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# 23rd Internet Seminar "Evolutionary Equations"

## Description of the Course

The 23rd Internet Seminar on Evolution Equations is devoted to the treatment of Evolutionary Equations. These are ordinary or partial differential equations written in the form

$$(\partial_t M(\partial_t) +A) U = F.$$

Such equations include, in particular, the heat equation, the wave equation, Maxwell's equations, but also mixed type equations.

The theory is based on a Hilbert space approach using the Fourier-Laplace transform. After introducing unbounded operators and the time derivative $\partial_t$ in this context, we first study ordinary differential equations, including time delays. We then present the abstract solution theory for partial differential equations and apply it to various examples. Further applications we have in mind are differential-algebraic equations, aspects of homogenization and – of course – the relation of the theory to $C_0$-semigroups.

This recent space-time framework to treat partial differential equations is based on the seminal paper by Picard in 2009 [Math. Methods Appl. Sci. 32 (2009), 1768–1803]. The developed insights have led to new results in control theory, stochastic partial differential equations, homogenisation theory, differential algebraic equations, delay differential equations and also non-linear partial differential equations.

We expect from the participants basic knowledge in functional analysis (bounded operators, uniform boundedness principle, closed graph theorem, Hahn-Banach theorem), foundations of Hilbert spaces as well as foundations in complex analysis of one variable.

## Structure of the Internet Seminar

The annual Internet Seminars introduce master, Ph.D. and postdoc students to varying subjects related to evolution equations. The course consists of three phases.

• Phase 1 (October-February): A weekly lecture will be provided via the ISem website. These lectures will be self-contained, and references for additional reading will be provided. The weekly lecture will be accompanied by exercises, and the participants are supposed to solve these problems.
• Phase 2 (March-June): The participants will form small international groups to work on diverse projects which supplement the theory of Phase 1 and provide some applications.
• Phase 3 (June 21 to June 27, 2020): Final one-week workshop at the Bundeshöhe in Wuppertal (Germany). There the project teams of Phase 2 will present their projects and additional lectures will be delivered by leading experts.

The ISem team of 2019/20 consists of

• Christian Seifert (Hamburg)
• Sascha Trostorff (Kiel)
• Marcus Waurick (Strathclyde)

The website of the 23rd ISem is http://www.mat.tuhh.de/isem23

## Registration

The registration for the 23rd ISem is now closed. (Registration was open until the end of October.) If you still want to register, please send us an e-mail to isem23@tuhh.de.

The first lecture will be delivered mid October.