### About the Internet Seminar

Organised by the European Consortium "Internet School on Evolution Equations", the Internet Seminar is an international academic event dedicated to modern analysis. It was started in 1997 by the functional analysis group of Tübingen (lead by Rainer Nagel) and, since then, has been held every year, seeing the participation of more and more universities from all around the world. The aim of the course is to introduce master students, Ph.D students and post-docs to subjects related to functional analysis and evolution equations. (For past iterations see here.)

This year's seminar is structured into three phases:

- In Phase 1 (October 2017 - February 2018), a weekly lecture will be provided, accompanied by exercises to be solved by the participants and by additional background material. For more details, the lectures and solutions you can click here or use the button on the left-hand side.
- In Phase 2 (April - June 2018), the participants will form small international groups to work on projects in which the material of Phase 1 is complemented, deepened or applied. For more information you can click here or use the button on the left-hand side.
- Finally, Phase 3 (30.06. - 06.07.2018) consists of the
**Final Workshop in Wuppertal (Germany)**. To learn more you can click here or use the button on the left-hand side.

### Programme of the course

In the 21st Internet Seminar we study the theory and applications of functional calculi of (bounded and unbounded) linear operators on Banach spaces. Roughly speaking, a functional calculus is a consistent way of defining operators of the form _{ }for a given operator and some class of scalar-valued functions _{} such that relations between the functions _{} translate into according relations of the operators _{} .

The most prominent example of such a calculus is the Borel calculus for a self-adjoint operator on a Hilbert space, but many important functional calculi (which allow to cover operator semigroups, fractional powers like the square root of an operator, the operator logarithm) can be constructed under much weaker assumptions on the operator. These functional calculi are of particular importance for the theory of evolution equations, where quite many problems can be reduced to the question whether an operator of the form _{} is bounded or not. Often, this question reduces to a vector-valued singular integral, and hence results from harmonic analysis enter the scene. (To download the programme click here.)

Your lecturer will be

### Prerequisites

The lectures will be on a beginning graduate level. Familiarity with functional analysis and basic complex analysis is assumed, some knowledge of Fourier analysis is helpful but not strictly necessary.

If you have questions or remarks you can contact us under the following address: