Scientists / Open questions

Curtis T. Mcmullen

Dynamical systems; hyperbolic geometry

1998 Fields Medal

29 questions

Explicit models for isoperiodic moduli spaces: the boundary of the periodic locus and genus-three cubic-field examples
McMullen's $M_d = 2$ question: infinitely many periodic continued fractions with partial quotients 1 and 2 in every real quadratic field
Asymptotic distribution of period geodesics in strata of Abelian differentials, including geodesics transverse to the absolute period foliation
Infinite generation of circle stabilizers in Kleinian groups
The compact-group Chebotarev law for knots: classification beyond semisimple targets
Divergence of complex-earthquake holonomy and holomorphic-function convexity of quasi-Fuchsian space in the character variety
Sharp lower bounds for the minimal growth rate of a graph with connected complement: McMullen's questions $\lambda(G) \geq |V(G)|$ and $\lambda(G) \geq 2^n$
Horocycle orbit-closure rigidity beyond the rigid acylindrical setting (McMullen–Mohammadi–Oh, 2016)
McMullen's open questions on cycle lengths for Blaschke and quasi-Blaschke products: critical points of the length functional, uniqueness of the binding minimum, a uniform simplicity threshold, compactness, and an intersection number for cycles
Which Salem numbers occur as dynamical degrees of automorphisms of K3 surfaces: the degree-10 non-projective case and the projective characterization
Effective and explicit gap statistics of $\sqrt{un}\bmod 1$ : the surviving open questions of Elkies and McMullen
The remaining case of McMullen's Conjecture 1.3: Cannon-Thurston maps from the Floyd boundary for Kleinian groups with degenerate compressible ends and parabolics
Exact realization of prescribed dynamics on prescribed postcritical sets of rational maps (DeMarco-Koch-McMullen, Question 1.2)
Margulis's conjecture on bounded diagonal orbits in the space of lattices, and the Woods covering conjecture for well-rounded lattices in dimensions 11-23
Classification of the realizable elements of the Weyl groups $W_n$ of blowups of the projective plane, and of the automorphisms they determine
Classifying primitive Teichmüller curves in genera three and four
For which $n$ and $q$ is the $A_n(q)$ complex reflection group a lattice in $U(r,s)$ ? (McMullen, Question 5.6)
Ergodic decomposition of vertical foliations under mass-loss degenerations in the moduli space of quadratic differentials
Veech groups beyond genus two: limit sets in higher genus, higher-degree trace fields, and cusp sets of triangle groups
Cyclotomic factors of Coxeter polynomials for diagrams obtained by decorating $A_n$ : a spherical-origin criterion and effective periodicity
Strong convergence, realization, and escape-rate control of geometric limits of degenerating Blaschke products
The sharp regularity threshold between Lipschitz and Hölder for prescribed-Jacobian realizability and separated nets in Euclidean space
Sharp $2p/(p+1)$ threshold for the Hausdorff dimension of Julia sets under horocyclic convergence of geometrically finite rational maps
No invariant line fields, density of hyperbolicity, and the geometry of the dynamical moduli space: the open core of the McMullen–Sullivan Teichmüller theory of rational maps

Back to open problems

Explicit models for isoperiodic moduli spaces: the boundary of the periodic locus and genus-three cubic-field examples

McMullen's Md=2M_d = 2Md​=2 question: infinitely many periodic continued fractions with partial quotients 1 and 2 in every real quadratic field

Asymptotic distribution of period geodesics in strata of Abelian differentials, including geodesics transverse to the absolute period foliation

Infinite generation of circle stabilizers in Kleinian groups

The compact-group Chebotarev law for knots: classification beyond semisimple targets

Divergence of complex-earthquake holonomy and holomorphic-function convexity of quasi-Fuchsian space in the character variety

Sharp lower bounds for the minimal growth rate of a graph with connected complement: McMullen's questions λ(G)≥∣V(G)∣\lambda(G) \geq |V(G)|λ(G)≥∣V(G)∣ and λ(G)≥2n\lambda(G) \geq 2^nλ(G)≥2n

Horocycle orbit-closure rigidity beyond the rigid acylindrical setting (McMullen–Mohammadi–Oh, 2016)

McMullen's open questions on cycle lengths for Blaschke and quasi-Blaschke products: critical points of the length functional, uniqueness of the binding minimum, a uniform simplicity threshold, compactness, and an intersection number for cycles

Which Salem numbers occur as dynamical degrees of automorphisms of K3 surfaces: the degree-10 non-projective case and the projective characterization

Effective and explicit gap statistics of un mod 1\sqrt{un}\bmod 1un​mod1: the surviving open questions of Elkies and McMullen

The remaining case of McMullen's Conjecture 1.3: Cannon-Thurston maps from the Floyd boundary for Kleinian groups with degenerate compressible ends and parabolics

Exact realization of prescribed dynamics on prescribed postcritical sets of rational maps (DeMarco-Koch-McMullen, Question 1.2)

Margulis's conjecture on bounded diagonal orbits in the space of lattices, and the Woods covering conjecture for well-rounded lattices in dimensions 11-23

Classification of the realizable elements of the Weyl groups WnW_nWn​ of blowups of the projective plane, and of the automorphisms they determine

Classifying primitive Teichmüller curves in genera three and four

For which nnn and qqq is the An(q)A_n(q)An​(q) complex reflection group a lattice in U(r,s)U(r,s)U(r,s)? (McMullen, Question 5.6)

Ergodic decomposition of vertical foliations under mass-loss degenerations in the moduli space of quadratic differentials

Veech groups beyond genus two: limit sets in higher genus, higher-degree trace fields, and cusp sets of triangle groups

Cyclotomic factors of Coxeter polynomials for diagrams obtained by decorating AnA_nAn​: a spherical-origin criterion and effective periodicity

Strong convergence, realization, and escape-rate control of geometric limits of degenerating Blaschke products

The sharp regularity threshold between Lipschitz and Hölder for prescribed-Jacobian realizability and separated nets in Euclidean space

Sharp 2p/(p+1)2p/(p+1)2p/(p+1) threshold for the Hausdorff dimension of Julia sets under horocyclic convergence of geometrically finite rational maps

No invariant line fields, density of hyperbolicity, and the geometry of the dynamical moduli space: the open core of the McMullen–Sullivan Teichmüller theory of rational maps

McMullen's $M_d = 2$ question: infinitely many periodic continued fractions with partial quotients 1 and 2 in every real quadratic field

Sharp lower bounds for the minimal growth rate of a graph with connected complement: McMullen's questions $\lambda(G) \geq |V(G)|$ and $\lambda(G) \geq 2^n$

Effective and explicit gap statistics of $\sqrt{un}\bmod 1$ : the surviving open questions of Elkies and McMullen

Classification of the realizable elements of the Weyl groups $W_n$ of blowups of the projective plane, and of the automorphisms they determine

For which $n$ and $q$ is the $A_n(q)$ complex reflection group a lattice in $U(r,s)$ ? (McMullen, Question 5.6)

Cyclotomic factors of Coxeter polynomials for diagrams obtained by decorating $A_n$ : a spherical-origin criterion and effective periodicity

Sharp $2p/(p+1)$ threshold for the Hausdorff dimension of Julia sets under horocyclic convergence of geometrically finite rational maps