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

1. Fundamental properties of strongly continuous semigroups. Connections between semigroups, their generators, and resolvents. On equicontinuous $C_0$-semigroups in locally convex spaces. Weak continuity.
2. Strong resolvent convergence, Trotter-Kato theorems, Chernoff and Lie–Trotter formulas.
3. The Chernoff theorem for equicontinuous semigroups in locally convex spaces.
4. The Kühnemund–Wacker theorem. Its applications in the theory of stochastic differential equations (SDEs) and in quantum mechanics.
5. Convergence in the operator norm for the Lie–Trotter formula.
6. Semigroups of positive operators on lattices. Generation of positive strongly continuous semigroups.
7. Bi-continuous semigroups and their applications in the theory of random processes. Trotter–Kato and Chernoff theorems for bi-continuous semigroups.
8. Sun-dual theory.
9. ntegrated semigroups.
10. Dissipative stochastic differential equations in Hilbert space. Existence and uniqueness of solutions in the weak topology.
11. 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.

Seminar organizers
Amosov Grigori Gennadievich
Utkin Andrey Vladimirovich

Financial support
The seminar is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, agreement no. 075-15-2025-303).

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center