## Sean Cleary

### Professor

Mathematics

#### Areas of Expertise/Research

• Group Theory
• Metrics on Spaces of Trees
• Computational Phylogentics
• Computational Mathematics
• Thompson's Groups

Marshak

MR 301C

212-650-5122

#### Website

http://cleary.ccnysites.cuny.edu

## 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 piecewise-linear homeomorphisms" supervised by Mladen Bestvina

### Recent Publications

• Sean Cleary and Roland Maio: Edge conflicts do not determine geodesics in the associahedron, SIAM Journal on Discrete Mathematics, Vol. 32 (2018), no. 2, 1003-1015.

There are no known efficient algorithms to calculate distance in the one-skeleta 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 rotation-distance 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 polynomial-time 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, 289-301..

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 quasi-isometry 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 Martinez-Perez: Undistorted embeddings of metabelian groups of finite Prüfer rank, New York Journal of Mathematics, 21 (2015) 1027-1054.

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.

Edge-flip 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 polynomial-time algorithms for computing edge-flip 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 edge-flip distance between triangular subdivisions of regular polygons. There are no known polynomial-time 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 randomly-selected 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) #8-1.

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), 809--828..

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 polynomial-time 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 randomly-selected 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. 3-4, 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 Baumslag-Solitar 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 2-cells adjacent to edges that do not lie on normal form paths.

• Sean Cleary and Katherine St. John: A Linear-Time 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 polynomial-time algorithms for computing rotation distance. We give an efficient, linear-time 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).

Higher-dimensional Thompson's groups nV are finitely presented groups described by Brin which generalize dyadic self-maps of the unit interval to dyadic self-maps of n-dimensional unit cubes. We describe some of the metric properties of higher-dimensional Thompson's groups. We give descriptions of elements based upon tree-pair 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 D-finite and not algebraic.

• Sean Cleary and Katherine St. John: Rotation distance is fixed-parameter tractable., Inform. Process. Lett. 109 (2009), no. 16, 918--922.

Rotation distance between trees measures the number of simple operations it takes to transform one tree into another. There are no known polynomial-time algorithms for computing rotation distance. In the case of ordered rooted trees, we show that the rotation distance between two ordered trees is fixed-parameter 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), 163--180..

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, 605--615.

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.

### Recent 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; Co-Principal 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; Co-Principal 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; Co-Principal 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; Co-Principal Investigator:Sean Cleary; Organization:CUNY City College;NSF Organization:DMS Start Date:06/01/2003; Award Amount:$10,000.00
• US-SPAIN Cooperative Research: Metric Properties of Thompson's Group Award Number:0305545; Principal Investigator:Sean Cleary; Co-Principal 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; Co-Principal 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, co-PI (PI Enric Ventura), “Geometric Methods in Group Theory,” 18,000 Euro , 1/1/2015 – 12/31/2017.
• Ministerio de Ciencia e Innovacion, Spanish Government, co-PI (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, 9/1/2012-8/31/2017.

CUNY and NYS Funding:

• PSC-CUNY 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 323, Advanced Calculus I
• Math 347, Elements of Modern Algebra
• Math 360, Introduction to Modern Geometry
• Math 392, Vector Analysis and Linear Algebra
• Math A4900, Graduate Abstract Algebra I
• Math B4900, Graduate Abstract Algebra II
• Math A6100, Graduate Differential Geometry