Abstracts of Talks at the Cornell Topology Festival, May 3-5, 2002
MEASURED LAMINATIONS AND GROUP THEORY
- Abstract not available.
THE COMBINATORICS OF SMOOTH
MANIFOLDS: ORIENTED MATROIDS IN TOPOLOGY
- The problem of finding a combinatorial formula for the rational Pontrjagin
classes was solved in the early 90's by Gelfand and MacPherson; their
solution makes essential use of combinatorial objects called oriented
matroids. We show that the oriented matroids in question actually
determine the tangent bundle of a smooth manifold; we will also discuss
potiential applications of this result to the topology of diffeomorphism
ISOPERIMETRIC INEQUALITIES FOR NILPOTENT GROUPS
- This is joint research with D. F. Holt and T. R. Riley.
- We prove that every finitely generated nilpotent group of class
c admits a polynomial isoperimetric function of degree c+1 and a
linear upper bound on its filling length function.
- Paper reference: arXiv:math.GR/0201261
THE TOPOLOGY OF GROUPS OF SYMPLECTOMORPHISMS
- A symplectomorphism is a diffeomorphism of a manifold
that preserves a symplectic form. Ever since Gromov showed that
the group of symplectomorphisms of the product of two 2-spheres of equal size
has the homotopy type of an extension of SO(3) x SO(3) by Z/2Z,
people have been interested in understanding the special properties of
groups of symplectomorphisms. This survey talk will describe some ways
in which the structure of a group of symplectomorphisms differs from that of
an arbitrary diffeomorphism group.
ON THURSTON'S ENDING LAMINATION CONJECTURE
- The classification theory of hyperbolic 3-manifolds (with finitely
generated fundamental group) hinges on Thurston's conjecture from the late
70's, that such a manifold is uniquely determined by its topological type
and a finite number of invariants that describe the asymptotic structure
of its ends.
- We will describe the recent proof of this conjecture, in the
"incompressible boundary case", in joint work with J. Brock and R. Canary.
HOLOMORPHIC DISKS AND LOW-DIMENSIONAL TOPOLOGY
- I will discuss recent work with Zoltan Szabo, in which we use techniques
from symplectic geometry -- holomorphic disks, and Lagrangian Floer homology
-- to construct topological invariants for three-and four-manifolds. These
invariants yield many of the four-dimensional
results which have been proved using their gauge-theoretic
and Seiberg-Witten theory), though the new invariants are constructed using
more topological and combinatorial input, rendering them easier to calculate.
I hope to focus on some of their applications to three-dimensional topological
questions which have not been addressed by gauge theory.
THE NOVIKOV CONJECTURE AND GEOMETRY OF GROUPS
- I will explain what is the Novikov conjecture, why it is
interesting and how it is related to geometry of groups.
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Last modified: Thu May 9 12:20:38 EDT 2002