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 L
_{n}denote n unknotted, unlinked circles in S^{3}. 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 L_{n}in S^{3}: 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 L^{2}-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(n
^{2}) 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
L ^{2} BETTI NUMBER OF A GROUP*

- This is a report on work in progress, jointly with Florian
Martin. Let
B
_{1}be the class of finitely generated groups G for which the first L^{2}-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 B_{1}. Our idea is to exploit systematically the isomorphism between the first L^{2}-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 B
_{1}, then G is in B_{1}; - wreath products are in B
_{1}; - if G acts on a tree with infinite edge stabilizers and
vertex stabilizers in B
_{1}, then G is in B_{1}.

- if N < H < G, where N is an infinite, normal
subgroup of G, and H is in B

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