Research
My research interests are in the
areas of number theory and its applications. Number theory has very rich
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, number theory and graph 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 working on some data clustering problems using techniques in theory of
graphs and hypergraphs. Along this direction, there are numerous applications
in bioinformatics and data science.
Publications and preprints
-The
Quantum Variance on the Modular Surface, joint with Peter Sarnak, to appear.
-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.
-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.
-Hypergraph Partitioning and Higher Order Markov
Tensors, joint with C. Zhao. Preprint.
-Optimal
Cluster Detection of a Certain Edge-weighted Bipartite Graph Using Markov
Matrix, joint with C. Zhao, etc. submitted.
-Reliance, Loss Mitigation, and Disgorgement
Damages for Breach of Contract, joint with John Liu, Ronen Avraham and Yue Qiao, submitted.