# Curtis Tracy McMullen

### Quick Info

Berkeley, California, USA

**Curtis McMullen**is an American mathematician who was awarded the Fields Medal in 1998 for his work in complex dynamics, hyperbolic geometry and Teichmüller theory.

### Biography

**Curtis McMullen**(known as Curt) moved around the United States a bit as he was growing up, but he basically grew up in Charlotte, Vermont. He first attended Windermere Elementary School in Upper Arlington, near Columbus, Ohio, then Charlotte Central School in Charlotte, Vermont, before studying at Champlain Valley Union High School in Hinesburg, Vermont, only about 12 km from Charlotte. He entered Williams College, Williamstown, Western Massachusetts, in 1976. He said that during his undergraduate years at Williams College [6]:-

... I went to Stanford for a year and took a great real analysis course from Benjamin Weiss who was a visiting professor from Jerusalem. And that really got me excited about analysis. Then I went back to Williams and I worked closely with Bill Oliver. He was very influential in my mathematical education; it was from him that I first learned this idea of using dictionaries in mathematics to use as a sort of analogy between different fields or different theoretical developments to try to guide my work.He graduated from Williams College with a B.A. in 1980 as a Valedictorian, Summa Cum Laude, with Highest Honours in Mathematics and an additional concentration in Physics. He was awarded a Herchel Smith fellowship to study at the University of Cambridge in England and he spent the academic year 1980-81 at Emmanuel College, Cambridge. He was placed first in Part II of the Mathematical Tripos. He then returned to the United States to begin research in 1981 at Harvard University. He explains in [6] how he managed to obtain a Ph.D. from Harvard with Dennis Sullivan as his thesis advisor, although Sullivan did not work at Harvard:-

I had been doing some computer work with David Mumford on Kleinian groups before I graduated, and I got interested in that subject. But I actually ended up writing my thesis with Dennis Sullivan, who at that time was a professor at City University in New York and the Institut des Hautes Études Scientiques at France. So I was very lucky that David Mumford introduced me to him in the last year of my graduate career, at which point I had no advisor and no thesis topic. And I went to France and worked with Sullivan at Institut des Hautes Études Scientiques for a semester, and I met Steve Smale there who gave me this nice thesis problem on solving polynomial equations by iteration.During each of the summers from 1980 to 1985 McMullen undertook combinatorial research and computer programming at the IBM Watson Research Center, Yorktown Heights, New York. His work there involved VLSI design problems: graph theory, logic synthesis, boolean minimization, and sparse matrix processing. At Yorktown Heights [6]:-

... Mandelbrot and Mumford were almost collaborating; Mandelbrot was furnishing access to computers at Yorktown Heights to Mumford, who was drawing these beautiful pictures of limit sets of Kleinian groups. As somebody who was conversant with the computer world at Yorktown, I started working for him as his computer programmer, helping him draw these pictures and so forth.As he mentions in the above quote, McMullen spent the autumn of 1984 as a visitor at the Institut des Hautes Études Scientiques, Bures-sur-Yvette near Paris. His first paper

*The Hausdorff dimension of general Sierpinski carpets*was published in 1984 before he submitted his thesis. In the academic year 1984-85 he held an Alfred P Sloan Doctoral Dissertation Fellowship. He submitted his thesis

*Families of Rational Maps and Iterative Root-Finding Algorithms*to Harvard University and was awarded his doctorate in June 1985.

The problems which McMullen considered in his thesis had long been important. Once a proper understanding was achieved of which polynomial equations could be solved by radicals, there remained the problem of finding the roots of a polynomial equation by an iterative procedure for those for which no formula existed. Newton had produced such a method and his iterative procedure generally converged for all quadratic polynomials and initial points, but this was not the case for polynomial equations of degree three. Stephen Smale had asked whether, for each $n$, there existed an iterative procedure which generally converged for all initial points. In his thesis McMullen produced such a procedure for polynomials of degree three, but showed that for degree greater than three no such iterative procedure existed.

After completing his doctorate, McMullen spent the academic year 1985-86 at the Massachusetts Institute of Technology having been appointed a C L E Moore Instructor in Mathematics. He submitted further papers for publication and

*Area and Hausdorff dimension of Julia sets of entire functions*was published in 1987. A paper with the same title as his thesis also appeared in 1987. He spent the year 1986-87 as a member of the Institute for Advanced Study at Princeton, then in 1987 he was appointed as an Assistant Professor at Princeton University and awarded a NSF Postdoctoral Fellowship. This fellowship ran until 1990 but he also held an Alfred P Sloan Fellowship in 1988 and was a Presidental Young Investigator from 1988 to 1993. In 1990 he was promoted to full professor at Princeton but, in the same year, accepted a full professorship at the University of California, Berkeley.

The Salem Prize, founded by the widow of Raphael Salem, is awarded every year to a young mathematician judged to have done outstanding work in Salem's field of interest, namely analysis. McMullen explains that [6]:-

... in 1991, I won the Salem Prize, which is a prize in Analysis; I was pleased to be recognized that way because I really love the field - it was my first, as a mathematician. In fact, I had written my minor thesis as a graduate student on Salem numbers, and this prize is in honour of Raphael Salem, so it has personal meaning for me. I had never expected to get any recognition of that kind ...In 1994 he was named Miller Professor at Berkeley and held a Chancellor's Professorship at Berkeley during 1996-1998. He left Berkeley in 1998 when appointed to a professorship at Harvard University. In 2001 he was named Maria Moors Cabot Professor at Harvard, a position he continues to hold. In each of the summers from 2001 to 2007 he held a visiting position at the Max-Planck-Institute für Mathematik in Bonn.

Let us mention two monographs written by McMullen.

*Complex dynamics and renormalization*(1994) was reviewed by Gregery T Buzzard who begins his review as follows:-

This book provides a very clear and readable presentation of many of the main ideas and techniques used in the study of complex dynamics in one variable.After giving details of the contents of each chapter, he ends his review:-

This book presents a great many ideas very clearly and should prove to be a valuable addition to the complex dynamics literature.In 1996 McMullen published

*Renormalization and 3-manifolds which fiber over the circle*which was reviewed by Athanase Papadopoulos:-

In this monograph, the author presents a comprehensive study of a theory which brings into parallel two recent and very deep theorems, involving geometry and dynamics. These are Thurston's theorem on the existence of hyperbolic metrics on three-manifolds which fiber over the circle with pseudo-Anosov monodromy, and Sullivan's theorem on the convergence of the renormalization map for real quadratic mappings. The proofs that are given here are new and use different arguments than Thurston's and respectively Sullivan's original proofs. The relation between the two theorems is difficult to state in a concise form, but one thing that can be said at least is that both theorems can be thought of as fixed-point theorems.In 1998 McMullen received a Fields Medal at the International Congress of Mathematicians in Berlin. In addition to the work in his thesis which we have described above, the citation mentions his work on the Mandelbrot set [2]:-

This set describes dynamic systems which can be used to model complicated natural phenomena such as weather or fluid flow. The point of interest is where a system drifts apart and which points move towards centres of equilibrium. The border between these two extremes is the so-called Julia set, named after the French mathematician Gaston Julia, who laid the foundations for the theory of dynamic systems early in the twentieth century. The Mandelbrot set shows the parameters for which the Julia set is connected, i.e. is mathematically attractive. This description is very crude, but a better characteristic of the boundary set was not available. Curtis T McMullen made a major advance, however, when he showed that it is possible to decide in part on the basis of the Mandelbrot set which associated dynamic system is "hyperbolic" and can therefore be described in more detail. For these systems a well-developed theory is available. McMullen's results were suspected already in the sixties, but nobody had previously been able to prove this exact characterization of the Julia set.In [6] he tells a nice story about the Fields Medal:-

I have a story about when I was coming back from Berlin. The security guard in the airport running the metal detector stopped me when my backpack went through the machine. She said, "Excuse me, what do you have in your backpack here?" I said, "It's a gold medal." She said, a little dubiously, "Mmm hmm." So I took it out of my pack. A little chagrined, she said "Oh, very nice; is it yours?" I said "Mmm hmm!"The paper [1] list the areas of McMullen's contributions:-

He has made important contributions to various branches of the theory of dynamical systems, such as the algorithmic study of polynomial equations, the study of the distribution of the points of a lattice of a Lie group, hyperbolic geometry, holomorphic dynamics and the renormalization of maps of the interval.In addition to the Salem Prize in 1991 and the Fields Medal he was awarded in 1998, McMullen had received a large number of other distinctions for his outstanding contributions. Among these we note that he was elected a Fellow of the American Academy of Arts and Sciences in 1998, was given an Honorary Doctor of Science Degree by Williams College in 1999, and was elected a Member of the National Academy of Sciences in 2007. He has also been invited to give special lectures such as (since 2000): Nevanlinna Colloquium, Helsinki (2000); American Mathematical Society Colloquium Lectures, Washington DC (2000); Distinguished Lecture Series, Brown University (2001); American Mathematical Society Ross Lecture, Boston (2002); Namboodiri Lectures, University of Chicago (2003); Alaoglu Lecture, California Institute of Technology, Pasadena (2003); Mathematische Arbeitstagung, Max-Planck-Institut, Bonn (2003); Bowen Lectures, University of California, Berkeley (2004); Kolchin Lecture, Columbia University, New York (2005); Hopf Lectures, Eidgenossische Technische Hochschule, Zürich (2005); Mathematische Arbeitstagung, Max-Planck-Institut, Bonn (2007); Ziwet Lectures, University of Michigan, Ann Arbor (2008) and Nielsen Lecture, CTQM, Aarhus, Denmark (2008).

Despite his stunning contributions to the latest mathematical research, McMullen has given excellent lectures to school children as for example the Arnold Ross Lecture he gave in 2002 on

*From Triangles to Infinity*. Here is a report of the lecture:-

McMullen motivated the talk by asking the audience what path a lion should take to capture a human, if both are in an enclosed ring. A little later in the talk, he asked students in the audience to assemble polyhedra using interlocking triangles, given the constraint that a fixed number of triangles have to meet at each vertex. As the title of the talk suggests, there were many different areas of mathematics touched on by McMullen, including: Fermat's Last Theorem, Zeno's Paradoxes, hyperbolic and spherical geometry, the harmonic series, and tiling. Near the end of his talk, McMullen showed a path that a human could take to elude the lion and used results about infinite series to demonstrate the path's effectiveness. The teachers and students who filled the Boston Museum of Science auditorium thoroughly enjoyed the subject of the talk and the manner in which it was delivered. Many students sought out McMullen after his talk to ask questions, and some even asked for his autograph.More recently he gave a lecture

*The Geometry of 3-Manifolds*to the annual meeting of the American Association for the Advancement of Science in Boston in February 2008:-

In this topical lecture, McMullen gave a wonderful description of the Poincaré conjecture ... McMullen explained many topological ideas in the conjecture, how it relates to the shape of the universe, and gave an overview of Grigori Perelman's proof of the conjecture.Let us end this biography by quoting Stephen Smale's words from [10]:-

I ... would like to emphasise that [McMullen's] work has encompassed a large realm of the kind of mathematics that lies at the cross-section of many paths of our rich culture. McMullen is not a dynamicist, not an analyst nor a geometer. He is a mathematician.

### References (show)

- A Douady and J-P Otal, Les travaux de Curtis McMullen,
*Gaz. Math.***80**(1999), 19-30. - Fields Medal Prize Winners (1998): Curtis Tracy McMullen (born 21 May 1958). http://www.icm2002.org.cn/general/prize/medal/1998.htm
- A Jackson, Borcherds, Gowers, Kontsevich, and McMullen Receive Fields Medals,
*Notices Amer. Math. Soc.***45**(10) (1998), 1358-1360. - J Lepowsky, J Lindenstrauss, Y I Manin, and J Milnor, The Mathematical Work of the 1998 Fields Medalists,
*Notices Amer. Math. Soc.***46**(1) (1999), 17-26. - K Matsuzaki, Y Takahashi and M Kato, Introducing two of the Fields medalists: the work of C T McMullen and W T Gowers (Japanese),
*Sugaku***51**(2) (1999), 186-191. - A-M Oreskovich and D Sagalovskiy, Harvard University Math Club Interview with Professor Curtis McMullen (Harvard University, 1999). http://www.math.harvard.edu/~ctm/expositions/interview.html
- Prize Winners at the 1998 International Congress of Mathematicians,
*American Mathematical Society Feature Columns*(September 1998). http://www.ams.org/featurecolumn/archive/index.html - A Quiros, The Fields Medals [of 1998] (Spanish),
*Gac. R. Soc. Mat. Esp.***1**(3) (1998), 439-446. - C S Rajan, R Bhatia, T R Ramadas and N A Shah, The work of the Fields medalists: 1998,
*Current Sci.***75**(12) (1998), 1290-1295. - S Smale, The work of Curtis T McMullen, in Proceedings of the International Congress of Mathematicians
**I**, Berlin, 1998,*Doc. Math. J. DMV*(1998), 127-132. - S Smale, The work of [Fields medalist] Curtis T McMullen,
*Mitt. Dtsch. Math.-Ver.***1998**(3) (1998), 48-51.

### Additional Resources (show)

Other websites about Curtis McMullen:

### Honours (show)

Honours awarded to Curtis McMullen

Written by J J O'Connor and E F Robertson

Last Update July 2009

Last Update July 2009