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The 5th William Rowan Hamilton Geometry and Topology Workshop

September 17-19, 2009

The Hamilton Mathematics Institute, Trinity College Dublin

Computational and Algorithmic Geometry

Schedule of Talks

All talks will be held in the Maxwell Lecture Theatre in the Hamilton Building, TCD (see campus map).


9.30 - 10.30
Nathan Dunfield (Urbana-Champaign)
Practical solutions to hard problems in 3-dimensional topology
10:45 - 11:45
Pat Hooper (Northwestern)
Symmetries of Generalized Staircases
11:45 - 1:30
1:30 - 2:30
Karen Vogtmann (Cornell)
Outer automorphism groups of right-angled Artin groups
2:45 - 3:45
János Pach (CUNY and Renyi Inst.) Towards a theory of topological graphs
4:00 - 5:00
David Eppstein (Irvine) Hyperconvexity and metric embedding

9:30 - 10:30 Robert Meyerhoff (Boston College)
Computer-Aided Analysis of Hyperbolic 3-Manifolds
10.30 - 10.45
10:45 - 11:45 Mikhail Belolipetsky (Durham)
Some computational problems from geometry of lattices
11:45 - 1:30 LUNCH
1:30 - 2:30 Walter Neumann (Barnard)
Quasi-isometry of 3-manifold groups
2:45 - 3:45 Volodymyr Nekrashevych (Texas A&M) Automata and topological models of expanding dynamical systems
4:00 - 5:00 Tara Brendle (Glasgow)
The symmetric Torelli group

9:30 - 10:30 Tim Riley (Cornell)
Hydra Groups
10.30 - 10.45
10:45 - 11:45 Abigail Thompson (Davis)
Knots in a (surface)x I and generalized Property R
11.45 - 1.30
1.30 - 2.30
Jon McCammond (Santa Barbara) Braid groups and buildings
2.45 - 3.45
Alexander Coward (Oxford)
Upper bounds on Reidemeister moves


Speaker: Misha Belolipetsky (Durham)

Title: Some computational problems from geometry of lattices.

Abstract: I will discuss a number of concrete problems which come from my previous work on geometry and arithmetic of lattices in semisimple Lie groups.

Speaker: Tara Brendle (Glasgow)

Title:  The symmetric Torelli group

Abstract:  We will discuss the subgroup of the mapping class group of a surface consisting of those elements which act trivially on homology and which also commute with a fixed hyperelliptic involution.  This group can also be viewed as a certain kind of "pure" braid group. Motivated by a conjecture of Hain on the generation of this group, we will discuss how the symmetric viewpoint leads to interesting new relations in the full Torelli group.  If time permits, we will also discuss a theory of "symmetric homology" which provides an approach to Hain's conjecture.
This is joint work with Dan Margalit.

Speaker: Alex Coward (Oxford)

Title: Upper bounds on Reidemeister moves
(Joint work with Marc Lackenby)

Abstract: Given any two diagrams of the same knot or link, we provide an explicit upper bound on the number of Reidemeister moves required to pass between them in terms of the number of crossings in each diagram. This provides a new and conceptually simple solution to the equivalence problem for knot and links.

Speaker: Nathan Dunfield (Urbana-Champaign)

Title: Practical solutions to hard problems in 3-dimensional topology.

Abstract:  Many fundamental problems in 3-dimensional topology are known to be algorithmically solvable, using tools ranging from normal surface theory to hyperbolic geometry.   However, most of these algorithms are too slow to be of use in any interesting example, and in fact some of these problems are known to be NP-hard or worse.   I  will discuss other methods, based on various heurisitics,  randomization, and hyperbolic geometry, which can effectively solve some of these problems in practice.   Particular questions of interest  will be whether a 3-manifold fibers over the circle, its  Heegaard genus, and the rank of its fundamental group.   I will conclude with a  brief demonstration of SnapPea, a modern Python interface to Jeff Weeks SnapPea.
This is joint work with (separately) Dinakar Ramakrishnan, Helen Wong, and Marc Culler.

Speaker: David Eppstein (Irvine)

Title: Hyperconvexity and metric embedding

Abstract: A metric is hyperconvex if its metric balls satisfy the Helly property: any pairwise intersecting family of balls has a common intersection. Examples of hyperconvex spaces include trees, and vector spaces with the L-infinity norm. We survey results related to hyperconvex spaces and the tight span, an embedding of an arbitrary metric space into a hyperconvex space that is closely related to rectilinear convex hulls. We describe how to use tight spans and hyperconvexity to efficiently find embeddings of distance matrices into the Manhattan plane and into star metrics.

Speaker: Pat Hooper (Northwestern)

Title: Symmetries of Generalized Staircases

Abstract: A translation surface is a surface built by gluing polygonal subsets of the plane together by translations. An affine automorphism of a translation surface is a homeomorphism which preserves the underlying affine structure underlying the translation surface structure. We study properties of the affine automorphism groups of periodic translation surfaces (surfaces equipped with an action of Z by translations). I will also explain why these groups are relevant to the study of geometric and dynamical properties of these surfaces. This is joint work with Barak Weiss.

Speaker: Jon McCammond  (Santa Barbara)

Title: Braid groups and buildings.

Abstract: In this talk I will survey several results about braid groups (and their generalizations such as Artin groups) and the  building-like geometric structures on which they  act.

Speaker: Robert Meyerhoff (Boston College)

Title: Computer-Aided Analysis of Hyperbolic 3-Manifolds

Abstract:  Some important, recent theorems about hyperbolic 3-manifolds (e.g., that the Weeks manifold is the minimum volume hyperbolic 3-manifold) have made fundamental use of computers in their discovery and proof.  We will discuss some of these uses of the computer.

Speaker: Volodymyr Nekrashevych (Texas A&M)

Title: Automata and topological models of expanding dynamical systems

Abstract: We will describe a notion of a topological automata, which can be used to construct converging approximations of the Julia set of an expanding dynamical system. Some examples from multi-dimensional holomorphic dynamics will be discussed.

Speaker: Walter Neumann (Barnard College)

Title: Quasi-isometry of 3-manifold groups

Abstract: The program, started in the early 90's, of classifying 3-manifold groups up to quasi-isometry, is now almost complete; the only cases remaining are some with "too many" arithmetic hyperbolic JSJ components. The non-geometric case will be described. The classification is in terms of equivalence classes under "bisimilarity" (a concept from computer science) of weighted graphs. (Joint work with Jason Behrstock.)

Speaker János Pach (City College, New York and Renyi Institute, Budapest)

Title: Towards a theory of topological graphs

Speaker: Tim Riley

Title: Hydra Groups

Abstract: I will describe a family of groups that are CAT(0), 1-relator, and free-by-cyclic, and yet have free subgroups of extraordinary (Ackermannian) distortion. This wild geometry, in an apparently benign setting, has its origins in a simple computational game --- a realisation of Hercules' battle with the hydra, played out in string rewriting.  This is work with Will Dison.  

Speaker: Abigail Thompson (UC Davis)

Title:  Knots in a (surface)x I and generalized Property R

Abstract: We examine surgery on knots in 3-manifolds with a natural product structure and relate it to manifolds that can be obtained by surgery on 2-component links in the 3-sphere.  This is joint work with Martin Scharlemann.

Speaker: Karen Vogtmann (Cornell)

Title: Outer automorphism groups of right-angled Artin groups.

Abstract:  Free groups and free abelian groups are examples of  right-angled Artin groups, and in general right-angled Artin groups  can be though of as interpolating between these two extremes.  The outer automorphism groups of free groups (Out(F_n)) and of free abelian groups (GL(n,Z)) have been extensively studied and been shown to have many properties in common; for example they are finitely-presented, have torsion-free subgroups of finite index, are residually finite, and satisfy the Tits alternative restricting the types of subgroups they contain.   We investivate which of these properties are in fact shared by the outer automorphism groups of all right-angled Artin groups. This is joint with Ruth Charney.



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