Abstracts of Talks at the Cornell Topology Festival, May 1-4, 2003
DROR BAR-NATAN
THE UNREASONABLE AFFINITY OF KNOT
THEORY AND THE ALGEBRAIC SCIENCES
- Click here for the abstract and some of the transparencies.
JOAN BIRMAN
STABILIZATION IN THE BRAID GROUPS
- Solved problems abound in topology where there is a notion of
"stabilization". Examples: the Reidemeister-Singer Theorem relating any
two Heegaard splittings of a 3-manifold, the Kirby Calculus relating any
two surgery presentations of a 3-manifold,and Markov's Theorem, relating
any two closed braid representatives of a knot. This talk will report on
new joint work with William Menasco. Our main result is "Markov's
Theorem Without Stabilization." We replace the stabilization move in the
Markov theorem by finitely many moves which are strictly "complexity"
reducing (and non-increasing on braid index). The statement of the
theorem is too messy for this abstract, which is another way of saying
that it's a very hard problem to decide when two knot diagrams
determine the same knot type! As an application we solve a classical open
problem about knots transverse to the standard contact structure.
FRED COHEN
BRAID GROUPS, THE TOPOLOGY OF
CONFIGURATION SPACES, AND HOMOTOPY GROUPS
- This talk explores how braid groups encode maps from spheres
to other natural spaces. One example given by elementary
"cabling of braids" encodes information about maps from large
dimensional spheres to the 2-sphere. An overview of these
structures, as well as connections to Vassiliev invariants
of pure braids and associated Lie algebras will
be given. This talk is based on joint work with J. Berrick,
Y. L. Wang, and J. Wu.
BENSON FARB
HIDDEN SYMMETRIES OF RIEMANNIAN MANIFOLDS
- Which Riemannian manifolds have the most symmetry? Does
each smooth manifold have a metric with biggest isometry
group? with most isometries of finite-sheeted covers? Are
such metrics unique? In this talk I will describe joint
work with Shmuel Weinberger which addresses these
questions. Our main result is essentially the computation
(up to finite index) of the isometry group of every
finite-volume Riemannian manifold, and that of its
universal cover.
GILBERT LEVITT
AUTOMORPHISMS OF CANONICAL SPLITTINGS
- Abstract not yet available.
JOHN MEIER
ASYMPTOTIC COHOMOLOGY FOR THE MOTION GROUP OF A TRIVIAL n-COMPONENT LINK
- Let Ln denote n unknotted, unlinked circles in
S3. Much as one can describe the n-strand braid group as
the motion group of n points in a disk, one can define the motion
group of Ln in S3: one starts a movie, where the
n circles are allowed to fly about in 3-space, never intersecting,
and returning to their original configuration after one unit of time.
In earlier work Brady, McCammond, Miller and myself have shown that
this group is a virtual duality group. The proof uses a spectral
sequence approach that ultimately depends on understanding something
about the combinatorics of a certain poset of trees, the
Whitehead poset. In recent work, McCammond and I have
returned to this group to compute its L2-Betti numbers,
which ultimately also boils down to interesting combinatorics of the
Whitehead poset. Both cohomology computations will be outlined, but
the focus will be on the connection with combinatorial ideas.
JUSTIN ROBERTS
ROZANSKY-WITTEN THEORY
- Rozansky and Witten proposed in 1996 a family of new
three-dimensional topological quantum field theories, indexed by
compact (or asymptotically flat) hyperkaehler manifolds. As a
byproduct they proved that hyperkaehler manifolds also give rise to
Vassiliev weight systems. (These may be thought of as invariants of
hyperkahler manifolds, so the theory is of interest to geometers as
well as to quantum topologists.) I will explain the geometrical
construction of the weight systems, how they may be integrated into
the framework of Lie algebra weight systems (joint work with Simon
Willerton), and an approach to a rigorous construction of the TQFTs
(joint work with Justin Sawon and Simon Willerton). I will also
describe Kontsevich's approach using Gelfand-Fuchs cohomology.
DYLAN THURSTON
HOW EFFICIENTLY DO 3-MANIFOLDS
BOUND 4-MANIFOLDS?
- It has been known since 1954 that every 3-manifold bounds a
4-manifold. Thus, for instance, every 3-manifold has a surgery
diagram. There are many proofs of this fact, including several
constructive ones, but they generally produce exponentially
complicated 4-manifolds. Given a 3-manifold M of complexity n, we
show how to construct a 4-manifold bounded by M of complexity
O(n2) for reasonable definitions of "complexity. (For instance,
one notion of complexity is the number of tetrahedra in a
triangulation of M.) It is an open question whether this quadratic
bound can be replaced by a linear bound.
ULRIKE TILLMANN
THE TOPOLOGY OF THE SPACE OF STRINGS
- Strings moving in a background space M give rise to a
category which is the underlying object of study in elliptic
cohomology, Gromov-Witten theory and Chas-Sullivan theory. I will
explain this and discuss the topology of its classifying space.
ALAIN VALETTE
VANISHING RESULTS FOR THE FIRST
L2 BETTI NUMBER OF A GROUP
- This is a report on work in progress, jointly with Florian
Martin. Let
B1 be the class of finitely generated groups G for which the first
L2-Betti number is zero. This research started from an attempt to give a
simplified proof for Gaboriau's result: if G has an infinite, finitely
generated subgroup which is normal and of infinite index, then G is in
B1. Our idea is to exploit systematically the isomorphism
between the first
L2-cohomology of G and the reduced Hochschild 1-cohomology of G with
coefficients in its left regular representation. Although the original attempt
failed so far, the approach already gave the following results:
- if N < H < G, where N is an infinite, normal
subgroup of G, and H is in B1, then G is in
B1;
- wreath products are in B1;
- if G acts on a tree with infinite edge stabilizers and
vertex stabilizers in B1, then G is in B1.
KAREN VOGTMANN
GRAPH HOMOLOGY AND OUTER SPACE
- Kontsevich's graph homology is motivated by considerations from
symplectic geometry and invariant theory, and is computed using chain
complexes of graphs. Different versions of graph homology are obtained
by decorating the vertices of graphs by elements of different cyclic
operads. The Lie version of Kontsevich's complex computes the
homology of the groups Out(F) of outer automorphisms of finitely
generated free groups. The connection with complexes of graphs is
given via Outer space, which is a contractible space on which Out(F)
acts, whose points correspond to isomorphism classes of finite metric
graphs. I will describe joint work with Jim Conant, in which we
carefully work out the link between graph homology and Outer space,
and then exploit it to gain new information about both chain complexes
of graphs and the homology of Out(F).
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Last modified: Thu Apr 24 15:03:26 EDT 2003