About

Professor Jon McCammond is associated with the field of geometric group theory and low-dimensional topology. The Geometric Group Theory Page provides information and resources about these areas, indicating his involvement and expertise in this field. The page is designed to help students, scholars, and interested laypersons orient themselves to this large and ever-expanding body of work, suggesting that Professor McCammond is engaged in research, teaching, or dissemination related to geometric group theory and its neighboring fields.

Research topics

• Pure mathematics
• Mathematics
• Computer Science
• Combinatorics
• Geometry
• Mathematical analysis

Selected publications

• The Geometry of Rectangular Multisets

  ArXiv.org · 2026-04-15

  articleOpen accessSenior author

  This article describes a natural piecewise Euclidean bi-simplicial cell structure for the space of $n$-element multisets in a fixed Euclidean rectangle. In particular, we highlight some connections with spaces of complex polynomials and permutahedra.

  PublisherOA PDF

• The Geometry of Rectangular Multisets

  arXiv (Cornell University) · 2026-04-15

  preprintOpen accessSenior author

  This article describes a natural piecewise Euclidean bi-simplicial cell structure for the space of $n$-element multisets in a fixed Euclidean rectangle. In particular, we highlight some connections with spaces of complex polynomials and permutahedra.

  PublisherDOI

• Extended Weak Order for the Rank 3 Universal Coxeter Group

  ArXiv.org · 2025-08-31

  preprintOpen access

  The weak order is a classical poset structure on a Coxeter group; it is a lattice when the group is finite but merely a meet-semilattice when the group is infinite. Motivated by problems in Kazhdan--Lusztig theory, Matthew Dyer introduced the extended weak order, a poset that contains a copy of the weak order as an order ideal, and he conjectured that the extended weak order for any Coxeter group is a lattice. We prove Dyer's conjecture for the rank 3 universal Coxeter group. This is the first non-spherical, non-affine Coxeter group for which Dyer's conjecture has been proven.

  PublisherOA PDFDOI

• Continuous Noncrossing Partitions and Weighted Circular Factorizations

  ArXiv.org · 2025-06-30

  preprintOpen accessSenior author

  This article examines noncrossing partitions of the unit circle in the complex plane; we call these continuous noncrossing partitions. More precisely, we focus on the degree-$d$ continuous noncrossing partitions where unit complex numbers in the same block have identical $d$-th powers. We prove that the degree-$d$ continuous noncrossing partitions form a topological poset whose uncountable set of elements can be indexed by equivalence classes of objects we call weighted linear factorizations of factors of a $d$-cycle. Moreover, the maximal elements in this poset form a subspace homeomorphic to the dual Garside classifying space for the $d$-strand braid group. The degree-$d$ continuous noncrossing partitions of the unit circle are a special case of a more general construction. For every choice of Coxeter element $c$ in any Coxeter group $W$ we define a topological poset of equivalence classes of weighted linear factorizations of factors of $c$ in $W$ whose elements we call continuous $c$-noncrossing partitions. The maximal elements in this poset form a subspace homeomorphic to the one-vertex complex whose fundamental group is the corresponding dual Artin group.

  PublisherOA PDFDOI

• Geometric Combinatorics of Polynomials II: Polynomials and Cell Structures

  arXiv (Cornell University) · 2024-10-04

  preprintOpen accessSenior author

  This article introduces a finite piecewise Euclidean cell complex homeomorphic to the space of monic centered complex polynomials of degree $d$ whose critical values lie in a fixed closed rectangular region. We call this the branched rectangle complex since its points are indexed by marked $d$-sheeted planar branched covers of the fixed rectangle. The vertices of the cell structure are indexed by the combinatorial "basketballs" studied by Martin, Savitt and Singer. Structurally, the branched rectangle complex is a full subcomplex of a direct product of two copies of the order complex of the noncrossing partition lattice. Topologically, it is homeomorphic to the closed $2n$-dimensional ball where $n=d-1$. Metrically, the simplices in each factor are orthoschemes. It can also be viewed as a compactification of the space of all monic centered complex polynomials of degree $d$. We also introduce a finite piecewise Euclidean cell complex homeomorphic to the space of monic centered complex polynomials of degree $d$ whose critical values lie in a fixed closed annular region. We call this the branched annulus complex since its points are indexed by marked $d$-sheeted planar branched covers of the fixed annulus.It can be constructed from the branched rectangle complex as a cellular quotient by isometric face identifications. And it can be viewed as a compactification of the space of all monic centered complex polynomials of degree $d$ with distinct roots. Finally, the branched annulus complex deformation retracts to the branched circle complex, which we identify with the dual braid complex. Our explicit embedding of the dual braid complex as a spine for the space of polynomials with distinct roots provides a direct proof that these two classifying spaces for the braid group are homotopy equivalent.

  PublisherOA PDFDOI

• Fixed points of parking functions

  Transactions of the American Mathematical Society · 2023-06-14 · 1 citations

  preprint1st authorCorresponding

  We define an action of words in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket m right-bracket Superscript n"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>m</mml:mi> <mml:msup> <mml:mo stretchy="false">]</mml:mo> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">[m]^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript m"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>m</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">{\mathbb {R}}^m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to give a new characterization of rational parking functions—they are exactly those words whose action has a fixed point. We use this viewpoint to give a simple definition of Gorsky, Mazin, and Vazirani’s zeta map on rational parking functions when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are coprime [Trans. Amer. Math. Soc. 368 (2016), pp. 8403–8445], and prove that this zeta map is invertible. A specialization recovers Loehr and Warrington’s sweep map on rational Dyck paths (see D. Armstrong, N. A. Loehr, and G. S. Warrington [Adv. Math. 284 (2015), pp. 159–185; E. Gorsky, M. Mazin, and M. Vazirani [Electron. J. Combin. 24 (2017), p. 29; H. Thomas and N. Williams, Selecta Math. (N.S.) 24 (2018), pp. 2003–2034]).

  PublisherDOI

• Connectivity at infinity for state spaces of complete bipartite graphs

  Rocky Mountain Journal of Mathematics · 2022-04-01

  article

  The state or configuration space for r vertices on a complete bipartite graph Km,n is a CAT(0) cube complex, which is sometimes interpreted as a parameter space for r robots moving on a Km,n. We combine an analysis of the topology of links of vertices in this complex, the description of a hidden symmetry among the parameters, and known results from the literature to explicitly compute the exact degree to which the universal covers of these complexes are connected at infinity.

  PublisherDOI

• Factoring isometries of quadratic spaces into reflections

  Journal of Algebra · 2022-03-29

  preprintOpen access1st author
  PublisherOA PDFDOI

• Geometric combinatorics of polynomials I: The case of a single polynomial

  Journal of Algebra · 2021-09-02

  preprintOpen accessSenior authorCorresponding

  There are many different algebraic, geometric and combinatorial objects that one can attach to a complex polynomial with distinct roots. In this article we introduce a new object that encodes many of the existing objects that have previously appeared in the literature. Concretely, for every complex polynomial p with d distinct roots and degree at least 2, we produce a canonical compact planar 2-complex that is a compact metric version of a tiled phase diagram. It has a locally metric that is locally Euclidean away from a finite set of interior points indexed by the critical points of p, and each of its 2-cells is a metric rectangle. From this planar rectangular 2-complex one can use metric graphs known as metric cacti and metric banyans to read off several pieces of combinatorial data: a chain in the partition lattice, a cyclic factorization of a d-cycle, a real noncrossing partition (also known as a primitive d-major), and the monodromy permutations for the polynomial. This article is the first in a series.

  PublisherDOI

• Geometric Combinatorics of Polynomials I: The Case of a Single\n Polynomial

  arXiv (Cornell University) · 2021-04-15

  preprintOpen accessSenior author

  There are many different algebraic, geometric and combinatorial objects that\none can attach to a complex polynomial with distinct roots. In this article we\nintroduce a new object that encodes many of the existing objects that have\npreviously appeared in the literature. Concretely, for every complex polynomial\n$p$ with $d$ distinct roots and degree at least 2, we produce a canonical\ncompact planar 2-complex that is a compact metric version of a tiled phase\ndiagram. It has a locally CAT(0) metric that is locally Euclidean away from a\nfinite set of interior points indexed by the critical points of $p$, and each\nof its 2-cells is a metric rectangle. From this planar rectangular 2-complex\none can use metric graphs known as metric cacti and metric banyans to read off\nseveral pieces of combinatorial data: a chain in the partition lattice, a\ncyclic factorization of a d-cycle, a real noncrossing partition (also known as\na primitive d-major), and the monodromy permutations for the polynomial. This\narticle is the first in a series.\n

  PublisherOA PDFDOI

Recent grants

• Geometric Group Theory via Geometric Combinatorics

  NSF · $115k · 2004–2008

• Discrete and continuous geometry in group theory

  NSF · $230k · 2008–2012

Frequent coauthors

• John Meier

  11 shared

• Michael Dougherty

  Lafayette College

  10 shared

• Noel Brady

  University of Oklahoma

  8 shared

• Henning Mortveit

  University of Virginia

  6 shared

• Matthew Macauley

  Clemson University

  6 shared

• Daniel T. Wise

  McGill University

  5 shared

• Nic Koban

  3 shared

• Damian Osajda

  3 shared