Reference number : GC-2017-Q1-02735, amendment number : , agreement type : G, recipient legal name : Kinzebulatov, Damir (Université Laval), recipient country : CA, recipient province : nan, recipient city : Quebec|Québec, recipient postal code : nan, program name english : Discovery Grants Program - Individual, program name french : Programme de subventions à la découverte - individuelles, program purpose english : My proposal is devoted to a number of interrelated problems originating in Mathematical Physics, playing a central role in several areas of modern Analysis, whose solution would lead to a significant progress and leadership of Canadian mathematicians in these areas. 
 I. A Brownian motion perturbed by a singular vector field (drift) is the principal component of many models of Mathematical Physics. It is constructed as a solution of the corresponding stochastic differential equation (SDE). The search for the maximal admissible singularities of the drift, i.e. such that the corresponding SDE has a unique solution, attracted the interest of many mathematicians, but is still far from being complete. I intend to substantially advance this search, reaching critical-order singularities, by applying new operator-theoretic techniques that recently allowed me to combine, for the first time, critical point and critical hypersurface singularities of the drift (in a weaker variant of this problem, i.e. constructing an associated Feller process). Next, I intend to develop the instruments needed to study solutions of such SDEs, including (non-Gaussian) two-sided bounds on the fundamental solution of the corresponding Kolmogorov backward operator. 
 II. I continue to work towards solving the long-standing problem of absence of positive eigenvalues of Schroedinger operators on R^d, in dimension d=3 or higher, and related problem of unique continuation (UC) for eigenfunctions of Schroedinger operators. I intend to obtain new, close-to-optimal results on the problem of absence of positive eigenvalues by exploiting an operator-theoretic technique that uses the link to the UC (extending my earlier work with L. Shartser), and a technique that does not rely on the UC (a new approach). 
 The goal of Projects I and II is to bring modern operator-theoretic techniques to the areas of diffusion processes and unique continuation. 
 III. Recently, I (jointly with A. Brudnyi) established the basic results of complex function theory within certain Fr&eacute;chet algebras of holomorphic functions on coverings of Stein manifolds by extending Cartan theorems A and B (Oka-Cartan theory) to coherent-type sheaves on the spectra of these algebras (model example: holomorphic almost periodic functions, arising in various problems of Analysis and Mathematical Physics, e.g. in Anderson localization). This work suggests that the Oka-Cartan theory, as an approach to complex function theory alternative to studying the d-bar equation, is valid beyond the classical setup of complex manifolds. I intend to extend the developed techniques to the algebras of holomorphic functions that have, in a sense, a similar local structure, but a different global structure, e.g. certain subalgebras of Hardy algebra on polydisk (obtaining a corona theorem for these algebras), aiming at determining the "natural domain" of Oka-Cartan theory., program purpose french : My proposal is devoted to a number of interrelated problems originating in Mathematical Physics, playing a central role in several areas of modern Analysis, whose solution would lead to a significant progress and leadership of Canadian mathematicians in these areas. 
 I. A Brownian motion perturbed by a singular vector field (drift) is the principal component of many models of Mathematical Physics. It is constructed as a solution of the corresponding stochastic differential equation (SDE). The search for the maximal admissible singularities of the drift, i.e. such that the corresponding SDE has a unique solution, attracted the interest of many mathematicians, but is still far from being complete. I intend to substantially advance this search, reaching critical-order singularities, by applying new operator-theoretic techniques that recently allowed me to combine, for the first time, critical point and critical hypersurface singularities of the drift (in a weaker variant of this problem, i.e. constructing an associated Feller process). Next, I intend to develop the instruments needed to study solutions of such SDEs, including (non-Gaussian) two-sided bounds on the fundamental solution of the corresponding Kolmogorov backward operator. 
 II. I continue to work towards solving the long-standing problem of absence of positive eigenvalues of Schroedinger operators on R^d, in dimension d=3 or higher, and related problem of unique continuation (UC) for eigenfunctions of Schroedinger operators. I intend to obtain new, close-to-optimal results on the problem of absence of positive eigenvalues by exploiting an operator-theoretic technique that uses the link to the UC (extending my earlier work with L. Shartser), and a technique that does not rely on the UC (a new approach). 
 The goal of Projects I and II is to bring modern operator-theoretic techniques to the areas of diffusion processes and unique continuation. 
 III. Recently, I (jointly with A. Brudnyi) established the basic results of complex function theory within certain Fr&eacute;chet algebras of holomorphic functions on coverings of Stein manifolds by extending Cartan theorems A and B (Oka-Cartan theory) to coherent-type sheaves on the spectra of these algebras (model example: holomorphic almost periodic functions, arising in various problems of Analysis and Mathematical Physics, e.g. in Anderson localization). This work suggests that the Oka-Cartan theory, as an approach to complex function theory alternative to studying the d-bar equation, is valid beyond the classical setup of complex manifolds. I intend to extend the developed techniques to the algebras of holomorphic functions that have, in a sense, a similar local structure, but a different global structure, e.g. certain subalgebras of Hardy algebra on polydisk (obtaining a corona theorem for these algebras), aiming at determining the "natural domain" of Oka-Cartan theory., agreement title english : Operator-theoretic approach to problems of Analysis and Partial Differential Equations, agreement title french : Operator-theoretic approach to problems of Analysis and Partial Differential Equations, agreement number : RGPIN, agreement value : 105000.0, agreement start date : 2017-05-10, agreement end date : nan, description english : nan, description french : nan, expected results english : nan, expected results french : nan, additional information english : Grant or Award spanning more than one fiscal year. (2017-2018 to 2022-2023), additional information french : Subvention ou bourse octroyée s'appliquant à plus d'un exercice financier. (2017-2018 à 2022-2023), owner organization : nserc-crsng, owner organization title : Natural Sciences and Engineering Research Council of Canada | Conseil de recherches en sciences naturelles et en génie du Canada