This project is to support a conference on Finitely presented solvable groups at The City College of New York, Fall 2010. In his address to the International Congress of Mathematicians in 1983, Gromov talked about groups as geometric objects. This followed on his proof in 1981 that finitely generated groups of polynomial growth contain a nilpotent subgroup of finite index, which has helped to focus attention on the extent to which the asymptotic properties of a finitely generated solvable group has on its algebraic structure. This has led to a number of results dealing with the quasi-isometries and rigidity of solvable groups.Some of these results have been motivated by ideas coming out of the theory of lie groups, where the semi-simple ones exhibit a rigidity that is not shared by the solvable ones. This ongoing geometric study of what are perhaps the simplest finitely generated solvable groups has put into sharp relief the very nature of finitely presented solvable groups.
Groups arise in many different areas of mathematics, in physics, in chemistry and in chrystallography among other disciplines. One of the reasons for this is that they can be used as a tool for understanding and defining symmetry. Floor patterns in many churches display a symmetry which can be analyzed and better understood by using group theory and it is this kind of symmetry that groups capture, but in a more abstract way. Groups also display an innate symmetry of their own and in recent years efforts have been made to connect geometry to group theory. This is an important area of current research. The objective of this conference is to join the existing combinatorial and theoretical approach of group theory to the geometric approach. The latter requires a great deal of mathematical machinery. The work that is currently being undertaken involves solvable groups introduced by Evariste Galois (who died in 1832 in a duel aged 21) to determine the nature of the solutions of everyday polynomial equations. The aim of the conference is to make the two aspects of group theory available to graduate students, postdocs and interested professionals.
This award supported a two-day conference on finitely-presented solvable groups at The City College of New York in Spring 2011. Mathematical groups are abstract objects that capture the essense of symmetry and are foundational part of abstract mathematics. Such groups can be classified into broad categories, each with interesting aspects and applications, requiring different expertise for understanding and analysis. Infinite finitely-presented solvable groups are important examples where treating abstract groups as geometric objects has been very helpful in better understanding, following the groundbreaking work and suggestions of Gromov more than 20 years ago. Solvable groups arose first in the work of Galois. His aim was to connect solving equations, the most well known of which are quadratic equations, with group theory. It is known to high school students that the solutions of such quadratic equations can be described by simple formulas involving the coefficients of the polynomials involved. Galois, in work that has revolutionized many parts of mathematics, showed that formulas do exist for the solutions of polynomial equations in many variables if a group that he associated with these equations is solvable. Solvable groups are more complicated analogues of so-called commutative groups, where the product is like that of the integers and independent of the order in which the product is carried out. Different communities of researchers have approached questions about finitely-presented solvable groups with different approaches, and this conference brought together researchers from a wide range of backgrounds to foster collaboration and better understanding. The two main approaches to understanding infinite finitely-presented solvable groups are geometric and combinatorial, with only limited interaction between researchers employing the different approaches. The ten speakers at the conference came from across the country and around the world and are top-level researchers at the forefront of the area. Particular effort was made to include mathematicians who use different tools to attack these problems, with researchers whose approaches being primarily combinatorial and primarily geometric both included. The presentations covered both their successful work and questions about possible directions for further research. The organization of the conference encouraged a great deal of informal interaction between the experts and the conference participants. Furthermore, great effort was made to ensure that their was sufficient background and context the conference to be valuable for graduate students in mathematics and mathematical researchers who were not already experts in the area. The bulk of the expense of the conference came from the City College of New York. This award was used to support graduate students in mathematics and early-career mathematical researchers at American institutions to attend the conference, giving them the opportunity to develop and learn about this cutting-edge research. It also helps to integrate students and postdoctoral researchers in the community of researchers in the field and foster new collaborations which may last for many years.
