian Systems and Partial Differential Equations

Our team is actively engaged in research on Hamiltonian systems and partial differential equations. We investigate hyperbolic singular flows and employ methods from nonlinear analysis, elliptic equations, and dynamical systems to systematically study vortex sheet solutions to the two-dimensional incompressible Euler equations, obtaining linear stability and instability conditions, with in-depth analyses of degenerate bifurcations, rigidity, and singularity set structures. We have also established multiplicity results for closed geodesics on compact space forms of Finsler metrics, and conducted systematic studies on the blow-up dynamics of Schrödinger-wave coupling systems. These results have appeared in journals including the Journal of Differential Equations, Communications in Mathematical Physics, Acta Mathematica Sinica (English Series), and Calculus of Variations and Partial Differential Equations.

4.3 Research Theme III: Group Theory

The team has long been working on finite groups and their representations, as well as Lie theory, with related work published in journals such as Acta Mathematica Sinica, Monatshefte für Mathematik, Journal of Algebra and Its Applications, and Communications in Algebra. Our main research topics include the arithmetic properties of character degrees and the influence of codimensions on group structure. In 2014, we improved Huppert’s ρ-σ conjecture for non-solvable groups; in 2015, we investigated the ρ-σ problem for p-Brauer characters and p-regular conjugacy classes; in 2019, in collaboration with Bessenrodt, Lu Ziqun, and Zhang Jiping, we studied the ρ-σ problem on blocks; and in 2021, in collaboration with Yong Yang, we improved the ρ-σ conjecture for solvable groups. In the past two years, we have made progress on character codimensions, including studying the codimension prime graphs of solvable groups and classifying non-solvable groups with small codimensions. These results are of significant importance for understanding the relationship between representations and group structure. In addition, we have obtained a new characterization of solvable Lie algebras using determinants of matrix pencils.

4.4 Research Theme IV: Quantum Information

In recent years, quantum science and technology have advanced rapidly, becoming a frontier field of the new technological and industrial revolution. The team has achieved a series of results in quantum information, published in journals including Frontiers of Physics, Quantum Information Processing, and Communications in Theoretical Physics. We have studied measures of quantum coherence based on modified trace distance, obtaining analytical expressions for the qubit case and for a class of maximally coherent mixed states in arbitrary finite-dimensional quantum systems, and establishing ordering relations with other coherence measures. We have also investigated the monogamy problem of multipartite entanglement in arbitrary finite-dimensional quantum systems, deriving a new set of weighted monogamy inequalities for multipartite entanglement by improving previous methods. We computed the total variance of quantum states under mutually unbiased bases and general SIC measurements, and elucidated the connection between this result and the Brukner–Zeilinger invariant information. Analytical expressions for the average coherence of quantum states under mutually unbiased bases and general SIC measurements were obtained, and their interrelations were studied. Furthermore, we investigated complementary measurement-induced quantum uncertainty based on metric adjusted skew information, and derived new criteria for quantum entanglement based on this measure.