Celebratio Mathematica

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.