These lecture notes are dedicated to the mathematical modelling, analysis and computation of interfaces and free boundary problems appearing in geometry and in various applications, ranging from crystal growth, tumour growth, biological membranes to porous media, two-phase flows, fluid-structure interactions, and shape optimization.

We first give an introduction to classical methods from differential geometry and systematically derive the governing equations from physical principles. Then we will analyse parametric approaches to interface evolution problems and derive numerical methods which will be thoroughly analysed. In addition, implicit descriptions of interfaces such as phase field and level set methods will be analysed. Finally, we will discuss numerical methods for complex interface evolutions and will focus on two phase flow problems as an important example of such evolutions.

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Eberhard B�nsch studied Mathematics at the University of Bonn and finished his PhD in 1990 as a student of Hans Wilhelm Alt. After that, he went to the University of Freiburg, where he received his habilitation in 1998. The same year, he went to Bremen for his first professorship. In 2000 he accepted an offer from Berlin, where he was appointed full professor at the Free University of Berlin and at the same time head of the research group Numerical Mathematics and Scientific Computing at the Weierstrass-Institute. He was also a member of the research center MATHEON in Berlin. Since 2004 he is full professor at the Friedrich-Alexander-University at Erlangen-Nuremberg.

The scientific interest of Eberhard B�nsch lies in the field of Numerical Analysis and Scientific Computing for free boundary and interface problems, in particular in computational fluid dynamics.

Klaus Deckelnick studied mathematics at the University of Bonn, where he received his PhD in 1990. He subsequently worked as a scientific assistant at the University of Freiburg completing his habilitation in 1996. In 1998, he joined the University of Sussex where he had previously spent one year as a research fellow. Since 2002, he is a professor at the University of Magdeburg.

Klaus Deckelnick works on the analysis and numerical analysis of nonlinear partial differential equations with a particular focus on geometric evolution equations.

Harald Garcke studied Mathematics at the University of Bonn and finished his PhD in 1993 as a student of Hans Wilhelm Alt. With an ESF-fellowship, he was in 1993/94 post-doc with Charles M. Elliott at the University of Sussex and from 1994 he was scientific assistant in Bonn where he finished his habilitation in 2000. In the year 2001, he got offers for professur-positions at the Universities Regensburg and Duisburg. Since 2002, he is full professor at the University of Regensburg where hewas dean of the Mathematics department from 2005 to 2007. He is DFG liaison officer at the University of Regensburg since 2011.

Harald Garcke works on nonlinear partial differential equations, free boundary problems, phase field equations, numerical analysis and geometric evolution equations.

Paola Pozzi received her undergraduate education in Mathematics at the University of Bologna in Italy. As a recipient of an Australian National University PhD Scholarship and an International Postgraduate Research Scholarship she could then move to Canberra to work on her dissertation under the supervision of John Hutchinson at the Australian National University. After receiving her PhD in 2004 and a postdoctoral fellowship at the ANU Mathematical Science Institute, she moved to Germany to work as a scientific assistant in the research group of Gerhard Dziuk at the University of Freiburg. In 2009-10 she was appointed for six-month as a substitute professor (Vertretungsprofessur) at the Technische Universit�t M�nchen. In 2011 she got offers for professor positions at the Freie Universit�t in Berlin and at the University of Duisburg-Essen. She has been working as a professor at the University of Duisburg-Essen since 2011 and has acted as vice-dean (Prodekanin) of the faculty of mathematics between 2018 and 2020.

Paola Pozzi works on analysis and numerical analysis for free boundary problems, geometric evolution equations, and nonlinear partial differential equations.

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