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. Although the work of Bieri and Strebel suggests that the structure of these finitely presented solvable groups is fairly restricted, very little is known in general about them, despite the existence of a number of negative algorithmic results. The special case of finitely presented metabelian groups, whose structure is closely connected to the structure of finitely generated modules over polynomial rings, algebraic geometry and seemingly also to Hilbert's Tenth Problem are themselves a particularly fascinating class of groups. There is an intriguing geometric invariant due to Bieri, Strebel and Neumann which distinguishes the finitely-presented metabelian groups from the finitely-generated, infinitely-related ones.

In view of the current situation, this suggests that a conference designed to join two rather different points of view, the geometric and the combinatorial, would lead to interesting new directions for research, joining these two points of view together. In addition, the possible connections with Hilbert's Tenth Problem provide the possibility of finding applications of recursive function theory to the study of finitely generated metabelian groups which are extremely promising. ]]>

Counting and distribution problems for word-metric spheres

It is a classical problem to study the growth of the number of words of length n in a word metric; in the 1980s and 90s, there was a surge of activity on the rationality of growth series for many classes of groups. Motivated by wanting to approach probabilistic problems in groups, however, one might want finer information: not only the number of points in the sphere of radius n, but also their distribution. For even the simplest example -- free abelian groups with finite generating sets -- this is already a surprisingly rich problem. I'll discuss the complete solution in that case and partial results for other groups. (Joint work with Lelièvre and Mooney)

The seminar meets at the Graduate Center of the City University of New York. The Graduate Center is located at 365 Fifth Avenue at 34th Street. If you are not affiliated with the City University of New York, you should allow a few extra minutes to sign in with the guards at the entrance. ]]>

**Abstract: **We start describing two infinite surfaces/billiards: The Ehrenfest Windtree model (billiard) and the folded plane, which is a half-translation surface essentially introduced by Panov. Those two models pretty much "look" the same, but carry very different dynamical behavior in certain directions. The unfolding of the Ehrenfest Windtree model and the folded plane are connected by a W-like covering diagram. If we fix certain parameters both surfaces contain the "same" pseudo-Anosov, the eigen-directions of which define path of very different bahavior: On one model such a path is escaping, on the other it is dense.

We describe the two models and the mechanism which gives the extremely different orbit behavior in the same direction.

This is joint work with Chris Johnson.

]]>**Title:** On Some Monoids Associated to Coxeter Groups

These geometric and combinatorial objects play a crucial role in the representation theory of W and other connections between W, geometry and algebraic combinatorics. In recent years it has been recognized that both C(W) and the Bruhat order have a natural structure of a monoid and that these monoids carry information about W, that the group structure alone can not see. We will define and explain the basic properties of these monoids and discuss their representation theory.

]]>**Title:** *About the Calabi-Yau theorem : a numerical approach.*

**Abstract:** To solve the Calabi conjecture, it is necessary to study a highly non linear PDE. S.T. Yau's proof is unfortunately non constructive. We will explain how S.K. Donaldson's iterative scheme provides actually a constructive solution and open a new way to investigate similar PDEs.

**Abstract: **In 1980 Smale studied the solution of a one complex polynomial in one variable. When we adapt his technique to a natural generalization for homogeneous polynomials many new open problems are encountered on the Riemann sphere. I will discuss the problems and their application to the complexity of equation solving.

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