Research Projects

A quasi-analytic class of functions is a generalization of the class of real analytic functions with respect to the strong unique continuation property, that is, the function vanishes identically on the domain if it vanishes of infinite order at a point. The class of quasi-analytic functions are well-known on one dimensional Euclidean spaces and their characterization was given by the celebrated Denjoy-Carleman theorem. While discussing the notion of quasi-analytic vectors on Hilbert spaces, P. Chernoff proved a variant of the Denjoy-Carleman theorem for functions on higher dimensional Euclidean spaces using the L^2 -norm of the iterates of the Laplacian. The theory of quasi-analytic functions has found several applications due to its profound connections with different branches of mathematics. Notable examples include: (i) Uncertainty principles in harmonic analysis (ii) moment determinacy problems (iii) uniqueness properties of solutions to partial differential equations (PDEs) (iv) observability and controllability in the context of solutions to the Schrodinger equation (v) Fractal Uncertainty principles [11] (vi) geometry of nodal sets of eigen functions on closed compact manifolds, to name a few. These applications illustrate the broad impact of quasi-analytic functions across various mathematical disciplines. We have been approaching these problems from several directions, particularly by extending the theory of quasi-analytic functions to the more general setting of connected semi-simple Lie groups. Our goal then is to apply this extended theory to the aforementioned problems within this broader context.