About

Danny Calegari is a Professor in the Department of Mathematics at The University of Chicago. He holds a PhD from the University of California, Berkeley, obtained in 2000. His research interests include geometry, topology, and dynamics in low dimensions; foliations and laminations; and extremal problems in topology and geometric group theory. His work encompasses various areas such as geometric topology, geometric group theory, dynamical systems, and ergodic theory. For more detailed information, his webpage can be visited at http://math.uchicago.edu/~dannyc/.

Research topics

• Mathematical analysis
• Geometry
• Mathematics
• Pure mathematics

Selected publications

• CaTherine wheels from trees and Liouville quantum gravity

  arXiv (Cornell University) · 2026-04-18

  articleOpen access1st authorCorresponding

  A CaTherine wheel is a space-filling curve $f : S^1\to S^2$ such that for every closed interval $J\subset S^1$, $f(J)$ is homeomorphic to a closed disk and $f(\partial J)$ is contained in $\partial f(J)$. A CaTherine wheel gives rise to a pair of disjoint, dense topological trees in $S^2$ which roughly speaking lie to the left and right of $f$. We give necessary and sufficient conditions for a topological tree in $S^2$ to arise as one of these trees for some CaTherine wheel $f$. We apply this result to show that there is a unique CaTherine wheel corresponding to the geodesic tree rooted at $\infty$ for the $γ$-Liouville quantum gravity (LQG) metric, for $γ\in (0,2)$. In other words, we construct the space-filling curve which is the contour exploration of the LQG geodesic tree.

  PublisherOA PDF

• CaTherine wheels

  arXiv (Cornell University) · 2026-04-27

  preprintOpen access1st authorCorresponding

  A CaTherine wheel is a surjective continuous map $f:S^1 \to S^2$ such that for every closed interval $I\subset S^1$ the image $f(I)$ is homeomorphic to a disk, and $f(\partial I)$ is contained in the boundary of this disk. CaTherine wheels arise in many areas of low-dimensional geometry and topology, including conformal dynamics (expanding Thurston maps, expanding origamis), probability theory (whole plane ${\rm SLE}_κ$ for $κ\ge 8$, LQG metric trees) and elsewhere. We develop their theory in generality, and explain how CaTherine wheels and their associated structures can serve as a dictionary between these various fields. Our most substantial applications are to the theory of hyperbolic 3-manifolds. If $M$ is a closed hyperbolic 3-manifold and $G=π_1(M)$, we show that there is a canonical bijection between four kinds of structures associated to $M$: 1. orbit-equivalence classes of pseudo-Anosov flows on $M$ without perfect fits; 2. $G$-equivariant CaTherine wheels up to conjugacy; 3. minimal $G$-zippers; and 4. connected components of the space of uniform quasimorphisms on $G$. This generalizes and amplifies the theory of fiberings of hyperbolic 3-manifolds over the circle and the Thurston norm.

  PublisherDOI

• CaTherine wheels

  ArXiv.org · 2026-04-27

  articleOpen access1st authorCorresponding

  A CaTherine wheel is a surjective continuous map $f:S^1 \to S^2$ such that for every closed interval $I\subset S^1$ the image $f(I)$ is homeomorphic to a disk, and $f(\partial I)$ is contained in the boundary of this disk. CaTherine wheels arise in many areas of low-dimensional geometry and topology, including conformal dynamics (expanding Thurston maps, expanding origamis), probability theory (whole plane ${\rm SLE}_κ$ for $κ\ge 8$, LQG metric trees) and elsewhere. We develop their theory in generality, and explain how CaTherine wheels and their associated structures can serve as a dictionary between these various fields. Our most substantial applications are to the theory of hyperbolic 3-manifolds. If $M$ is a closed hyperbolic 3-manifold and $G=π_1(M)$, we show that there is a canonical bijection between four kinds of structures associated to $M$: 1. orbit-equivalence classes of pseudo-Anosov flows on $M$ without perfect fits; 2. $G$-equivariant CaTherine wheels up to conjugacy; 3. minimal $G$-zippers; and 4. connected components of the space of uniform quasimorphisms on $G$. This generalizes and amplifies the theory of fiberings of hyperbolic 3-manifolds over the circle and the Thurston norm.

  PublisherOA PDF

• CaTherine wheels from trees and Liouville quantum gravity

  arXiv (Cornell University) · 2026-04-18

  preprintOpen access1st authorCorresponding

  A CaTherine wheel is a space-filling curve $f : S^1\to S^2$ such that for every closed interval $J\subset S^1$, $f(J)$ is homeomorphic to a closed disk and $f(\partial J)$ is contained in $\partial f(J)$. A CaTherine wheel gives rise to a pair of disjoint, dense topological trees in $S^2$ which roughly speaking lie to the left and right of $f$. We give necessary and sufficient conditions for a topological tree in $S^2$ to arise as one of these trees for some CaTherine wheel $f$. We apply this result to show that there is a unique CaTherine wheel corresponding to the geodesic tree rooted at $\infty$ for the $γ$-Liouville quantum gravity (LQG) metric, for $γ\in (0,2)$. In other words, we construct the space-filling curve which is the contour exploration of the LQG geodesic tree.

  PublisherDOI

• Surgery sequences and self-similarity of the Mandelbrot set

  Algebraic & Geometric Topology · 2025-09-17

  articleOpen access1st authorCorresponding
  PublisherOA PDFDOI

• Combinatorics of the tautological lamination

  Pacific Journal of Mathematics · 2024-06-12

  articleOpen access1st authorCorresponding

  The tautological lamination arises in holomorphic dynamics as a combinatorial model for the geometry of 1-dimensional slices of the shift locus.In each degree q the tautological lamination defines an iterated sequence of partitions of 1 (one for each integer n) into numbers of the form 2 m q -n .Denote by N q (n, m) the number of times 2 m q -n arises in the n-th partition.We prove a recursion formula for N q (n, 0), and a gap theorem: N q (n, n) = 1 and N q (n, m) = 0 for ⌊n/2⌋ < m < n.

  PublisherOA PDFDOI

• Zippers

  Open MIND · 2024-01-01

  article1st authorCorresponding

  If $M$ is a hyperbolic 3-manifold fibering over the circle, the fundamental group of $M$ acts faithfully by homeomorphisms on a circle (the circle at infinity of the universal cover of the fiber), preserving a pair of invariant (stable and unstable) laminations. Many different kinds of dynamical structures (e.g. taut foliations, quasigeodesic or pseudo-Anosov flows) are known to give rise to universal circles -- a circle with a faithful $π_1(M)$ action preserving a pair of invariant laminations -- and these universal circles play a key role in relating the dynamical structure to the geometry of $M$. In this paper we introduce the idea of zippers, which give a new and direct way to construct universal circles, streamlining the known constructions in many cases, and giving a host of new constructions in others. In particular, zippers (and their associated universal circles) may be constructed directly from uniform quasimorphisms or from uniform left orders.

• Baguenaudy, Baguenaudier, Baguenaudiest

  Notices of the American Mathematical Society · 2024-01-01

  articleOpen access1st authorCorresponding

  The red hook in the figure must be extricated from the rings by sliding it left or right (when the rings are not in the way) and by moving the rings on their side through the hook to remove them or put them back.

  PublisherDOI

• 4. Disappointment by Danny Calegari

  Notices of the American Mathematical Society · 2023-10-01

  article1st authorCorresponding
  PublisherDOI

• Almost Sufficiently Large

  Notices of the American Mathematical Society · 2023-09-07

  articleOpen access1st authorCorresponding

  The long and meandering route that led from the resolution of the unknot recognition problem to the Virtual Haken Conjecture winds in and out of topology, geometry, combinatorics, group theory, PDE, and many other fields.

  PublisherDOI

Recent grants

• Stable commutator length

  NSF · $404k · 2010–2013

• Groups and Group Actions in Low-Dimensional Topology

  NSF · $258k · 2004–2007

Frequent coauthors

• Alden Walker

  CCI Reprographics (United States)

  25 shared

• Shicheng Wang

  Peking University

  10 shared

• Hongbin Sun

  Rutgers, The State University of New Jersey

  10 shared

• Koji Fujiwara

  Doshisha University

  9 shared

• Nathan M. Dunfield

  8 shared

• Lvzhou Chen

  Purdue University West Lafayette

  8 shared

• Michael Freedman

  5 shared

• Cameron Gordon

  Australian National University

  5 shared

Education

• Ph.D.

  University of California, Berkeley

  2000