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Papers by Mike Sullivan

$Research paper thumbnail of A Mathematician Reads \u3cem\u3eSocial Text\u3c/em\u3e$

An analysis of the issue of the postmodern journal Social Text in which physicist Alan Sokal publ... more An analysis of the issue of the postmodern journal Social Text in which physicist Alan Sokal published his spoof article

$Research paper thumbnail of Visually Building Smale Flows in S\u3csup\u3e3\u3c/sup\u3e$

A Smale flow is a structurally stable flow with one dimensional invariant sets. We use informatio... more A Smale flow is a structurally stable flow with one dimensional invariant sets. We use information from homology and template theory to construct, visualize and in some cases, classify, Smale flows in the 3-sphere

$Research paper thumbnail of Knots about Stokes\u27 Theorem$

Connections between knot theory and Stokes\u27 Theorem as presented to students in a sophomore le... more Connections between knot theory and Stokes\u27 Theorem as presented to students in a sophomore level vector calculus class

$Research paper thumbnail of Positive Knots and Robinson\u27s Attractor$

We study knotted periodic orbits which are realized in an attractor of a certain ODE. These knots... more We study knotted periodic orbits which are realized in an attractor of a certain ODE. These knots can be presented so as to have all positive crossings, but may not be restricted to positive braids

This is an expository account of a theorem of Louise Moser that describes the types of manifolds ... more This is an expository account of a theorem of Louise Moser that describes the types of manifolds that can be constructed via Dehn surgery along a trefoil in the 3-sphere. These include lens spaces, connected sums of two lens spaces, and certain Seifert fibered spaces with three exceptional fibers. Various concepts from the topological theory of three dimensional manifolds are developed as needed.

Suppose that φ is a nonsingular (fixed point free) C1 flow on a smooth closed 3-dimensional manif... more Suppose that φ is a nonsingular (fixed point free) C1 flow on a smooth closed 3-dimensional manifold M with H2(M)=0. Suppose that φ has a dense orbit. We show that there exists an open dense set N ⊆ M such that any knotted periodic orbit which intersects N is a nontrivial prime knot.

Abstract. The aim of this paper is to show that any two knots can be realized as an attractor and... more Abstract. The aim of this paper is to show that any two knots can be realized as an attractor and repeller pair for some nonsingular Smale flow on S 3 with any linking number. We view this as progress, albeit limited, to the conjecture that all two component links can be realized as an attractorrepeller pair in a nonsingular Smale flow on S 3 with just one other basic set of saddle type. 1.

Open Problems in Topology II, 2007

After a brief survey of various types of flows (Morse-Smale, Smale, Anosov, & partially hyperboli... more After a brief survey of various types of flows (Morse-Smale, Smale, Anosov, & partially hyperbolic) we focus on Smale flows on S 3. However, we do give some consideration to Smale flows on other three-manifolds and to Smale diffeomorphisms.

The aim of this paper is to show that any two knots can be realized as an attractor and repeller ... more The aim of this paper is to show that any two knots can be realized as an attractor and repeller pair for some nonsingular Smale flow on S with any linking number. We view this as progress, albeit limited, to the conjecture that all two component links can be realized as an attractor-repeller pair in a nonsingular Smale flow on S with just one other basic set of saddle type.

Topology and its Applications, 2000

Topology and its Applications, 2014

We study simple Smale flows on S 3 and other 3-manifolds modeled by the Lorenz template and anoth... more We study simple Smale flows on S 3 and other 3-manifolds modeled by the Lorenz template and another template with four bands but that still has cross section a full 2-shift.

Algebraic and Topological Dynamics, 2005

Contemporary Mathematics, 1992

Ill. Series: Contemporary mathematics (American Mathematical Society); v. 135. QA611.5.S96 1992 5... more Ill. Series: Contemporary mathematics (American Mathematical Society); v. 135. QA611.5.S96 1992 514 1 74-dc20 92-20203 CIP Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Manager of Editorial Services, American Mathematical Society,

Topology and its Applications, 1995

A zeta function for a map f : A4-+ A4 is a device for counting periodic orbits. For a topological... more A zeta function for a map f : A4-+ A4 is a device for counting periodic orbits. For a topological flow however, there is not a clear meaning to the period of a closed orbit. We circumvent this for flows which have positive templates by counting the "twists" in the stable manifolds of the periodic orbits.

Topology and its Applications, 1994

We study an Anosov flow 4, in S"-(figure-8 knot}. Birman and Williams conjectured that the knot t... more We study an Anosov flow 4, in S"-(figure-8 knot}. Birman and Williams conjectured that the knot types of the periodic orbits of this flow could have at most two prime factors. Below, we give a geometric method for constructing knots in this flow with any number of prime factors.

Topology and its Applications, 2006

Suppose that ϕ is a nonsingular (fixed point free) C 1 flow on a smooth closed 3-dimensional mani... more Suppose that ϕ is a nonsingular (fixed point free) C 1 flow on a smooth closed 3-dimensional manifold M with H 2 (M) = 0. Suppose that ϕ has a dense orbit. We show that there exists an open dense set N ⊆ M such that any knotted periodic orbit which intersects N is a nontrivial prime knot.

Qualitative Theory of Dynamical Systems, 2008

In this note we apply results of Goodman, Yano and Wada to determine which nonsingular Morse-Smal... more In this note we apply results of Goodman, Yano and Wada to determine which nonsingular Morse-Smale flows on S 3 have transverse foliations. We then observe that there is a connection to flows arising from certain Hamiltonian systems and from certain contact structures.

Proceedings of the London Mathematical Society, 2005

Let G be a finite group. We classify G-equivariant flow equivalence of nontrivial irreducible shi... more Let G be a finite group. We classify G-equivariant flow equivalence of nontrivial irreducible shifts of finite type in terms of (i) elementary equivalence of matrices over ZG and (ii) the conjugacy class in ZG of the group of G-weights of cycles based at a fixed vertex. In the case G = Z/2, we have the classification for twistwise flow equivalence. We include some algebraic results and examples related to the determination of E(ZG) equivalence, which involves K 1 (ZG). Contents 1. Introduction 1 2. G-flow equivalence and SFTs 3 3. Positive equivalence 9 4. The weight class 12 5. Equivalence through very positive matrices 15 6. The main results 21 7. Twistwise flow equivalence 25 8. E(ZG)-equivalence 27 9. E(ZG)-equivalence of injective matrices 32 References 36