Research Interests
The research questions I investigate fall at the nexus of geometry, spectral analysis,
probability theory, optimization, and partial differential equations (PDEs), with recent
applications of spectral techniques in Computer Vision and Machine Learning.
In particular, my research focuses on isoperimetric, and geometric functional inequalities.
These optimal inequalities play a central role in many fields of science such as acoustics,
quantum mechanics, visualization and machine learning, fluid mechanics, and geometry.
Inequalities of isoperimetric type are useful for getting bounds for solutions of elliptic
equations.
In the popular culture, these sorts of questions are related to the old problem of
Queen Dido and the ruse at the heart of the foundational myth of the city of Carthage.
These are the major themes summarizing past, current, and future research interests:
Prove and improve universal and isoperimetric inequalities for
various spectral functions of operators modeled on the Laplacian, and study the interplay
between sum rules and universal bounds;
Study the isoperimetric problem on weighted manifolds, particularly focusing
on the persistence of classical functional inequalities;
Prove embedding properties for weighted Laplacians and nonlocal Riesz and
Bessel operators, and apply spectral techniques in machine learning in problems
related to dimensionality reduction, shape recognition, and clustering;
integrate these research interests in educational programs aimed at both
undergraduate and graduate students, with local, national and international reach.
