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watching recorded videos, to register at this link.
This seminar is dedicated to classical and recent results related to one-parameter operator
semigroups on linear spaces. The primary focus is on strongly continuous (or $C_0$-) semigroups
on Banach spaces. However, other classes of semigroups will also be covered, such as bi-
continuous semigroups, Gibbs semigroups, and others.
Applications of semigroup theory include Markov operators and semigroups, stochastic
differential equations, evolution equations in quantum mechanics, and more. A number of
important results concern semigroup approximations, Lie–Trotter formulas, and Chernoff
iteration formulas.
The seminar encourages the presentation and discussion of research papers by students
interested in semigroup theory and its applications. The program can be adjusted according
to the interests of the participants.
Program
- Fundamental properties of strongly continuous semigroups. Connections between
semigroups, their generators, and resolvents. On equicontinuous $C_0$-semigroups in
locally convex spaces. Weak continuity.
- Strong resolvent convergence, Trotter-Kato theorems, Chernoff and Lie–Trotter
formulas.
- The Chernoff theorem for equicontinuous semigroups in locally convex spaces.
- The Kühnemund–Wacker theorem. Its applications in the theory of stochastic
differential equations (SDEs) and in quantum mechanics.
- Convergence in the operator norm for the Lie–Trotter formula.
- Semigroups of positive operators on lattices. Generation of positive strongly
continuous semigroups.
- Bi-continuous semigroups and their applications in the theory of random processes.
Trotter–Kato and Chernoff theorems for bi-continuous semigroups.
- Sun-dual theory.
- ntegrated semigroups.
- Dissipative stochastic differential equations in Hilbert space. Existence and uniqueness
of solutions in the weak topology.
- Averaging of random semigroups.
Literature
[1] V.I. Bogachev, O.G. Smolyanov, Real and Functional Analysis: A University Course.
Moscow-Izhevsk: NIC Regular and Chaotic Dynamics, Institute of Computer Science, 2009.
(In Russian)
[2] K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution
Equations (Graduate Texts in Mathematics, Vol. 194). Springer, New York, 2000.
[3] K. Yosida, Functional Analysis (Classics in Mathematics). Springer-Verlag, Berlin, 1995.
[4] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis.
Academic Press, 1980.
 RSS: Forthcoming seminars
 Seminar organizers
Amosov Grigori Gennadievich
Utkin Andrey Vladimirovich
Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center |
