Methods of Functional and Harmonic Analysis and PDE with Singularities

Research plan includes the following topics: Analysis of hypoellipticity of equations through the theory of Gelfand-Shilov spaces; Local analysis of generalized function spaces and algebras of generalized functions through the development of geometric theory of generalized functions used for solving various classes of PDE with singularities; Tauber theorems for the wavelet transformation and for the regularization transformation, including a new approach to the 2nd microlocalization; Study of measure valued solutions of hyperbolic systems, the interaction of waves with applications in fluid dynamics; Chaos expansion and Malliavin calculus with applications in stochastic differential equations with singular coefficients and data and applications in economy and the risk valuation in insurance; Fractional differential equations within the calculus of variations with applications in audio and visual signal processing; Fractional calculus in Colombeau spaces with values in white noise spaces; Microlocal analysis of wave front sets in time-frequency analysis and discrete wave front through Gabor expansions; Evolution equations and different classes of semigroups and cosine functions on spaces of distributions, ultradistributions and hyperfunctions; Development of frame theory for translational invariant spaces and Fr\' echet spaces; Investigations in real and complex analysis related to inequalities and quasiconformal mappings of various domains important in PDE.