My research interests are in the areas of number theory and its applications. Number theory has very rich applications, such as Cryptography, and interaction of ideas across such diverse fields as analysis, algebra, geometry and representation theory, now even quantum mechanics and data science. For instance, there are more and more fascinating collaborations among physicists pursuing the implications of quantum chaos and number theorists attracted by Riemann Hypothesis, one of the deepest problems in mathematics.
Recently, I am interested in studying the interactions between data science and number theory. For instance, we can apply the method in data mining to the computational aspect of number theory problems. On the other hand, in data analysis, the discrete structures of big data can be well represented by p-adic number systems and p-adic numbers have many powerful applications on data analysis and data clustering.
I am also interested in Applied Math related with biomathematics. Currently, we are working on some projects, which study the clustering analysis of microRNA and mRNA expression data using various techniques from graph theory, number theory, statistics and data mining, etc.
Publications and preprints
- The Quantum Variance on the Modular Surface, joint with Peter Sarnak, accepted by Annales scientifiques de l'ENS.
- Quantum Variance on Maass-Hecke Cusp Forms, Comm. in Math. Phy., 475- 514, Vol. 297, 2010.
- Quantum Variance on Maass-Hecke Cusp Forms, PhD thesis, Advisor: Wenzhi Luo, Ohio State University, 2009.
- Clustering Analysis of microRNA and mRNA Expression Data from TCGA, joint with Lizhong Ding, etc., submitted.
- Optimal Cluster Detection of A Certain Directed Edge-weighted Bipartite Graph, joint with Cheng Zhao, etc., preprint.
- Non-Gaussian Distribution of Higher Moments of Matrix Elements, preprint.
- A Large Sieve Type Inequality and Its Applications on Waring-Goldbach Problem, preprint.
- Visible Points in Lattice Parallelograms, joint with Mizan Khan, preprint.
- Clustering Analysis of Bipartite Graph and Markov Matrix, joint with Cheng Zhao, in progress.