# Titles and Abstracts

**John Milnor, "The Dynamics of Evolution"**

*Abstract:* Passive evolution (= genetic drift) can best be described as an unbiased random walk on a "genetic space"' which is a cartesian product of spherical simplexes. Driven evolution (or adaptation) can be described as a gradient flow on this same genetic space.

**Michael Shub, "Random versus Mean Exponents for Families of Maps with Symmetry"**

*Abstract:* We examine random versus mean exponents for **SU(n)** invariant families of linear maps and an **SO(3)** invariant family of "twist maps". In the linear case there is an inequality (joint work with J.P.Dedieu), in the twist map case the analogous inequality is false for small values of the parameter, but experimentally appears to be true after a small transitional region. There are partial results along the way concerning the behaviour "at infinity" in the twist map case, and the average of invariant measures in the linear case (joint work with François Ledrappier, Carles Simo and Amie Wilkinson).

**Etienne Ghys, "Commutators and Diffeomorphisms of Surfaces"**

*Abstract:* Many groups of diffeomorphisms are simple groups. In particular, in these groups, any element can be written as a product of commutators. How many commutators does one need? I will focus on the case of area preserving diffeomorphisms of closed surfaces and describe some new dynamical invariants related to this "commutator length".

**Mikhail Lyubich, "Low Dimensional Renormalization: from Conjectures to Theorems"**

*Abstract:* We will discuss developments in the renormalization theory during the past decade.

**Nicola Teleman, "Manifold Structures and Operators on Topological Manifolds"**

*Abstract:* One considers different relevant manifold structures (quasi conformal, Lipschitz, combinatorial, smooth) and corresponding Fredholm operators with the purpose of extracting fundamental invariants of topological manifolds.

**John Morgan, "One dimensional families of Calabi-Yau threefolds"**

*Abstract:* In this talk we discuss complex threefolds which are Calabi-Yau and have $hˆ{2,1}=1$. These come in one-dimensional families. Associating to each variety its period point in the space of all Hodge structures with the given numerical invariants associates to each family a curve in the period space. We shall discuss the possible monodromy representations for these curves as wellas the known examples. We are especially interested in the relationship of these examples to hypergeometric differential equations, and to conjectures arising out of homological mirror symmetry.

**Zhenghan Wang, "Quantum Invariants and Quantum Algorithms"**

*Abstract:* An equivalent model of quantum computing based on topological quantum field theories (TQFTs) has been proposed in the work of Freedman, Kitaev, Larsen and Wang. This new way of looking at quantum computation provides efficient quantum algorithms to approximately compute quantum invariants of links and 3-manifolds.

Most other known efficient quantum algorithms such as Shor's factoring algorithm are based on Fourier sampling of quantum states and can be formulated into a hidden subgroup problem. In this framework, Shor's algorithm corresponds to the cyclic groups. It is known that the hidden subgroup problem can be solved for abelian groups, but it is open for non-abelian groups. We will also discuss the possibility of solving the hidden subgroup problem for non-abelian groups using TQFTs.

**Moira Chas, "Lie bialgebras of curves on surfaces and their computation"**

*Abstract:* On the vector space generated by the free homotopy classes of curves on a surface, Goldman found that a Lie bracket is naturally defined using the intersection points of representatives and the usual loop product at these intersections. Later on, Turaev found a Lie coalgebra structure on the same vector space, using self-intersection points and loop coproduct. Moreover, he proved that the Goldman Lie bracket and the cobracket he discovered satisfy a compatibility equation, yielding a Lie bialgebra. We will present a combinatorial description of this Lie bialgebra, which can be implemented by a computer program. In this way, we found an answer to a question about simple curves posed by Turaev. We may also explore some of the possibilities that this program offers to find new three-manifolds.

**Yair Minsky, "Ends of Hyperbolic 3-Manifolds: Flexibility, Rigidity and Classification"**

*Abstract:* The ends of an infinite-volume hyperbolic 3-manifold have a rich and mysterious geometric structure, which has been studied using methods of complex analysis, dynamics, topology and geometry. Thurston conjectured in the 1980's that this structure is completely classified by "end invariants" which describe its asymptotic properties. Recently, in joint work with J. Brock and R. Canary, we were able to prove this conjecture (in the incompressible-boundary case), using in an essential way the combinatorial structure of the set of closed curves on a surface. I will give an overview of the structure of this field and of these and other developments.

**Graeme Segal, "The Structure Of Manifolds"**

*Abstract:* Our understanding of the structure of high-dimensional manifolds is a beautiful mathematical edifice to which many people have contributed, notably Dennis Sullivan in his earliest works. I shall try to present a short unified perspective on it, and at the same time explain how it fits in with some current ideas in string theory.

**For information and registration:** Please contact Ms. Karen Duhart at (212) 817-8578, fax: (212) 817-1584, email: kduhart@gc.cuny.edu