#### by Michael C. Sullivan

I started my Ph.D. at the University of Texas at Austin in 1986, the year
before
Professor Williams moved there from Northwestern University. I was
drawn to dynamical system from talks he
gave, and I asked to
take a reading course with him. We worked through a book by
John Guckenheimer
and
Philip Holmes,
*Nonlinear Oscillations, Dynamical
Systems, and Bifurcations of Vector Fields*
[e1].
The first paper I read was
*Knotted periodic orbits in suspensions of Smale’s horseshoe: torus
knots and bifurcation sequences* by
Williams and Philip Holmes
[4].
This got me hooked on knots in dynamical systems.

After working together for just over a year, Professor Williams realized my name was not Paul, but Mike. In another year I would start call him Bob.

My dissertation built on Bob’s two papers with
Joan Birman
[1],
[2]
and his paper showing that Lorenz knots are prime
[3].
My first
theorem came from reading
[3].
Bob had shown that the knotted
periodic orbits on the __\( L(0,n) \)__ Lorenz-like templates, for __\( n \geq 0 \)__,
were prime. He had an example of a composite knot on the __\( L(0,-1) \)__ template.
I showed that the __\( L(0,n) \)__ templates also had composite knots when __\( n \)__ was
negative and odd, and I thought I had shown that when __\( n \)__ was negative and
even the knotted orbits were prime. I showed this to Bob while we were at
a conference. (I do not remember which one.)
He had just gotten back from
a night out while I had stayed back in order to work. He was very enthusiastic
about these results.
As it turned out, the second result did not
hold up the next morning when the effects of the beers wore off. Since I do
not drink myself, I was at first surprised by the relationship between beer
and mathematical truth.

A couple of weeks later, I was able to show that the __\( L(0,n) \)__ templates for __\( n < 0 \)__ all have
composite knots
[e2].
Rob Ghrist
would
subsequently show they contain all
knots
[e3].
Rob was a student of Phil Holmes at Cornell who would do
a postdoc with Bob at UT Austin. Rob initiated a book project on template
theory with Phil Holmes that I also become involved with
[e4].
We
dedicated the book to Bob:
*Dedicated to Robert F. Williams, teacher, colleague, and friend.*

I graduated in 1992, a difficult time in the mathematics job market. Bob was very supportive. He helped get me visiting positions at Cornell with Phil Holmes and at CUNY with Dennis Sullivan and then a postdoc at Northwestern. Bob and I have kept in touch. We share a number of personal and political interests.