Solving Algebraic Equations II

This project shall pursue various objectives on solving algebraic equations: synthetic geometry and visualization, minimal model programming, geometry of arc spaces, resolution of singularities in positive characteristic, Lie algebra methods. The first topic aims at doing algebraic geometry as close to geometry as possible. The power of algebra has occasionally marginalized geometric intuition, geometric curiosity and a geometric way of communication. Nevertheless there are still very intricate geometric questions, often simple to formulate, but generally hard to solve. The "minimal model program" has been initiated by Mori and others as a program to classify varieties birationally. By "minimal model programming", we mean the constructive side of this theoretical program. Arc spaces are sets of solutions of a system of algebraic equations in the domain of power series in one variable. This leads to a study of ideals in a polynomial ring with countably many variables. The relation between the geometry of an algebraic singularity and the geometry of its arc space should be investigated. Resolution of singularities in positive characteristic has become a hot topic by recent results of Bravo/Villamayor, Cossart/Piltant, Cutkosky, Hauser/Wagner and Kawanoue/Matsuki. Our group developed powerful techniques to study this still unsolved problem, and we hope to apply them to get significant results. The purpose of the Lie algebra method is to use Lie algebras of vector fields to determine the isomorphism class of varieties.

Summary for public relations workThe main goal of this project was to develop new methods to get insight into geometric phenomena, so-called singularities, which appear when solving algebraic equations. A singularity of an algebraic variety (=a zeroset of polynomial equations) is a point where the variety is not smooth, i.e., has intersections or cusps. Algebraic varieties are geometric objects which are a main object of study of the field of algebraic geometry.In this project several theoretic techniques were developed to study singular algebraic varieties: infinite dimensional geometry (arc spaces on varieties) was employed to show combinatorial identities; particular singularities, so-called normal crossing divisors, were characterized in a purely algebraic way; the celebrated minimal model program, which seeks for simple models for singularities was made effective in certain cases (minimal model programing). To foster communication between experts we moreover organized in 2011 a three-week long research program about Algebraic Geometry vs. Analytic Geometry at the Erwin Schrödinger International Institute for Mathematical Physics in Vienna.The other main research direction was the problem of resolution of singularities: the problem asks to find for any singular variety a paramatrization by a smooth variety. The problem was solved 50 years ago for the characteristic zero case by Heisuke Hironaka, who was then awarded a Fields medal for his seminal result. But the general case is still open up to the current date. In our project new approaches for the two-dimensional case of this problems were developed and useful studies in higher dimensions (the Kangaroo phenomenon) were carried out. The importance of this problem is highlighted by the fact that in summer 2012 we were able to host the 4 week long Clay summer school The Resolution of Singular Algebraic Varieties in Obergurgl in the Tyrolean Alps. In this international summer school, which was organized by Herwig Hauser and the prestigious Clay Mathematics Institute, about 100 young international researchers came together to learn about this topic.Another goal of this project was to communicate our research field to a general public. Through the IMAGINARY initiative of the Mathematisches Forschungsinstitut Oberwolfach, which started 2008 with a traveling exhibition that still goes on, many school children and members of the interested public could be reached in meanwhile over 60 cities around the world. This exhibition contains in particular visualizations and sculptures of singular algebraic varieties made by our research group. In 2013 the new building for the Faculties of Mathematics and Economic Sciences of the Universität Wien was opened. The sculpture Dodekaederstern, which is an algebraic variety defined by one algebraic equation and illustrates the interplay of various fields of singularity theory, was erected in front of this building after plans of Herwig Hauser.