Background

I got my Ph.D. in mathematics from the University of Toronto. In my Ph.D. study, I worked in differential geometry. I used quantitative topology to study different variational problems in differential geometry including closed geodesics and minimal surfaces. I also applied techniques in differential geometry to study and develop manifold learning algorithms.

Research Interests

My research focuses on manifold learning and its applications. Manifold learning consists of techniques exploring the geometric structure of complicated data. Specifically, my research interests include the following directions.

  1. Nonlinear dimension reduction
    I establish the mathematical foundation for nonlinear dimension reduction algorithms. I also develop new methods for dimension reduction. Some nonlinear dimension reduction algorithms that I study include Locally Linear Embedding and Diffusion Maps.
  2. Exploration of the geometric structure of the data
    I develop methods and theories to explore the geometric structure of the data, e.g. denoising and reconstruction of high dimensional data.
  3. Applications of manifold learning
    I apply techniques in manifold learning to study problems in nonparametric statistics and stochastic dynamics.