|September 17||Jason Cantarella|
Jason Behrstock, Lehman College, “Quasi-isometric classification of right angled Artin groups”.
Any finitely generated group can be endowed with a natural metric which is unique up to maps of bounded distortion (quasi-isometries). A fundamental question is to classify finitely generated groups up to quasi-isometry. Surprisingly, for a large family of right angled Artin groups the quasi-isometric classification can be described in terms of a concept in computer science called “bisimulation.” We will describe this classification and a geometric interpretation of bisimulation. (Joint work with Walter Neumann and Tadeusz Januszkiewicz.)
Rob Schneiderman, Lehman College, “Geometric filtrations of link concordance”.
This talk will explain how a tree-valued intersection theory of Whitney towers on immersed disks in the 4-ball bounded by a link in the 3-sphere provides a geometric interpretation and unification of Milnor's link invariants, the Arf invariant and the Sato-Levine invariant. Recent work has led to the formulation of what appear to be new concordance invariants in terms of a quadratic enhancement given by Whitney-disk framing obstructions.
Alan Reid, UT Austin, “Grothendieck's problem for 3-manifold groups”.
Chris Leininger, UIUC, “Small dilatation pseudo-Anosovs and 3-manifolds”.
I will discuss joint work with Farb and Margalit, the main theorem of which states: Up to removing a finite invariant set, all small dilatation pseudo-Anosov homeomorphisms arise as the monodromies of fibers of a FINITE list of fibered 3-manifolds.
Greg Schneider, SUNY Buffalo, “Box-Dot Diagrams for "Regular" Rational Tangles”.
We introduce a new presentation for rational tangles which illustrates a geometric connection to the number theory of positive regular continued fractions. This presentation also admits a suitable extension to the contact setting, allowing us to define a natural Legendrian embedding of a particular class of rational tangles into the standard contact Euclidean 3-space. We will briefly discuss how these box-dot diagrams, along with an associated construction, can be used to determine when the Legendrian flyping operation yields tangles which are not Legendrian isotopic, further refining an earlier result of Traynor.
Sumio Yamada (Tohoku University) “Geometry of Teichmuller-Coxeter complexes”.
In this talk, I will introduce a new space, which is a Coxeter complex constituted by simplices which are themselves Teichmuller spaces. The construction is made possible by the facts that with respect to the Weil-Petersson distance, the Teichmuller space can bee seen as a simplex spanned by the vertex set representing maximally pinched nodal surfaces, and that meetings of two facets occur always at the right angle. The resulting cubical Coxeter complex has several nice geometric properties including being CAT(0), and being geodesically complete. I will also indicate how the space can be isometrically embedded in the universal Teichmuller space canonically
Paul Kirk, “Instantons, Chern-Simons invariants, and Whitehead doubles of $(2,2^k-1)$ torus knots”.
Abstract: Using SO(3) instanton moduli spaces and estimates of Chern-Simons invariants of flat SO(3) connections on 3-manifolds we show that the infinite family of untwisted positive clasped Whitehead doubles of the $(2, 2^k-1)$ torus knots are linearly independent in the smooth knot concordance group. (joint work with Matt Hedden)
Helge Moeller Pedersen (Heidelberg) “Splice diagrams and Universal Abelian Covers of Graph Manifolds”.
To a rational homology sphere graph manifold one can associate an invariant called the "splice diagram." It has been proven that the splice diagram determines the universal abelian cover, but the proof aqctually leads to an algorithm to construct the universal abelian cover explicitely. This construction will be described.
Diane Vavricheck “Quasi-isometry invariant subgroups”.
A subgroup H of a group G is "invariant under quasi-isometries" if, for any quasi-isometry f from G to a group G', f(H) is a finite Hausdorff distance from a subgroup of G'. I will discuss recent results that give sufficient conditions for certain subgroups to be invariant under quasi-isometries.
Slava Krushkal “Topological arbiters”.
Abstract: In this talk I will introduce the notion of a topological arbiter, and I will discuss several construction of arbiters in 4 and higher dimensions. Given an n-dimensional manifold W, a topological arbiter associates a value 0 or 1 to codimension zero submanifolds of W, subject to natural topological and duality axioms. For example, there is a unique arbiter on the real projective plane, induced by homology. In contrast, I will show that there exists an uncountable collection of topological arbiters in dimension 4. (Joint work with Michael Freedman)
Moira Chas “String topology and three manifolds”.
This is joint work with Siddharta Gadgil.
Consider an oriented manifold M. We recall the definition of the string bracket: Start with two (oriented) families of loops in M which determine two cycles in the equivariant homology of the free loop space of M. The string bracket yields a new cycle as follows: Suppose that the images of the cycles intersect transversally in M. At each intersection point consider the loop that runs along the loop of the first family and then along the loop of the second family. Forgetting the basepoints of the loops and orienting the parameter space of this new family using the orientation of M, one obtains a new cycle, the string bracket of the two initial cycles. The string bracket is well defined up to homology, skew-symmetric and satisfies the Jacobi identity.
Keiko Kawamuro, “Characteristic polynomials of pseudo-Anosov maps”.
This is a joint work with Joan Birman and Peter Brinkmann.
Thurston-Nielsen classifies the mapping classes of surfaces into three kinds; periodic, reducible, and pseudo-Anosov. Pseudo-Anosov (PA) type is the most interesting because its mapping tori admits hyperbolic geometry. The "dilatation" is an important invariant for PA maps. Geometrically, the dilatation is the ``stretching rate'' of a foliation in the surface under the action of the PA map. Algebraically, the dilatation is the largest real root of the characteristic polynomial for certain matrix, associated to the PA map. Thus it is an algebraic integer grater than 1.
It is interesting to note that the characteristic polynomial is not an invariant for the PA map because it often factorizes over Z. In this talk, I focus on this factorization phenomena. I show how the vector space, on which the PA map acting, decomposes into direct summands compatible with the factorization. I will discuss which subspace would be expected to be a new invariant of the PA map. In addition, if a PA map is associated to a hyperbolic fibered knot, then the minimal polynomial for the dilatation, that is obviously an invariant of the PA map, sometimes coincides with the Alexander polynomial. I explain this fact is related to the orientability of a train track graph for the pseudo-Anosov map.
Other relevant information.Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006.
- Columbia Symplectic Geometry/Gauge Theory Seminar
- CUNY Geometry and Topology Seminar
- CUNY Complex Analysis & Dynamics Seminar
- CUNY Magnus Seminar
- Princeton Topology Seminar.
- All Columbia Math Dept Seminars