Sean Cleary
Professor
Areas of Expertise/Research
 Computational Mathematics
 Computational Phylogentics
 Group Theory
 Metrics on Spaces of Trees
 Thompson's Groups
Building
Marshak
Office
MR 301C
Phone
2126505122
Website
Sean Cleary
Education

A.B., Cornell University, Mathematics and Physics. Mathematics major concentration in pure mathematics, physics major concentration in astrophysics.

Ph.D. University of California, Los Angeles, Mathematics. Specializations in geometric group theory and topology. Thesis ``Groups of piecewiselinear homeomorphisms" supervised by Mladen Bestvina
Recent Publications
 Sean Cleary, Random explorations in Thompson's groups, Proceedings of Beyond Hyperbolicity, Seminaires et Congres, Societe Mathematique de France, Vol. 34 (2024), 4757.
 Sean Cleary, Restricted rotation distance between kary trees, Journal of Graph Algorithms and Applications, Vol. 27 No. 1 (2023) 1933.
We study restricted rotation distance between ternary and highervalence trees using approaches based upon generalizations of Thompson's group F. We obtain bounds and a method for computing these distances exactly in linear time, as well as a lineartime algorithm for computing rotations needed to realize these distances. Unlike the binary case, the highervalence notions of rotation distance do not give Tamari lattices, so there are fewer tools for analysis in the highervalence settings. Highervalence trees arise in a range of database and filesystem applications where balance is important for efficient performance.
 Sean Cleary and Roland Maio, An efficient algorithm for sampling difficult rotation distance instances Acta Cybernetica, Vol. 25 No. 3 (2022), 629646.
It is an open question whether there exists a polynomialtime algorithm for computing the rotation distances between pairs of extended ordered binary trees. The problem of computing the rotation distance between an arbitrary pair of trees, (S, T), can be efficiently reduced to the problem of computing the rotation distance between a difficult pair of trees (S', T'), where there is no known first step which is guaranteed to be the beginning of a minimal length path. Of interest, therefore, is how to sample such difficult pairs of trees of a fixed size. We show that it is possible to do so efficiently, and present such an algorithm that runs in time O(n^{4}).
 Sean Cleary and Ariadna Fossas Tenas, Elements with northsouth dynamics and free subgroups have positive density in Thompson's groups T and V, Munster Journal of Mathematics, Vol. 14 (2021) No. 2, 485496.
We show that the density of elements in Thompson's groups T and V which have northsouth dynamics acting on the circle is positive with respect to a stratification in terms of size. We show that the fraction of element pairs which generate a free subgroup of rank two is positive in both groups with respect to a stratification based on diagram size, which shows that the associated subgroup spectra of T and V contain free groups.
 Sean Cleary and Harris Nadeem Distributions of restricted rotation distances, The Art of Discrete and Applied Mathematics, Vol. 5 No. 1 (2022), No. 1.03, 116.
Rotation distances measure the differences in structure between rooted ordered binary trees. The onedimensional skeleta of associahedra are rotation graphs, where two vertices representing trees are connected by an edge if they differ by a single rotation. There are no known efficient algorithms to compute rotation distance between trees and thus distances in rotation graphs. Limiting the allowed locations of where rotations are permitted gives rise to a number of notions related to rotation distance. Allowing rotations at a minimal such set of locations gives restricted rotation distance. There are lineartime algorithms to compute restricted rotation distance, where there are only two permitted locations for rotations to occur. The associated restricted rotation graph has an efficient distance algorithm. There are linear upper and lower bounds on restricted rotation distance with respect to the sizes of the reduced tree pairs. Here, we experimentally investigate the expected restricted rotation distance between two trees selected at random of increasing size and find that it lies typically in a narrow band well within the earlier proven linear upper and lower bounds.
 Sean Cleary and Roland Maio Counting difficult tree pairs with respect to the rotation distance problem Journal of Combinatorial Mathematics and Combinatorial Computing, Vol. 115 (2020). 199213.
Rotation distance between rooted binary trees is the minimum number of simple rotations needed to transform one tree into the other. Computing the rotation distance between a pair of rooted trees can be quickly reduced in cases where there is a common edge between the trees, or where a single rotation introduces a common edge. Tree pairs which do not have such a reduction are difficult tree pairs, where there is no generally known first step. Here, we describe efforts to count and estimate the number of such difficult tree pairs, and find that the fraction decreases exponentially fast toward zero. We also describe how knowing the number of distinct instances of the rotation distance problem is a helpful factor in making the computations more feasible
 Sean Cleary and Roland Maio:Edge conflicts do not determine geodesics in the associahedron, SIAM Journal on Discrete Mathematics, Vol. 32 (2018), no. 2, 10031015.
There are no known efficient algorithms to calculate distance in the oneskeleta of associahedra, a problem that is equivalent to finding rotation distance between rooted binary trees or the flip distance between polygonal triangulations. One measure of the difference between trees is the number of conflicting edge pairs, and a natural way of trying to find short paths is to minimize successively this number of conflicting edge pairs using flip operations in the corresponding triangulations. We describe examples that show that the number of such conflicts does not always decrease along geodesics. Thus, a greedy algorithm that always chooses a transformation that reduces conflicts will not produce a geodesic in all cases. Further, for any specified amount, there are examples of pairs of all large sizes showing that the number of conflicts can increase by that amount along any geodesic between the pairs.
 Sean Cleary: Thompson's Group, Office hours with a geometric group theorist, Princeton University Press, 2017.
Book chapter covering the important example of Thompson's group F and its metric properties.
 Jose Burillo, Sean Cleary, and Claas E. Röver: Obstructions for subgroups of Thompson's group V, Geometric and cohomological group theory, London Math. Soc. Lecture Note Ser., 444, Cambridge Univ. Press, 2018.
Thompson's group V has a rich variety of subgroups, containing all finite groups, all finitely generated free groups and all finitely generated abelian groups, the finitary permutation group of a countable set, as well as many wreath products and other families of groups. Here, we describe some obstructions for a given group to be a subgroup of V.
 Sean Cleary and Roland Maio: Edge conflicts can increase along minimal rotationdistance paths, Proceedings of the 27th Annual Fall Workshop on Computational Geometry, Stony Brook University, (2017).
Rotation distance between binary trees measures differences in tree shape by counting the minimal number of rotations needed to transform one tree to another. There are no known polynomialtime algorithms for calculating rotation distance. The number of conflicts between edges is a rough measure of tree distance and has been useful in comparing trees for the extent of similarity. Here we show that the number of conflicts can rise along minimal length paths for rotation distance, which precludes finding such paths by examining edge conflicts alone.
 Jose Burillo, Sean Cleary, Armando Martino, and Claas Roever: Commensurations and Metric Properties of Houghton's Groups, Pacific Journal of Mathematics, Vol. 285 (2016), No. 2, 289301..
We describe the automorphism groups and the abstract commensurators of Houghton's groups. Then we give sharp estimates for the word metric of these groups and deduce that the commensurators embed into the corresponding quasiisometry groups. As a further consequence, we obtain that the Houghton group on two rays is at least quadratically distorted in those with three or more rays.
 Sean Cleary and Conchita MartinezPerez: Undistorted embeddings of metabelian groups of finite Prüfer rank, New York Journal of Mathematics, 21 (2015) 10271054.
General arguments of Baumslag and Bieri guarantee that any metabelian group of finite Pr\"ufer rank can be embedded in a metabelian constructible group. Here, we consider the metric behavior of a rich class of examples and analyze the distortions of specific embeddings.
 Timothy Chu and Sean Cleary: Expected maximum vertex valence in pairs of polygonal triangulations, Involve, a journal of mathematics Vol. 8 (2015), No. 5, 763–770.
Edgeflip distance between triangulations of polygons is equivalent to rotation distance between rooted binary trees. Both distances measure the extent of similarity of configurations. There are no known polynomialtime algorithms for computing edgeflip distance. The best known exact universal upper bounds on rotation distance arise from measuring the maximum total va lence of a vertex in the corresponding triangulation pair obtained by a duality construction. Here we describe some properties of the distribution of maximum vertex valences of pairs of triangulations related to such upper bounds.
 Sean Cleary, Andrew Rechnitzer, and Thomas Wong: Common Edges in Rooted Trees and Polygonal Triangulations, Electronic Journal of Combinatorics, Volume 20, Issue 1 (2013).
Rotation distance between rooted binary trees measures the degree of similarity of two trees with ordered leaves and is equivalent to edgeflip distance between triangular subdivisions of regular polygons. There are no known polynomialtime algorithms for computing rotation distance. Existence of common edges makes computing rotation distance more manageable by breaking the problem into smaller subproblems. Here we describe the distribution of common edges between randomlyselected triangulations and measure the sizes of the remaining pieces into which the common edges separate the polygons. We find that asymptotically there is a large component remaining after sectioning off numerous small polygons which gives some insight into the distribution of such distances and the difficulty of such distance computations, and we analyze the distributions of the sizes of the largest and smaller resulting polygons.
 Sean Cleary, John Passaro and Yasser Toruno: Average reductions between random tree pairs, Involve, a journal of mathematics (2015) #81.
There are a number of measures of degrees of similarity between rooted binary trees. Many of these ignore sections of the trees which are in complete agreement. We use computational experiments to investigate the statistical characteristics of such a measure of tree similarity for ordered, rooted, binary trees. We generate the trees used in the experiments iteratively, using the Yule process modeled upon speciation.
 Jose Burillo, Sean Cleary and Claas Roever: Addendum to ‘Commensurations and finite index subgroups of Thompson’s group F.’, Geometry & Topology 17 (2013) 1199–1203.
We show that the abstract commensurator of Thompson's group F is composed of four building blocks: two isomorphism types of simple groups, the multiplicative group of the positive rationals and a cyclic group of order two. The main result establishes the simplicity of a certain group of piecewise linear homeomorphisms of the real line.
 José Burillo and Sean Cleary: The automorphism group of Thompson's group F: subgroups and metric properties, Revista Matemática Iberoamericana, Vol. 29, #3 (2013), 809828..
We describe some of the geometric properties of the automorphism group Aut(F) of Thompson's group F. We give realizations of Aut(F) geometrically via periodic tree pair diagrams, which lead to natural presentations and give effective methods for estimating the word length of elements. We study some natural subgroups of Aut(F) and their metric properties. In particular, we show that the subgroup of inner automorphisms of F is at least quadratically distorted in Aut(F), whereas other subgroups of Aut(F) isomorphic to F are undistorted.
 Timothy Chu and Sean Cleary: Expected conflicts in pairs of rooted binary trees, Involve, a journal of mathematics #6 (2013).
Rotation distance between rooted binary trees measures the extent of similarity of two trees with ordered leaves. There are no known polynomialtime algorithms for computing rotation distance. If there are common edges or immediately changeable edges between a pair of trees, the rotation distance problem breaks into smaller subproblems. The number of crossings or conflicts of a tree pair also gives some measure of the extent of similarity of two trees. Here we describe the distribution of common edges and immediately changeable edges between randomlyselected pairs of trees via computer experiments, and examine the distribution of the amount of conflicts between such pairs.
 Sean Cleary, Susan Hermiller, Melanie Stein, and Jennifer Taback: Tame combing and almost convexity conditions, Math. Zeit., 269 (2011), no. 34, 879–915.
We give the first examples of groups which admit a tame combing with linear radial tameness function with respect to any choice of finite presentation, but which are not minimally almost convex on a standard generating set. Namely, we explicitly construct such combings for Thompson's group F and the BaumslagSolitar groups BS(1, p) with p \ge 3. In order to make this construction for Thompson's group F, we significantly expand the understanding of the Cayley complex of this group with respect to the standard finite presentation. In particular we describe a quasigeodesic set of normal forms and combinatorially classify the arrangements of 2cells adjacent to edges that do not lie on normal form paths.
 Sean Cleary and Katherine St. John: A LinearTime Approximation Algorithm for Rotation Distance, Journal of Graph Algorithms and Applications, 2010, Vol. 14, no. 2.
Rotation distance between rooted binary trees measures the number of simple operations it takes to transform one tree into another. There are no known polynomialtime algorithms for computing rotation distance. We give an efficient, lineartime approximation algorithm, which estimates the rotation distance, within a provable factor of 2, between ordered rooted binary trees. .
 Jose Burillo and Sean Cleary: Metric properties of higher dimensional Thompson's groups, Pacific Journal of Mathematics, 248, #1 (2010).
Higherdimensional Thompson's groups nV are finitely presented groups described by Brin which generalize dyadic selfmaps of the unit interval to dyadic selfmaps of ndimensional unit cubes. We describe some of the metric properties of higherdimensional Thompson's groups. We give descriptions of elements based upon treepair diagrams and give upper and lower bounds for word length in terms of the size of the diagrams. Though these upper and lower bounds are somewhat separated, we show that there are elements realizing the lower bounds and that the fraction of elements which are close to the upper bound converges to 1, showing that the bounds are optimal and that the upper bound is generically achieved.
 Sean Cleary, Murray Elder, Andrew Rechnitzer, and Jennifer Taback: Random subgroups of Thompson's group , Groups, Geometry and Dynamics, Vol 4, #1, 2010..
We consider random subgroups of Thompson's group $F$ with respect to two natural stratifications of the set of all $k$ generator subgroups of this group. We find that the isomorphism classes of subgroups which occur with positive density vary greatly between the two stratifications. We give the first known examples of {\em persistent} subgroups, whose isomorphism classes occur with positive density within the set of $k$generator subgroups, for all $k$ greater than some $k_0$. Additionally, Thompson's group provides the first example of a group without a generic isomorphism class of subgroup. In $F$, there are many isomorphism classes of subgroups with positive density less than one. Elements of $F$ are represented uniquely by reduced pairs of finite rooted binary trees. We compute the asymptotic growth rate and a generating function for the number of reduced pairs of trees, which we show is Dfinite and not algebraic.
 Sean Cleary and Katherine St. John: Rotation distance is fixedparameter tractable., Inform. Process. Lett. 109 (2009), no. 16, 918922.
Rotation distance between trees measures the number of simple operations it takes to transform one tree into another. There are no known polynomialtime algorithms for computing rotation distance. In the case of ordered rooted trees, we show that the rotation distance between two ordered trees is fixedparameter tractable, in the parameter, k, the rotation distance. The proof relies on the kernalization of the initial trees to trees with size bounded by 5k.
 Blake Fordham and Sean Cleary: Minimal length elements of Thompson's groups F(p), Geom. Dedicata 141 (2009), 163180..
We describe a method for determining the minimal length of elements in the generalized Thompson's groups F(p). We compute the length of an element by constructing a tree pair diagram for the element, classifying the nodes of the tree and summing associated weights from the pairs of node classi¯cations. We use this method to e®ectively ¯nd minimal length representatives of an element.
 Jose Burillo and Sean Cleary: Metric properties of braided Thompson's groups, Indiana Univ. Math. J. 58 (2009), no. 2, 605615.
Braided Thompson's groups are finitely presented groups introduced by Brin and Dehornoy which contain the ordinary braid groups $Bn$, the finitary braid group $B{\infty}$ and Thompson's group $F$ as subgroups. We describe some of the metric properties of braided Thompson's groups and give upper and lower bounds for word length in terms of the number of strands and the number of crossings in the diagrams used to represent elements.
Research Awards
National Science Foundation research award funding:
 Experimental and Theoretical Analyses of Tree Distance Distributions Award Number:1417820; Principal Investigator:Sean Cleary; Organization:CUNY City College;NSF Organization:DMS Start Date:07/01/2014; Award Amount:$240,000.00
 Finitely presented solvable groups at The City College of New York, Fall 2010 conference Award Number:1061232; Principal Investigator:Gilbert Baumslag; CoPrincipal Investigator:Sean Cleary; Organization:CUNY City College;NSF Organization:DMS Start Date:01/15/2011; Award Amount:$20,000.00
 Experimental and Theoretical Approaches for Efficient Tree Distance Algorithms Award Number:0811002; Principal Investigator:Sean Cleary; Organization:CUNY City College;NSF Organization:DMS Start Date:07/01/2008; Award Amount:$120,989.00
 Finitely presented groups at The City College of New York, Spring 2009 conference Award Number:0854902; Principal Investigator:Gilbert Baumslag; CoPrincipal Investigator:Sean Cleary; Organization:CUNY City College;NSF Organization:DMS Start Date:03/01/2009; Award Amount:$15,000.00
 Parametric Computation in Axiom Towards Indefinite Symbolic Computing Award Number:0430722; Principal Investigator:Gilbert Baumslag; CoPrincipal Investigator:Timothy Daly, William Sit, Sean Cleary, Douglas Troeger; Organization:CUNY City College;NSF Organization:CCF Start Date:08/01/2004; Award Amount:$170,000.00
 New York Group Theory Seminar and Symbolic Computation Workshops Award Number:0330802; Principal Investigator:Gilbert Baumslag; CoPrincipal Investigator:Sean Cleary; Organization:CUNY City College;NSF Organization:DMS Start Date:06/01/2003; Award Amount:$10,000.00
 USSPAIN Cooperative Research: Metric Properties of Thompson's Group Award Number:0305545; Principal Investigator:Sean Cleary; CoPrincipal Investigator:Jennifer Taback; Organization:CUNY City College;NSF Organization:DMS Start Date:07/15/2003; Award Amount:$14,140.00
 MRI: Parallel Computing Environment for Computational Mathematics Award Number:0215942; Principal Investigator:Katherine St. John; CoPrincipal Investigator:Gilbert Baumslag, Sean Cleary, Damian Rouson; Organization:Research Foundation Of The City University Of New York (Lehman);NSF Organization:DMS Start Date:08/01/2002; Award Amount:$173,673.00
International Funding:
 Ministerio de Ciencia e Innovacion, Spanish Government, coPI (PI Enric Ventura), “Geometric Methods in Group Theory,” 18,000 Euro , 1/1/2015 – 12/31/2017.
 Ministerio de Ciencia e Innovacion, Spanish Government, coPI (PI Enric Ventura, other coPIs Jose Burillo and Warren Dicks), “Geometric Methods in Group Theory,” 60,000 Euro , 1/1/2012 – 12/31/2014.
Private Foundation Funding:
 Simons Foundation, PI, “Metric properties of Thompson’s groups”, Collaboration Grants for Mathematics, $35,000, 9/1/20128/31/2017.
CUNY and NYS Funding:
 PSCCUNY research awards
 Graduate Research Technology Initiative awards
 CUNY Faculty Development Program Grants
Books and Book Chapters

Thompson's group, in "Office hours with a geometric group theorist", Princeton University Press, 2017.

"Geometric Methods in Group Theory", editor, with Jose Burillo, Murray Elder, Jennifer Taback and Enric Ventura, Contemporary Mathematics series of American Mathematical Society, 2005, 230 pages.

"Combinatorial and Computational Group Theory", editor, with Robert Gilman, Alexei G. Miasnikov, and Vladimir Shpilrain, Contemporary Mathematics series of American Mathematical Society, 2002, 275 pages.

"A Mathematica Mystery Tour: An Elementary Primer and Reference on the use of Mathematica" (with Larry Cusick), Kennel Press, 1998, 60 pages.

"Vector and Multivariate Calculus: Some Mathematica Explorations", Kennel Press, 1998, 120 pages.

"Differential Equations, Linear Algebra, and Fourier Series: A Mathematica Workbook for Scientists and Engineers", Kennel Press, 1998, 108 pages.
Recent Courses Taught
 Math 203: Calculus III
 Math 212: Calculus II with introduction to Multivariate Functions
 Math 213: Calculus III with Vector Analysis
 Math 213: Calculus III with Planar Vector Analysis
 Math 323, Advanced Calculus I
 Math 347, Elements of Modern Algebra
 Math 360, Introduction to Modern Geometry
 Math 392, Vector Analysis and Linear Algebra
 Math 461, Differential Geometry
 Math A4900, Graduate Abstract Algebra I
 Math B4900, Graduate Abstract Algebra II
 Math A6100, Graduate Differential Geometry
Webpages and publication links

Some arXiv preprints.

MathSciNet reviews of publications (subscription required)