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A Celebration of the 20th anniversary
of
Dennis Sullivan's appointment as
the Albert Einstein Chair in Science at
The City University of New York
September 20-21, 2002
CUNY Graduate Center
365 Fifth Avenue, New York
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
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