Goals:
(1) Global Analysis and Geometric Analysis: characterization of functions initially defined on subsets of Euclidean space which admit differentiable convex extensions; absolutely minimal extensions on metric spaces; sphericalization and flattening in metric spaces; self-contracting curves in non-Euclidean spaces; nonsmooth analysis in Riemannian and Finslerian manifolds; subdifferential extension theorems; global inversion of nonsmooth functions in infinite dimensions; Hamilton-Jacobi equations and Myers-Nakay type theorems in Finslerian manifolds; Sard-type theorems for nonsmooth functions with some subdifferentiability properties.
(2) Differentiability, Convexity and geometry of Banach spaces: structure of derivatives of Gâteaux-differentiable functions between Banach spaces; convexity and differentiability properties in function spaces; completely continuous operators and representable operators; bornologies in metric spaces; limit points of analytic functions.
(3) Lineability and Operator Theory: We will continue with what our research group has been doing recently. Fristly, we’ll try to extend lineability to other areas. We shall try to obtain "Baire Category"-type results in lineability (and algebrability) general frameworks. We’ll delve into the new concept of polynomiable set. Within Operator Theory, we intend to focus on the study of hypercyclic operators and distributional chaos.
(4) Polynomials in Banach spaces, Inequalities, and Complex Analysis: It is intended to continue the study of the constants in inequalities of Bernstein, Markov, Bohnenblust-Hille, and Hardy-Littlewood, and continue research on optimal values for the n-dimensional Bohr radius K_n. We also aim to deepen the broad practical applications of the FFT (fast Fourier transform) and KLT (Karhunen-Loeve transform) within Signal Processing.