Orbifolds and commensurability

These are notes based on a series of talks that the author gave at the "Interactions between hyperbolic geometry and quantum groups" conference held at Columbia University in June of 2009.Comment: 11 page

Unfixing knowledges: Queering the literacy curriculum

In the literacy classroom, students have few opportunities to use their literacy practices to contest narratives of race, class, gender and sexuality. Instead, extensive time is spent completing literacy activities associated with what 'good' readers and writers do. Students' literacy practices are often formulaic, repetitive, and serve classroom management strategies producing a mythic narrative of good literacy teaching. This paper introduces a queer literacy curriculum that poses pedagogy as a series of questions: What does being taught, what does knowledge do to students? How does knowledge become understood in the relationship between teacher/text and student? (Lusted, 1986) It emphasizes developing critical analyses of heterosexism, heteronormativity and normativity with the goal of helping students understand binary categories are not givens, rather social constructions we are often forced to perform (Butler, 1990) through available discourses. The paper highlights an interruption into the literacy curriculum where, through collective memory work, students investigated, analysed and contested the usually-not-noticed ways a small understanding of heterosexuality has come to structure their lives

Enumeration of Hypermaps of a Given Genus

This paper addresses the enumeration of rooted and unrooted hypermaps of a given genus. For rooted hypermaps the enumeration method consists of considering the more general family of multirooted hypermaps, in which darts other than the root dart are distinguished. We give functional equations for the generating series counting multirooted hypermaps of a given genus by number of darts, vertices, edges, faces and the degrees of the vertices containing the distinguished darts. We solve these equations to get parametric expressions of the generating functions of rooted hypermaps of low genus. We also count unrooted hypermaps of given genus by number of darts, vertices, hyperedges and faces.Comment: 42 page

Coxeter groups, hyperbolic cubes, and acute triangulations

Let $C(L)$ be the right-angled Coxeter group defined by an abstract triangulation $L$ of $\mathbb{S}^2$. We show that $C(L)$ is isomorphic to a hyperbolic right-angled reflection group if and only if $L$ can be realized as an acute triangulation. The proof relies on the theory of CAT(-1) spaces. A corollary is that an abstract triangulation of $\mathbb{S}^2$ can be realized as an acute triangulation exactly when it satisfies a combinatorial condition called "flag no-square". We also study generalizations of this result to other angle bounds, other planar surfaces and other dimensions.Comment: 27 pages, 9 figures. Accepted for publication by the Journal of Topolog

The big Dehn surgery graph and the link of S^3

In a talk at the Cornell Topology Festival in 2005, W. Thurston discussed a graph which we call "The Big Dehn Surgery Graph", B. Here we explore this graph, particularly the link of S^3, and prove facts about the geometry and topology of B. We also investigate some interesting subgraphs and pose what we believe are important questions about B.Comment: 15 pages, 4 figures, 4 ancillary files. Reorganized and shortened from previous versions, while correcting one error in the proof of Theorem 5.4. Also, ancillary files detailing our computations with the computer program ORB have been provide

HR: A System for Machine Discovery in Finite Algebras

We describe the HR concept formation program which invents mathematical definitions and conjectures in finite algebras such as group theory and ring theory. We give the methods behind and the reasons for the concept formation in HR, an evaluation of its performance in its training domain, group theory, and a look at HR in domains other than group theory

The automorphism group of the free group of rank two is a CAT(0) group

We prove that the automorphism group of the braid group on four strands acts faithfully and geometrically on a CAT(0) 2-complex. This implies that the automorphism group of the free group of rank two acts faithfully and geometrically on a CAT(0) 2-complex, in contrast to the situation for rank three and above.Comment: 7 pages, 2 figures. The manuscript has been modified in minor ways in accordance with a referee's recommendations, and a misattribution of the result "Aut F_2 is biautomatic" has been correcte