As part of a , the MCA will host a mini-course of lectures
鈥淎nalogues of the Lebesgue measure and boundary value problems in infinite-dimensional spaces鈥�. It will be read by the leading researcher of the Keldysh Institute of Applied Mathematics RAS, Professor of the Department of Higher Mathematics of the Moscow Institute of Physics and Technology
Vsevolod Sakbaev.
Lectures will be held on
December 4, 6, 11 and 13 at 16:20 in Zoom at zoom.us/j/93847824313?pwd=VTNUam1UMWZnK1NiWFdGWDUvZjlXUT09 (ID: 938 4782 4313, password: 036211)
Course program:
"Translationally and rotationally invariant measures on an infinite-dimensional Hilbert space and their applications in mathematical physics"
According to A. Weil's theorem, there is no measure on an infinite-dimensional normed linear space that is invariant under the group of shifts by vectors of this space. Finitely additive measures on a real Hilbert space that are invariant under shifts and orthogonal transformations will be considered. In the space of functions squarely integrable with respect to one of these measures, we study groups of shifts by random vectors and their averages.
"Random walks in Hilbert space and semigroup approximations"
The mathematical expectation of a shift operator by a random vector whose distribution is given by a semigroup (with respect to convolution) of Gaussian measures. Let us establish that such mathematical expectations form a semigroup of self-adjoint contractions in the space of functions that are square-integrable with respect to a translation-invariant measure. Let us obtain a criterion for the strong continuity of such semigroups. Self-adjoint operators of such semigroups are defined as Laplace operators. Let us introduce analogs of Sobolev spaces and spaces of smooth functions. Let us show that the introduced Laplace operators are Gross-Volterra Laplacians. Let us obtain approximations of semigroups by mathematical expectations from random processes.
"Scales of Sobolev spaces and boundary value problems"
Conditions for embedding and dense embedding of spaces of smooth functions in Sobolev spaces. Existence of traces of functions from the Sobolev space on subspaces of codimension 1. Analogue of the Ostrogradsky-Gauss formula. Statement of the first boundary value problem for the Poisson equation and obtaining a variational method for its solution.
