Title: Configuration spaces and braid groups of graphs

Abstract: The classical braid groups can be described as fundamental groups of certain topological spaces, namely configuration spaces of points in the plane. I will discuss a theory of braid groups of graphs, developed from this point of view. The configuration spaces of graphs have lots of natural structure: algebraically, their fundamental groups split as graphs of groups; geometrically, the spaces themselves have non-positive curvature; and topologically, the spaces are aspherical. Some beautiful examples will emerge, including a configuration space which is homeomorphic to a closed orientable surface. Flows on these spaces and applications will briefly be discussed.

Title: Morse theory and taut foliations

Abstract: We develop a Morse theory for taut foliations on 3-manifolds. In particular we describe the Godbillon-Vey invariants of these foliations purely in terms of the morse theoretic deta. For foliations transversal to the Þbers of a circle bundle over a surface, it coincides with the Bott-Thurston cocycle.

Title: Symbolic dynamics from low dimensional expansion

Abstract: The Conley index is an index for isolated invariant sets. It is an algebraic topological index based on homology. We use the spectral radius of the Conley index to bound the topological entropy of a smooth map. When the spectral radius of the index on the Þrst level of homology is greater than one and there is a neighborhood of the invariant set with no intrinsic homology, we show the existence of symbolic dynamics.

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Title: Aspects of Minimum Volume in Higher Rank

Abstract: I will begin by presenting metric rigidity results in the case of a certain class of higher rank symmetric spaces. The proof further develops some of the techniques introduced by Besson, Courtois and Gallot. I will then show how these techniques can be altered to prove positivity of the minimum volume of manifolds of higher rank. This provides an alternate approach to that of Gromov's Þlling methods.

Title: Legendrian Knots, Contact Homology and Orientations.

Abstract: Contact homology is a powerful new invariant of contact structures and Legendrain submanifolds. Given a contact structure one counts various interesting þow lines in a Reeb vector Þeld associated the the contact structure. This information is then assembles into a differential algebra whose homology does not depend on the choice of Reeb vector Þeld. Though there are many technical difÞculties in establishing a general theory of contact homology one may reformulate it in combinatorial terms for Legendrian knots.

Until recently this combinatorial version has been done in terms of algebras over $Z_2$ (in other words, orientations were not taken into account). In joint work with Sabloff, I have introduced orientations into this combinatorial contact homology.

In this talk I will give a brief overview of the general theory of contact homology and then discuss the oriented combinatorial theory. As time permits I will discuss applications and potential generalizations.

Title: R-covered foliations in 3-manifolds and transverse pseudo-Anosov flows

Abstract: A codimension one foliation in a 3-manifold M is R-covered if the lifted foliation in the universal cover has leaf space homeomorphic to the set of real numbers. For example 1) Þbrations over the circle, 2) large classes of stable/unstable foliations associated to Anosov þows, 3) foliations induced by slitherings over the circle and many others. We prove a general result about the transverse geometry of such foliations: If M is aspherical, then either there is a Z+Z subgroup of the fundamental group of M or there is a pseudo-Anosov þow transverse to the foliation. In this sense general R-covered foliations behave very much like Þbrations. Using a result of Gabai and Kazez, this theorem implies that the underlying manifolds satisfy the weak hyperbolization conjecture. To prove the theorem we use the universal circle of the foliation which encodes the variation of the ideal geometry from leaf to leaf. In the atoroidal case we produce a pair of laminations transverse to the original foliation and transverse to each other. The complementary regions of the union of the laminations can be blowed down to produce a þow which is transverse to the original foliation and which is dynamically a pseudo-Anosov þow.

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Title: 3-Manifolds Which Admit Quasigeodesic Flows

Abstract: We will discuss some ways to construct quasigeodesic flows on certain classes of 3-manifolds.

Title: Periodic orbits of a particle in a magnetic field

Abstract: We will discuss the question of existence of periodic orbits for a family of Hamiltonian dynamical systems generalizing those which describe the motion of a charged particle in a magnetic Þeld. In particular, we will focus on some recent progress by V.L. Ginzburg and the speaker for the problem on low energy levels in dimensions greater than two. This work also leads to a Þrst extension of the now classical Weinstein-Moser Theorem to higher dimensional critical submanifolds.

Title: Homeomorphisms inducing all link types

Abstract: Let $\phi: D^2 \rightarrow D^2$ be an orientation preserving homeomorphism of the disk into itself, and $\Phi = \{ \phi_t\}_{0 \le t \le 1}$ an isotopy with $\phi_0 = id_{D^2}$ and $\phi_1 = \phi$. Then for a Þnite union of periodic orbits $P$ of $\phi$, the set ${\cal S}_{ \Phi } P = \bigcup_{0 \le t \le 1 } (\phi_t(P) \times \{t\}) / (x,0) \sim (x,1)$ is a link in $D^2 \times S^1$. We say that $\phi$ {\it induces all link types} ({\it for} $\Phi$) if there exists a homeomorphism $h$ of $D^2 \times S^1$ into a standardly embedded solid torus in the $3$-sphere $S^3$ such that any link $L$ in $ S^3$ can be realized by a Þnite union of periodic orbits $P_L$ of $\phi$ so that $L$ and $h({\cal S}_{ \Phi } P_L)$ are equivalent. We will show that any pseudo-Anosov homeomorphism relative three points in the interior of $D^2$ induce all link types.

Title: Knots, Links and Symbolic Dynamics II: Mahler Measure and Alexander Polynomials (with S. Williams)

Abstract: Let $l$ be an oriented link of $d$ components. We deÞne the {\it Alexander group} $\cal A$ of $l$, a close relative of the commutator subgroup of the link group. For any topological group $S$, the space $Hom(A,S)$ of representations of $\cal A$ into $S$ admits a natural $Z^d$-action. When $S$ is the circle group ${\bf R}/{\bf Z}$, the periodic point structure gives information about the homology of branched Þnite abelian covers of $l$. Using the action we give a topological interpretation of the Mahler measure of the Alexander polynomial of the link.

Title: Twist-wise flow equivalence

Abstract: A self homeomorphism the Cantor set, h:C &emdash;> C, can be suspended to produce a þow on a one-dimensional space. (Such objects arise as basic sets of Smale þows on a manifold.) Two such þows are topologically equivalent if there is an orbit preserving homeomorphism between them. A square nonnegative integral matrix can be associated to a Markov partition for h. Two such matrices are said to be þow equivalent if they give rise to topologically equivalent suspension þows. John Franks has devised a complete set of computable invariants to detect þow equivalence of irreducable square nonnegative integral matrices.

We ask what happens when the matrix encodes additional information about h. In particular, orientation data is considered: in Smale þows there are stable and unstable manifolds for each orbit in a basic set. These can twist around. This twisting data is encoded in matrices with entries of the form a+bt, where a and b are nonnegative integers and all calculations are done modulo t^2 = 1. Then one can deÞne twist-wise þow equivalence of such matrices. Several invariants will be deÞned, but they are not known to be complete.

Title: Topology of cyclic configuration spaces and periodic trajectories of multi-dimensional Birkhoff billiards

Abstract: This is a report on a joint work with M. Farber. We give lower bounds on the number of periodic trajectories in strictly convex smooth multi-dimensional billiards in Euclidean space. For plane billiards such bounds were obtained by G. Birkhoff in the 1920's. The proof is based on topological methods of calculus of variations - equivariant Morse and Lusternik-Schnirelman theories. Namely, we compute the equivariant cohomology ring of the cyclic conÞguration space of the sphere.

Title: Morse theory on the space of braids and an application to Lagrangian dynamics

Abstract: For a large class of second order Lagrangian systems one may reformulate the problem of Þnding periodic solutions as a problem in solving second-order recurrence relations satisfying a monotonicity condition. We project periodic solutions of such recurrence relations onto the space of closed braid diagrams. Under this reformulation, one can construct a þow on the space of closed piecewise linear braid diagrams. The monotonicity condition shows that the word metric in the braid group corresponds to a weak Lyapunov function for the induced þow, which in turn gives rise to a Morse theory on the space of closed braid diagrams.

We will explain this Morse theory in detail and give applications to second order Lagrangian systems. For general conditions, one obtains results of the form ``any periodic solution(s) which gives rise to a nontrivial braid implies inÞnitely many other periodic solutions.'' We reÞne this into information about Conley indices and braid types of periodic solutions.

Title: Knots, Links and Symbolic Dynamics I: Knot Group Representations and Shifts of Finite Type (with D. Silver)

Abstract: Let $K$ be the commutator subgroup of a knot group. The set $Hom(K, S)$ of representations of $K$ into a Þnite group $S$ has the structure of a shift of Þnite type $\Phi_S$, a special type of dynamical system completely described by a Þnite directed graph. Invariants of $\Phi_S$, such as its topological entropy or the number of periodic points of a given period, determine invariants of the knot. A similar construction for oriented links yields a $Z^d$-shift of Þnite type.