** Markus Brodmann**

University of Zuerich, Switzerland

July 7, 2016

**Projective Varieties of Maximal Sectional Regularity**:
We give a survey on joint work with Wanseok Lee, Euisung Park and Peter Schenzel. By Gruson-Peskine-Lazarsfeld, projective curves of maximal (Castelnuovo-Mumford) regularity were classified. Our aim is, to extend this classification to projective varieties of higher dimension. We therefore study $n$-dimensional irreducible non-degenerate projective varieties $X subset mathbb{P}^r$ with the property that for a general linear subspace $mathbb{L} in mathbb{G}(r-n+1, mathbb{P}^r)$ of $mathbb{P}^r$ with dimension $r-n+1$ the intersection $X cap mathbb{L} subset mathbb{L}$ is a curve of maximal regularity. We classify such varieties and describe their geometric, homological and cohomological properties. We also mention a few open problems concerning our subject.