The Weizmann Institute of Science Faculty of Mathematics and Computer Science Algebraic Geometry and Representation Theory Seminar Lecture Hall, Room 1, Ziskind Building on Monday, May 21, 2012 at 15:00 Note the unusual time and place Maxim Leyenson will speak on On ramified covers of algebraic surfaces, and combinatorics of some infinite discrete groups. Abstract: Bernhard Riemann studied (complex) algebraic curves by considering ramified covers of the projective line. In dimension two, one of the classical approaches of the Italian school of algebraic geometry was to study (complex) algebraic surfaces by considering ramified covers of the simplest surface -- the projective plane. Such ramified covers are classified, in any dimensions, by suitable representations of some (usually very large!) discrete group $\Gamma$ into a symmetric group $Sym_n$. Here $\Gamma$ is the fundamental group of a complement to the branch divisor. Now, a striking difference appears between dimensions one and two. It was discovered by Enriques, Castelnuovo and, especially, Zariski. First, any collection of points on a projective line is a ramification locus of some ramified cover. The corresponding fundamental group is the free group $F_n$, which always has lots of representations into the symmetric group $Sym_n$. In particular, the construction gives lots of complex curves. There were many studies of the corresponding representations. The case of dimension two is very different. Only very special plane curves are ramification divisors, and only very special groups are fundamental groups. Moreover, there are restrictions on representations. We argue that the groups $\Gamma$ appearing here are (a very broad) generalizations of the Artin's braid group, and define the corresponding class of discrete groups. We call them twisted Artin-Brieskorn groups. Then we discuss combinatorics of their representations into symmetric groups $Sym_n$, and discuss the related Chisini conjecture of algebraic geometry. --------------------------------------------------------- Technion Math Net-2 (TECHMATH2) Editor: Michael Cwikel <techm@math.technion.ac.il> Announcement from: Yaeli Malka <yaeli.malka@weizmann.ac.il>