Curtis McMullen

Curtis T. McMullen
Born	Curtis Tracy McMullen
Nationality	American
Occupation	Mathematician, university professor
Title	Maria Moors Cabot Professor of the Natural Sciences
Employer	Harvard University
Known for	Work in complex dynamics, hyperbolic geometry, and Teichmüller theory
Awards	Fields Medal (1998)

Curtis Tracy McMullen is an American mathematician and the Maria Moors Cabot Professor of the Natural Sciences at Harvard University. He is known for his deep contributions to complex dynamics, hyperbolic geometry, Teichmüller theory, and related areas of mathematics. In 1998, McMullen received the Fields Medal, widely deemed the most prestigious award in mathematics, for his work on complex dynamics and related topics. His research has spanned a remarkable range of mathematical fields, connecting ideas from dynamics, geometry, topology, number theory, and algebra. McMullen's work has also extended into the visual realm, with his mathematical illustrations helping to render infinite and abstract structures in vivid, accessible form. His public lectures, including an Arnold Ross Lecture at the Museum of Science in Boston titled "From Triangles to Infinity," have brought advanced mathematical ideas to broader audiences. At Harvard, McMullen has been a central figure in the mathematics department, known for his research, teaching, and mentorship. Among his doctoral students was Maryam Mirzakhani, who in 2014 became the first woman to receive the Fields Medal, a distinction that drew international attention to the mathematical lineage McMullen helped foster.

Career

Academic career and research

McMullen holds the position of Maria Moors Cabot Professor of the Natural Sciences at Harvard University, where he has been a member of the mathematics faculty for decades. His research has addressed fundamental questions across multiple branches of mathematics, with particular emphasis on complex dynamics, hyperbolic geometry, and Teichmüller theory. These fields deal with the behavior of iterated functions on complex number systems, the geometry of spaces with constant negative curvature, and the study of deformations of geometric structures on surfaces, respectively.

McMullen's work in complex dynamics contributed to the understanding of the structure and behavior of Julia sets, Mandelbrot sets, and related fractal objects that arise from iterating polynomial and rational maps on the complex plane. His research in this area, along with contributions to hyperbolic three-manifolds and the rigidity of dynamical systems, formed the core of the work recognized by the Fields Medal in 1998.

Beyond these central areas, McMullen has made contributions linking dynamics to number theory and algebraic geometry. His ability to draw connections across mathematical disciplines has been a hallmark of his career. In an interview with The Harvard Crimson in 2025, McMullen discussed his perspective on mathematics, touching on themes of shared truths, unsolved problems, and the challenge of illustrating infinity — reflecting his broad intellectual interests and his belief in the unity of mathematical knowledge.

Visualization of mathematical structures

A distinctive aspect of McMullen's work has been his creation of mathematical illustrations and visual representations of abstract structures. In 2020, Harvard Magazine profiled his work on what it described as "interwoven tapestries" that help make the infinite visible. The article focused on McMullen's visualizations of negatively curved crystals — structures related to hyperbolic geometry that exhibit intricate, repeating patterns extending infinitely in space. These images serve both as tools for mathematical research and as striking visual art, providing a window into geometric worlds that cannot be directly perceived.

McMullen's approach to illustration reflects a broader commitment to making abstract mathematics accessible. His public lecture "From Triangles to Infinity," delivered as part of the American Mathematical Society's Arnold Ross Lecture series at the Museum of Science in Boston, exemplified this approach by guiding a general audience from elementary geometric concepts to the frontiers of mathematical research.

Mentorship and influence

McMullen's influence on mathematics extends significantly through his role as a doctoral advisor. Among his most notable students was Maryam Mirzakhani, who completed her Ph.D. under McMullen's supervision at Harvard. Mirzakhani went on to become a professor at Stanford University and, in 2014, was awarded the Fields Medal — the first woman ever to receive the honor. Her work on the dynamics and geometry of Riemann surfaces and their moduli spaces built upon and extended themes central to McMullen's own research.

Mirzakhani's untimely death from breast cancer in July 2017, at the age of 40, prompted tributes from across the mathematical world. Colleagues spoke of her brilliance and the depth of her contributions, which included work on shapes and surfaces unconstrained by the limitations of the real, physical world. Siobhan Roberts, writing in The New Yorker, spoke with Mirzakhani's colleagues about her legacy and the mathematical tradition she emerged from — a tradition in which McMullen played a formative role as her advisor. The fact that both McMullen and his student Mirzakhani received the Fields Medal is an extraordinary distinction in the history of mathematics, underscoring the depth and influence of the research program McMullen cultivated at Harvard.

Public engagement

McMullen has participated in outreach activities aimed at fostering interest in mathematics among students and the general public. His Arnold Ross Lecture, organized by the American Mathematical Society, was part of a program that also included the "Who Wants to Be a Mathematician" competition, designed to engage high school students with mathematical problem-solving. McMullen's lecture, "From Triangles to Infinity," was delivered at the Museum of Science in Boston and introduced audiences to the sweep of geometric and dynamic ideas that have shaped modern mathematics.

In his 2025 interview with The Harvard Crimson, McMullen reflected on the nature of mathematical truth and the pursuit of unsolved problems, offering insights into his approach to research and the intellectual motivations that drive his work. He discussed the challenge of illustrating infinity — a recurring theme in his career — and the ways in which visual representation can deepen mathematical understanding.

Recognition

McMullen's most prominent recognition is the 1998 Fields Medal, awarded once every four years to mathematicians under the age of forty and considered the highest honor in the field of mathematics. McMullen received the medal for his work on complex dynamics, hyperbolic geometry, and related areas.

The American Mathematical Society has featured McMullen in its public outreach programs, including inviting him to deliver the Arnold Ross Lecture, a distinction reserved for leading mathematicians who can communicate advanced ideas to broad audiences.

McMullen's mathematical illustrations, particularly his visualizations of negatively curved crystals and hyperbolic structures, have been profiled in Harvard Magazine, which highlighted their dual significance as research tools and as works of mathematical art that make abstract concepts visible and tangible.

The achievement of having mentored a Fields Medalist — Maryam Mirzakhani — further distinguishes McMullen's career. The advisor-student pair of Fields Medal winners represents a rare occurrence in mathematical history, reflecting the significance of the intellectual environment McMullen fostered at Harvard.

Legacy

Curtis McMullen's legacy in mathematics is defined by the breadth and depth of his research contributions, the influence of his mentorship, and his efforts to communicate mathematical ideas beyond the academic community. His work in complex dynamics, hyperbolic geometry, and Teichmüller theory has shaped the development of these fields and opened new avenues of inquiry connecting dynamics, geometry, topology, and number theory.

The lineage of Fields Medal recipients from McMullen to Mirzakhani represents a striking legacy of mathematical mentorship. Mirzakhani's groundbreaking work on Riemann surfaces and moduli spaces carried forward and expanded the research agenda that McMullen had helped define, and her recognition as the first female Fields Medalist brought worldwide attention to the mathematical community and the traditions of research and training within it.

McMullen's visualizations of mathematical structures have contributed to a broader understanding of how abstract mathematical objects can be represented and appreciated. His images of negatively curved crystals and hyperbolic patterns have appeared in prominent publications and have served to bridge the gap between advanced mathematical research and public comprehension.

As a professor at Harvard, McMullen has contributed to the training of numerous mathematicians and to the intellectual culture of one of the world's leading mathematics departments. His public lectures and interviews have articulated a vision of mathematics as a unified, aesthetic, and deeply human endeavor — one in which shared truths, unsolved problems, and the challenge of illustrating infinity remain central concerns.