I am interested in a variety of problems in the mathematical foundations of Data Science, motivated by the need to exploit data to advance scientific discovery and build generalizable, interpretable predictive models. These problems require ideas and techniques from a variety of areas, including Harmonic Analysis, Approximation Theory, Probability and Statistics. Scalable algorithms are a requirement in applications, and I often use multiscale techniques to develop near-linear time algorithms.
I apply these techniques to the study of physical systems, e.g. to the analysis of molecular dynamics data in order to automatically learn reduced models or speed up simulations, or to infer models of complex agent-based systems (e.g. to model cell dynamics), to the study of hyperspectral images (e.g. for unsupervised segmentation or anomaly detection), to reinforcement learning (to learn optimal policies for automated agents).
See my Research page for more info about:
- Learning Interaction Kernels in interacting particle- or agent-based systems.
- Diffusion Wavelets: a construction of families of wavelets and Multi-resolution Analyses on graphs, manifolds and point clouds. Pictures, papers and presentations available.
- Diffusion Geometries: here are some links to the use of diffusion geometries in data analysis.
- Multiscale Geometric Methods for Data: various techniques for studying geometry of high-dimensional data in a multiscale fashion.
- Analysis of Molecular Dynamics Data: in collaboration with Cecilia Clementi and her lab, we use the geometric structure of data generated from molecular dynamics data to construct observables that provide reaction coordinates and reduced, low-dimensional dynamics that well-approximates the long-time dynamics of the original system.
- Multiscale Analysis of Markov Decision Processes.
- Visualization of large data sets.
- Harmonic Analysis and Wavelets: here I talk a bit about Harmonic Analysis and provide links to related web pages.
- HyperSpectral Imaging and Pathology: hyper-spectral imaging applied to pathology.
Academic year 2020-2021: I am on sabbatical, and will be slower in replying to e-mails, especially those on administrative matters.
Postdocs and students
My research group may have open positions for graduate students and postdocs. Areas of interest include stochastic dynamical systems, statistical signal processing, statistical/machine learning, high-dimensional probability and geometry, spectral graph theory and signal processing on graphs, reinforcement learning. Please use mathjobs to apply, and also consider the J.J. Sylvester Asst. Prof. positions in Math.
Past students and postdocs
It is necessary to teach both to accept and to reject the past with a kind of balance that takes considerable skill. Science alone of all the subjects contains within itself the lesson of the danger of belief in the infallibility of the greatest teachers of the preceding generation.