Research Projects

Graduate work

Descriptions Under Construction.

Undergraduate work

Non-asymptotic Equipartition Properties for Hidden Markov Processes [PDF]

Adviser: Professor Guangyue Han, The University of Hong Kong

Abstract: A hidden Markov process (HMP) is a discrete-time finite-state homogeneous Markov chain observed through a discrete-time memoryless invariant channel. The non-asymptotic equipartition property (NEP) is a bound on the probability of the sample entropy deviating from the entropy rate of a stochastic process, which can be viewed as a refinement of Shannon-McMillan-Breiman theorem. In this report, we start from the basic concept and properties of a hidden Markov process, the Renyi entropy rate and a review of Shannon-McMillan-Breiman theorem. Then we will study a NEP for i.i.d. sources, a NEP for hidden Markov processes and a conjectured stronger NEP for hidden Markov processes which relates the Renyi entropy rate surprisingly. Finally, we conclude with possible future works on this project.   

Generalizations of Schwarz Lemma [PDF]

Adviser: Professor Ngaiming Mok, The University of Hong Kong

Abstract: In this project, we study generalized Schwarz lemma in complex geometry. The Ahlfors-Schwarz’s lemma and its generalization to holomorphic maps between the unit disk and Kahler manifolds are investigated. In particular, we study the case when equality in Schwarz lemma holds at a certain point for holomorphic maps between the unit disk and classical bounded symmetric domains of type I, II and III. We also investigate two higher-dimensional generalizations of the Ahlfors-Schwarz lemma for holomorphic maps from a compact Kahler manifold to another Kahler manifold. Finally, we conclude with two applications of various versions of Ahlfors-Schwarz lemma.