Curtis T. McMullen

How to use this course

Curtis McMullen works in complex dynamics, hyperbolic geometry, Teichmüller theory, and, more recently, the dynamics of algebraic surfaces and the arithmetic of lattices. What unites these is a method. He builds dictionaries between fields, so that a theorem about Kleinian groups suggests one about rational maps, and a question about billiards becomes a question about Hilbert modular surfaces. This course follows that method through seven technical lessons, framed by an account of his education and career.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, the key idea of each proof, and exercises with folded solutions. Readers who want only the career can read Parts I and III and the italic openings of Part II. McMullen is a living mathematician; this course covers his work, positions and public honours only.

Timeline

Vermont and Williams, 1958–1981

Curtis Tracy McMullen was born in Berkeley, California, on 21 May 1958. His family moved several times during his childhood, including a period near Columbus, Ohio, and he grew up mainly in Charlotte, Vermont, a small town on Lake Champlain. He has said he regards himself as being from Vermont. He attended Charlotte Central School and Champlain Valley Union High School in Hinesburg.

In 1976 he entered Williams College in Williamstown, Massachusetts. He spent one year of his undergraduate studies at Stanford, where he took real analysis from Benjamin Weiss, then visiting from Jerusalem. At Williams he worked closely with Bill Oliver, who taught him to look for analogies between subjects and to use them as a guide to research. McMullen later described these analogies as dictionaries, and the word fits his whole career. He graduated in 1980 as valedictorian, summa cum laude, with highest honours in mathematics and an additional concentration in physics.

Cambridge

A Herchel Smith Fellowship took him to Emmanuel College, Cambridge, for 1980–81. He took Part II of the Mathematical Tripos and placed first. Emmanuel made him an Honorary Fellow in 2014.

Harvard, IBM and the IHÉS, 1981–1985

From 1980 to 1985 McMullen spent his summers at the IBM Watson Research Center in Yorktown Heights, New York, doing combinatorial research and programming. The work touched VLSI design, logic synthesis, Boolean minimization and sparse matrices. The programming skill shaped his entry into research.

He began graduate work at Harvard in 1981. David Mumford hired him to write programs drawing the limit sets of Kleinian groups, the fractal curves on which a discrete group of Möbius transformations accumulates. In those years he worked in FORTRAN over a slow terminal and a long-distance modem, and printed pictures sometimes arrived by mail a week later. The pictures interested him in Hausdorff dimension. His first paper, published in 1984, computed the dimension of a class of self-affine fractals posed by Heisuke Hironaka (Lesson 04).

Sullivan and Smale

In his last year he had neither an advisor nor a thesis topic. Mumford introduced him to Dennis Sullivan, who was then based at the City University of New York and the IHÉS near Paris, and McMullen spent the autumn of 1984 at the IHÉS. Sullivan became his advisor and, by McMullen’s account, his greatest mathematical influence. Sullivan had begun to build a dictionary between Kleinian groups and the iteration of rational maps. At the IHÉS, Stephen Smale suggested the thesis problem: can polynomial equations be solved reliably by iterating a rational function?

The thesis, Families of rational maps and iterative root-finding algorithms, was accepted in June 1985 and published in the Annals of Mathematics in 1987. It answered Smale’s question completely (Lesson 05). As a graduate student he also wrote a minor thesis on Salem numbers, a topic that returned twenty years later in his work on surface automorphisms (Lesson 09).

MIT, Princeton, Berkeley, 1985–1998

After the doctorate McMullen held a C. L. E. Moore Instructorship at MIT (1985–86), a postdoctoral fellowship at MSRI in Berkeley, and membership of the Institute for Advanced Study (1986–87). He joined Princeton as an assistant professor in 1987, with an NSF postdoctoral fellowship, a Sloan Fellowship (1988) and a Presidential Young Investigator award (1988–93).

The Princeton years were productive. With Peter Doyle he showed how to solve the quintic by a tower of iterative algorithms. He proved Irwin Kra’s theta conjecture on Poincaré series, gave a new route to Thurston’s hyperbolization theorem through iteration on Teichmüller space, and proved Lipman Bers’ conjecture that cusps are dense in the boundary of Teichmüller space (Lesson 06). With Alex Eskin he proved counting and equidistribution theorems for lattice orbits on affine symmetric varieties, published in 1993. He was an invited speaker at the 1990 International Congress in Kyoto.

In 1990 he was promoted to professor at Princeton and also accepted a professorship at the University of California, Berkeley, where he taught until 1998, holding a Miller Professorship in 1994 and a Chancellor’s Professorship from 1996 to 1998. He won the Salem Prize in 1991. At Berkeley he wrote his two monographs on renormalization (Lesson 07) and supervised his first doctoral students, among them Kevin Pilgrim, Jeremy Kahn, Jeffrey Brock and Kenneth Bromberg.

Fractal dimension: carpets and Julia sets

For the general reader. A fractal can be too large to be a curve and too thin to have area. Hausdorff dimension measures where it sits between the two. McMullen’s first papers computed this dimension for fractals built by stretching unequally in two directions, and showed that some Julia sets, though they have no interior, are as large as a set in the plane can be.

Self-affine carpets

Divide the unit square into \(n\) columns and \(m\) rows of congruent rectangles, with \(n\ge m\ge2\). Choose a set \(D\) of these rectangles, and let \(t_j\) be the number chosen in row \(j\). The affine maps sending the square onto the chosen rectangles contract by \(1/n\) horizontally and \(1/m\) vertically. Their attractor \(K\) is the unique non-empty compact set with \(K=\bigcup_{R\in D}\phi_R(K)\). When \(n=m\) the maps are similarities and \(\dim_H K=\log|D|/\log n\). When \(n>m\) the problem is genuinely harder, because a covering by squares must treat long thin rectangles differently in the two directions.

Idea of the proof. Covering \(K\) by approximate squares of side \(n^{-k}\) gives the box dimension directly. For the Hausdorff dimension, McMullen put a Bernoulli measure on \(K\) giving each rectangle in row \(j\) weight proportional to \(t_j^{\log_n m-1}\), and computed its local dimension at almost every point with the law of large numbers. The Hausdorff dimension is the dimension of this optimal measure. The sets are now called Bedford–McMullen carpets, after the independent work of Tim Bedford.

Julia sets of entire functions

The Julia set \(J(f)\) of an entire function is the set where the iterates \(f^n\) fail to form a normal family. For transcendental functions it is unbounded and often contains the points that escape to infinity.

So a Julia set with no interior can still have full dimension, and in the sine family even positive measure. The proofs estimate how the escaping set is distributed in long horizontal strips, where \(e^z\) grows fast enough to make the escaping points fill space at every scale.

Solving equations by iteration

For the general reader. Newton’s method finds a root of an equation by repeatedly improving a guess. For some cubic equations it fails on a whole region of starting guesses, which fall into a cycle instead of converging. Smale asked whether any rational iteration could avoid this. McMullen proved that for cubics the answer is yes, for equations of degree four or more it is no, and that the reason is a rigidity theorem in complex dynamics.

Definitions

Let \(\mathrm{Poly}_d\) be the space of polynomials of degree \(d\). A purely iterative algorithm is a rational map \(p\mapsto T_p\) from \(\mathrm{Poly}_d\) to rational functions of \(z\), whose coefficients are rational functions of the coefficients of \(p\). Newton’s method \(N_p(z)=z-p(z)/p'(z)\) is the basic example. The algorithm is generally convergent if \(T_p^n(z)\) converges to a root of \(p\) for \((p,z)\) in an open dense set of full measure.

Newton’s method is generally convergent for quadratics but not for cubics. For \(p(z)=z^3-2z+2\) the points 0 and 1 form a superattracting 2-cycle of \(N_p\) (Exercise 5.1), so an open set of starting points never approaches a root, and this persists for nearby cubics.

Why rigidity

If the algorithm converges almost everywhere for an open set of \(p\), the family \(T_p\) has no bifurcations: it is stable in the sense of Mañé, Sad and Sullivan. McMullen proved that a stable algebraic family of rational maps is either trivial, with every member conjugate to one map \(f\), or a family of Lattès examples coming from complex tori, and the second case cannot occur for a root-finding algorithm. The attracting fixed points of \(T_p\) that converge to roots correspond under the conjugacy to fixed points of \(f\). So the roots of \(p\) are the images of a fixed finite set under a Möbius map \(M_p\). For four or more points this forces the cross-ratios of the roots to be constant, which is false as \(p\) varies. For three roots there is no constraint, since any three points are Möbius-equivalent to any other three.

The quintic

Peter Doyle and McMullen pushed further in 1989. Allowing a tower of algorithms, in which the output of one iteration feeds the next, they proved that a field extension can be computed this way if and only if its Galois group is “within \(A_5\) of solvable”: every composition factor is cyclic or the alternating group \(A_5\). The quintic is therefore solvable by a tower, using a rational map with icosahedral symmetry, while the general sextic is not. This is a dynamical analogue of the Abel–Ruffini theorem.

Kleinian groups and Teichmüller space

For the general reader. The shapes a surface can take form a space, Teichmüller space. Hyperbolic three-dimensional spaces are built by gluing such surfaces, and their boundaries record the shapes of the surfaces at infinity. In three papers between 1989 and 1991 McMullen proved a conjecture of Irwin Kra on how shapes behave under coverings, gave a new proof of a central case of Thurston’s hyperbolization theorem, and proved Bers’ conjecture that the most degenerate boundary points are dense.

Poincaré series and amenability

Let \(X=\Delta/\Gamma\) be a hyperbolic Riemann surface of finite type, uniformized by a Fuchsian group \(\Gamma\) acting on the disc \(\Delta\). For an integrable holomorphic quadratic differential \(\phi\) on \(\Delta\), the Poincaré series

is an integrable quadratic differential on \(X\). The triangle inequality gives \(\|\Theta\|\le1\) in the \(L^1\) norm. Kra conjectured the strict inequality. More generally, a covering \(Y\to X\) gives a push-forward \(\Theta_{Y/X}:Q(Y)\to Q(X)\), the coderivative of the inclusion \(\mathrm{Teich}(X)\hookrightarrow\mathrm{Teich}(Y)\).

A covering is amenable when the lift of a finite graph carrying \(\pi_1(X)\) has expansion constant \(\inf|\partial V|/|V|=0\) over finite vertex sets \(V\). The bound on \(\|\Theta\|\) is not uniform: it tends to 1 as the shortest geodesic on \(X\) shrinks.

Iteration on Teichmüller space

Thurston proved that an atoroidal Haken 3-manifold is hyperbolic by cutting it along incompressible surfaces and regluing. The key step studies the skinning map \(\sigma\): given a hyperbolic structure on a manifold \(M\) with boundary, it records the conformal structure seen on the other side of each boundary surface. Regluing by a map \(\tau\) gives a hyperbolic structure exactly when \(\tau\circ\sigma\) has a fixed point in Teichmüller space.

The contraction comes from the theta theorem: the derivative of \(\sigma\) factors through a Poincaré series for a non-amenable covering. McMullen’s argument replaced an existence proof with an iteration that converges.

Cusps are dense

Fix a closed surface \(S\) of genus \(g\ge2\). Quasi-Fuchsian groups uniformize two surfaces at once, so their space is \(\mathrm{Teich}(S)\times\mathrm{Teich}(\bar S)\) by Bers’ simultaneous uniformization. Fixing the first factor gives the Bers slice, a bounded embedding of Teichmüller space in a space of Kleinian groups. Its boundary groups are degenerate in various ways. A cusp is a boundary group in which some non-peripheral simple closed curve has become parabolic.

The proof estimates how a quasiconformal deformation of unit size, supported in the thin part of a hyperbolic surface, moves the corresponding group. Bers had asked the question. It was a step toward the density conjecture of Bers, Sullivan and Thurston, completed in 2011–12 using the ending lamination theorem.

Renormalization and the Mandelbrot set

For the general reader. The Mandelbrot set catalogues the behaviour of the maps \(z\mapsto z^2+c\). Zooming in near certain points shows smaller copies of the whole set, again and again. Renormalization is the study of this self-similarity. McMullen used it, together with ideas taken from hyperbolic 3-manifolds, to prove that the real slice of the Mandelbrot set behaves as conjectured.

The quadratic family

Let \(f_c(z)=z^2+c\). The Mandelbrot set \(M\) is the set of \(c\) for which the critical orbit \(0,c,c^2+c,\dots\) stays bounded, equivalently for which the Julia set \(J(f_c)\) is connected. A map is hyperbolic if its critical orbit tends to an attracting cycle; hyperbolic parameters form open components of the interior of \(M\).

• HD: hyperbolic maps are dense in the quadratic family.
• MLC: the Mandelbrot set is locally connected. Douady and Hubbard showed MLC implies HD.
• NILF: \(J(f_c)\) carries no measurable invariant line field. By Mañé, Sad and Sullivan, NILF for all \(c\) is equivalent to HD in the quadratic family.

Renormalization

A quadratic-like map \(g:U\to V\) is a proper holomorphic map of degree 2 between topological discs with \(\overline U\subset V\). Douady and Hubbard’s straightening theorem says it is quasiconformally conjugate near its filled Julia set to some \(f_c\). The map \(f_c\) is renormalizable with period \(n\) if there is a disc \(U\ni0\) on which \(f_c^n:U\to V\) is quadratic-like with connected filled Julia set. It is infinitely renormalizable if this happens for infinitely many \(n\). The Feigenbaum parameter \(c\approx-1.4011552\), the limit of period doubling, is the basic real example.

The dictionary at work. An invariant line field is the dynamical analogue of a deformation of a Kleinian group supported on its limit set, which Sullivan had ruled out. McMullen took limits of renormalizations, as one takes geometric limits of hyperbolic manifolds, and showed that a line field would have to become almost parallel at small scales near deep points of the Julia set, which the limiting dynamics forbids. His second monograph (1996) made the parallel explicit: it treats the convergence of renormalization for real quadratics alongside the hyperbolization of 3-manifolds that fibre over the circle, both through geometric limits and a quantitative form of Mostow rigidity called inflexibility.

Billiards and Teichmüller curves

For the general reader. A billiard ball bouncing in a polygon whose angles are rational multiples of \(\pi\) is the same as a straight line on a surface glued from copies of the polygon. In 1989 William Veech found polygons where the motion is perfectly organized: in each direction, every path either closes up or spreads evenly. McMullen found infinitely many new examples, L-shaped tables with side lengths in quadratic fields, by recognizing them as special curves on Hilbert modular surfaces, and then classified everything that can happen in genus two.

Translation surfaces and Veech groups

A translation surface \((X,\omega)\) is a compact Riemann surface with a non-zero holomorphic 1-form. Integrating \(\omega\) gives charts to \(\mathbb C\) whose transitions are translations, so the surface has a flat metric with cone points at the zeros of \(\omega\). The group \(\mathrm{SL}_2(\mathbb R)\) acts by post-composing the charts. The stabilizer \(\mathrm{SL}(X,\omega)\), the Veech group, is discrete. When it is a lattice, the orbit \(\mathrm{SL}_2(\mathbb R)\cdot(X,\omega)\) projects to a closed algebraic curve in the moduli space \(\mathcal M_g\), isometric for the Teichmüller metric: a Teichmüller curve.

L-shaped tables

Let \(L(a,b)=[0,a]\times[0,1]\,\cup\,[0,1]\times[0,b]\) with \(a,b>1\). Gluing opposite sides by translations gives a genus-two surface with a single zero of \(\omega\), of order 2.

Real multiplication. The Jacobian of a genus-two surface is a 2-dimensional abelian variety. Say it has real multiplication by the quadratic order \(\mathcal O_D\) of discriminant \(D\) if \(\mathcal O_D\) acts on it by self-adjoint endomorphisms. Such Jacobians fill a Hilbert modular surface \(X_D\cong(\mathbb H\times\mathbb H)/\Gamma\). McMullen proved that if \((X,\omega)\) generates a Teichmüller curve whose trace field is \(\mathbb Q(\sqrt D)\), then the Jacobian has real multiplication and \(\omega\) is an eigenform. Adding the condition that \(\omega\) has a double zero cuts a surface down to a curve: the Weierstrass curve \(W_D\).

Orbit closures in genus two

Every orbit closure is cut out by linear equations in period coordinates. This was the first complete case of the picture that Eskin, Mirzakhani and Mohammadi established in all genera in 2015, which is why McMullen’s genus-two work is the template for the “magic wand” theorem.

Lehmer’s number and complex surfaces

For the general reader. In 1933 Derrick Lehmer found an algebraic number slightly larger than 1, about 1.17628, and asked whether any smaller one of its kind exists. The question is still open. McMullen showed that Lehmer’s number is exactly the smallest possible rate of chaos for a holomorphic symmetry of a complex surface, and built surfaces where this minimum is reached.

Salem numbers and entropy

A Salem number is a real algebraic integer \(\lambda>1\) whose other conjugates include \(1/\lambda\) and otherwise lie on the unit circle. Lehmer’s number \(\lambda_L\approx1.17628081\) is the largest real root of

For an automorphism \(F\) of a compact complex surface, the topological entropy equals \(\log\) of the spectral radius of \(F^*\) on cohomology, by the theorems of Gromov and Yomdin. Because \(F^*\) preserves the intersection form, of signature \((1,n)\) on the relevant part of \(H^2\), positive entropy forces this spectral radius to be a Salem number or a quadratic unit.

In 2002 McMullen also constructed automorphisms of non-projective K3 surfaces with positive entropy and a Siegel disc, an open set on which the map is holomorphically conjugate to an irrational rotation. The construction runs entirely through lattices and Hodge theory: one finds a Salem polynomial of degree 22 and an integral matrix with that characteristic polynomial preserving the K3 lattice, then uses the Torelli theorem to realize it geometrically. Brandhorst and Elkies later wrote explicit equations for the projective K3 surface of minimal entropy.

Three-manifolds and lattices

For the general reader. A three-dimensional space can sometimes be swept out by a moving surface, like the pages of a book whose last page is glued back to the first. Such spaces fibre over a circle. McMullen found polynomials that record, all at once, every way a 3-manifold fibres and how much stretching each fibration involves. Related ideas about flows on spaces of lattices settled a case of a century-old conjecture of Minkowski and revealed rigidity for planes inside hyperbolic 3-manifolds.

Norms on cohomology

Let \(M\) be a compact orientable 3-manifold with empty or toroidal boundary. For \(\phi\in H^1(M;\mathbb Z)\), the Thurston norm \(\|\phi\|_T\) is the least value of \(\chi_-(S)=\sum\max(0,-\chi(S_i))\) over embedded surfaces dual to \(\phi\). The Alexander norm \(\|\phi\|_A\) is the width of the Alexander polynomial \(\Delta_M=\sum a_g g\) in the direction \(\phi\): the maximum of \(\phi(g-h)\) over \(g,h\) with non-zero coefficients.

The Teichmüller polynomial

Thurston showed that the fibred classes of \(M\) fill the open cones over certain top-dimensional fibred faces \(F\) of the Thurston norm ball. Each fibred class \(\phi\) has a pseudo-Anosov monodromy with stretch factor \(K(\phi)>1\).

Lattices and planes

Minkowski conjectured that for every unimodular lattice \(\Lambda\subset\mathbb R^n\) and every \(x\in\mathbb R^n\) there is \(y\in\Lambda\) with \(\prod_i|x_i-y_i|\le2^{-n}\). It was known for \(n\le5\). McMullen (2005) proved that the closure of every bounded orbit of the diagonal group on the space of lattices contains a well-rounded lattice, and deduced the conjecture for \(n=6\). The proof uses topological dimension theory, through the combinatorics of open covers of \(\mathbb R^n\) and of tori.

With Amir Mohammadi and Hee Oh he began the study of totally geodesic planes in hyperbolic 3-manifolds of infinite volume. They proved (2017) that when \(M\) is acylindrical and its convex core has totally geodesic boundary, the closure of every immersed geodesic plane is a properly immersed submanifold, and that this rigidity fails for quasi-Fuchsian manifolds. In the same year, with Ronen Mukamel and Alex Wright, he found the first primitive totally geodesic subvariety of moduli space of dimension greater than one, a complex surface inside \(\mathcal M_{1,3}\) built from the flex points of cubic curves.

The Fields Medal and Harvard, 1998–

At the International Congress of Mathematicians in Berlin in August 1998, McMullen received the Fields Medal, together with Richard Borcherds, Timothy Gowers and Maxim Kontsevich. The citation recognized his contributions to the theory of holomorphic dynamics and to the geometrization of three-manifolds, including the proofs of Bers’ density conjecture for cusps and Kra’s theta conjecture. Dennis Sullivan, who had been his advisor, had built the dictionary between the two fields that McMullen’s work exploited. The same year he was elected a Fellow of the American Academy of Arts and Sciences and returned to Harvard as professor.

He became Maria Moors Cabot Professor in 2001. Much of his work since then has been on Teichmüller dynamics and algebraic geometry: the genus-two classification (Lesson 08), the surface automorphisms of Lesson 09, and later Galois orbits and arithmetic questions about billiards. From 2001 to 2014 he spent regular periods at the Max Planck Institute for Mathematics in Bonn, and he has visited MSRI and the IHÉS many times. He gave an invited lecture at the ICM in Seoul in 2014, and held the Mazur Chair at the IHÉS in 2024.

His later honours include a Guggenheim Fellowship (2004), election to the National Academy of Sciences (2007), a Humboldt Research Award (2011) and election to the American Philosophical Society (2023). He chaired the Harvard mathematics department from 2017 to 2020, organized the international online seminar BiSTRO during 2020–22, and has served as an editor of the Annals, the Journal of the AMS and Forum of Mathematics.

Students, style and influence

McMullen has supervised about twenty doctoral students at Berkeley, Harvard and MIT. They include Kevin Pilgrim (1994), Jeremy Kahn (1995), Jeffrey Brock (1997), Kenneth Bromberg (1998), Laura DeMarco (2002), David Dumas (2004), Maryam Mirzakhani (2004), Matthew Bainbridge (2006), Ronen Mukamel (2011), Thomas Koberda (2012), Giulio Tiozzo (2013) and Oleg Ivrii (2014). Mirzakhani, who attended his seminar from her first year at Harvard, received the Fields Medal in 2014; McMullen wrote the account of her work for the Congress proceedings. Bainbridge computed the Euler characteristics of the Weierstrass curves of Lesson 08, extending his advisor’s programme.

Style

In interviews McMullen has called himself a dilettante who enjoys learning new subjects, and has said he prefers to build on fields he already understands, moving from real analysis to one complex variable and then to Kähler geometry. He does his best research alone, while testing arguments on experts. His papers are compact and rich in computer-drawn figures, a habit that goes back to the Kleinian-group pictures of his student years. He posts his course notes and lecture slides on his Harvard web page, where graduate students use them as introductions to the field.

Influence

• Complex dynamics: rigidity, renormalization and the no-invariant-line-field method shaped the attack on MLC and on density of hyperbolicity.
• Kleinian groups: the theta theorem and the skinning contraction became standard tools; density of cusps fed into the solution of the Bers–Sullivan–Thurston density conjecture.
• Teichmüller dynamics: the genus-two classification is the model for the Eskin–Mirzakhani–Mohammadi theorem and for the classification programme for Teichmüller curves in higher genus.
• Algebraic dynamics: minimal entropy and Lehmer’s number opened a line of work on automorphisms of K3, Enriques and rational surfaces.

Connections

People linked to Curtis T. McMullen across this collection. A name without a link has no page here yet. See Curtis T. McMullen in the network of all 337 lives →

Teachers and advisors

• Dennis Sullivan 1984-1985

Students

• Maryam Mirzakhani 2004
• David Dumas 2004
• Giulio Tiozzo 2013
• Jeffrey Brock 1997
• Jeremy Kahn 1995
• Kenneth Bromberg 1998
• Kevin Pilgrim 1994
• Laura DeMarco 2002
• Matthew Bainbridge 2006
• Oleg Ivrii 2014
• Ronen Mukamel 2011
• Thomas Koberda 2012

Influenced by

• David Mumford
• Howard Masur
• Lars Ahlfors
• William Thurston
• Irwin Kra
• Lipman Bers
• Stephen Smale
• William Veech

Colleagues and collaborators

• Alex Eskin 1993
• Alex Wright
• Amir Mohammadi
• Hee Oh
• Peter Doyle 1989

Also mentioned in this life

• Niels Henrik Abel
• Leonhard Euler
• Évariste Galois
• Mikhail Gromov
• Felix Hausdorff
• David Hilbert
• Alexander von Humboldt
• Maxim Kontsevich
• Sophus Lie
• Hermann Minkowski
• August Ferdinand Möbius
• Isaac Newton
• Max Planck
• Henri Poincaré
• Bernhard Riemann
• Carl Ludwig Siegel
• Wacław Sierpiński
• Oswald Teichmüller
• Karl Weierstrass
• Hermann Weyl
• Oscar Zariski

Fields

• hyperbolic geometry 3 lives
• teichmüller theory 3 lives
• kleinian groups 4 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Curtis Tracy McMullen, MacTutor History of Mathematics.
2. C. T. McMullen, Curriculum vitae, Harvard University.
3. Math Club interview with Professor Curtis McMullen, Harvard.
4. Curtis Tracy McMullen, Mathematics Genealogy Project.
5. Heidelberg Laureate Forum, Curtis T. McMullen.
6. The work of Curtis T. McMullen, Fields Medal laudatio, Documenta Mathematica, Extra Volume ICM I (1998), 127–132.

Mathematics

1. C. T. McMullen, The Hausdorff dimension of general Sierpiński carpets, Nagoya Math. J. 96 (1984), 1–9.
2. C. T. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. AMS 300 (1987), 329–342.
3. C. T. McMullen, Families of rational maps and iterative root-finding algorithms, Ann. of Math. 125 (1987), 467–493.
4. A. Eskin and C. T. McMullen, Mixing, counting and equidistribution in Lie groups, Duke Math. J. 71 (1993).
5. P. Doyle and C. T. McMullen, Solving the quintic by iteration, Acta Math. 163 (1989).
6. C. T. McMullen, Amenability, Poincaré series and quasiconformal maps, Invent. Math. 97 (1989), 95–127.
7. C. T. McMullen, Iteration on Teichmüller space, Invent. Math. 99 (1990), 425–454.
8. C. T. McMullen, Cusps are dense, Ann. of Math. 133 (1991), 217–247.
9. C. T. McMullen, Riemann surfaces and the geometrization of 3-manifolds, Bull. AMS 27 (1992), 207–216.
10. C. T. McMullen, Complex Dynamics and Renormalization, Annals of Math. Studies 135, Princeton, 1994; Frontiers in complex dynamics, Bull. AMS 31 (1994).
11. C. T. McMullen, Renormalization and 3-Manifolds which Fiber over the Circle, Annals of Math. Studies 142, Princeton, 1996.
12. C. T. McMullen, Polynomial invariants for fibered 3-manifolds and Teichmüller geodesics for foliations, Ann. Sci. ÉNS 33 (2000), 519–560; The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology, Ann. Sci. ÉNS 35 (2002), 153–171.
13. C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. AMS 16 (2003).
14. C. T. McMullen, Teichmüller curves in genus two: discriminant and spin, Math. Ann. 333 (2005), 87–130; The decagon and beyond, J. reine angew. Math. 582 (2005).
15. C. T. McMullen, Dynamics of SL₂(ℝ) over moduli space in genus two, Ann. of Math. 165 (2007), 397–456.
16. C. T. McMullen, Minkowski’s conjecture, well-rounded lattices and topological dimension, J. AMS 18 (2005), 711–734.
17. C. T. McMullen, Dynamics on K3 surfaces: Salem numbers and Siegel disks, J. reine angew. Math. 545 (2002); Dynamics on blowups of the projective plane, Publ. Math. IHÉS 105 (2007), 49–89; Automorphisms of projective K3 surfaces with minimum entropy, Invent. Math. 203 (2016), 179–215.
18. C. T. McMullen, A. Mohammadi, H. Oh, Geodesic planes in hyperbolic 3-manifolds, Invent. Math. 209 (2017), 425–461.
19. C. T. McMullen, R. Mukamel, A. Wright, Cubic curves and totally geodesic subvarieties of moduli space, Ann. of Math. 185 (2017), 957–990.
20. C. T. McMullen, Billiards and Teichmüller curves, Bull. AMS 60 (2023), 195–250.
21. S. Friedl, A survey of the Thurston norm, arXiv:2108.08645.
22. S. Brandhorst and N. D. Elkies, Equations for a K3 Lehmer map, arXiv:2103.15101.