Buffalo Geometry and Topology Seminar

Spring 2011

Unless noted, all seminars are Friday at 4pm, in Mathematics 122.

Date Speaker Organization Title Abstract

January 21st Mark Hagen McGill University Contact graphs of CAT(0) cube complexes I will briefly define CAT(0) cube complexes and their contact graphs. The main theorem states that the contact graph of a cube complex is quasi-isometric to a tree. I will discuss this result using the example of regular tilings of the hyperbolic plane by squares, and give some applications to relative hyperbolicity and asymptotic finite dimensionality of groups acting on cube complexes, as well as to embeddings of cube complexes in products of trees.

January 28th Sinem Onaran University of Waterloo Knots in Contact Structures and Open Book Decompositions Due to Alexander, it is well known that every closed oriented 3-manifold has an open book decomposition. In this talk, we will define open book decompositions and discuss various examples. Further, we will discuss the importance of the open books in manifold theory, in particular in contact geometry. Then, we will focus on a class of knots in contact 3-manifolds called Legendrian knots. We will define a new invariant for Legendrian knots using open book decompositions. The invariant is called the support genus of a Legendrian knot. We will discuss the applications of this invariant and list several open problems related to the support genus of Legendrian knots.

April 1st

April 8th Will Cavendish Princeton University Finite sheeted covering spaces of 3-manifolds and the cohomology of solenoids In this talk I will discuss a cohomological approach to studying the finite sheeted covering spaces of a 3-manifold $M$. A well studied question in this setting is the following: given a $\pi_1$-injective immersion $f$ from a manifold $N$ into $M$, when can $f$ be lifted to a non-trivial finite sheeted covering space of $M$? I will describe how the inverse limit of the collection of all finite sheeted covering spaces of $M$, the universal solenoid over $M$, can be used to study this question. Topologically this solenoid is a Cantor set bundle $E\to M$, and recent work of Wilton-Zalesskii can be used to show, modulo the virtual fibered conjecture for Haken hyperbolic 3-manifolds recently announced by Wise, that this bundle has trivial cohomology over finite coefficients groups for any compact 3-manifold $M$. I will then show how these results can be used to resolve Grothendieck's problem for the fundamental groups of many compact 3-manifolds, and how this in turn can be used as a tool to give a positive answer to the lifting problem stated above under suitable hypotheses.

April 15th Jozef Przytycki George Washington University Homology of distributive magmas While homology theory of associative structures, such as groups and rings, has been extensively studied in the past beginning with the work of Hopf, Eilenberg, and Hochschild, the non-associative structures, such as quandles, were neglected until recently. The distributive structures have been studied for a long time and even C.S. Peirce in 1880 emphasized the importance of (right) self-distributivity in algebraic structures. However, homology for such universal algebras was introduced only fifteen years ago by Fenn, Rourke and Sanderson. I will develop this theory in the historical context and describe relations to topology and similarity with some structures in logic. I will also speculate how to define homology for Yang-Baxter operators and how to relate our work to Khovanov homology and categorification. We use here the fact that Yang Baxter equation can be thought of as a generalization of self-distributivity.

April 22nd Owen Baker Cornell University The Jacobian Map on Outer Space Culler-Vogtmann Outer Space X_n is a space on which the outer automorphism group of a rank n free group acts nicely. I will define the Jacobian map that associates a positive definite quadratic form to each point of Culler-Vogtmann Outer Space. I will show how this Jacobian map can be used to study the homology of the kernel of the natural map from Out(F_3) to GL_3(Z).

April 29th Gregory Schneider University at Buffalo (Dissertation Defense)

Old seminar pages:

Spring 2007

Fall 2007

Spring 2008

Fall 2008

Fall 2009

Fall 2010

Date	Speaker	Organization	Title	Abstract
January 21st	Mark Hagen	McGill University	Contact graphs of CAT(0) cube complexes	I will briefly define CAT(0) cube complexes and their contact graphs. The main theorem states that the contact graph of a cube complex is quasi-isometric to a tree. I will discuss this result using the example of regular tilings of the hyperbolic plane by squares, and give some applications to relative hyperbolicity and asymptotic finite dimensionality of groups acting on cube complexes, as well as to embeddings of cube complexes in products of trees.
January 28th	Sinem Onaran	University of Waterloo	Knots in Contact Structures and Open Book Decompositions	Due to Alexander, it is well known that every closed oriented 3-manifold has an open book decomposition. In this talk, we will define open book decompositions and discuss various examples. Further, we will discuss the importance of the open books in manifold theory, in particular in contact geometry. Then, we will focus on a class of knots in contact 3-manifolds called Legendrian knots. We will define a new invariant for Legendrian knots using open book decompositions. The invariant is called the support genus of a Legendrian knot. We will discuss the applications of this invariant and list several open problems related to the support genus of Legendrian knots.
April 1st
April 8th	Will Cavendish	Princeton University	Finite sheeted covering spaces of 3-manifolds and the cohomology of solenoids	In this talk I will discuss a cohomological approach to studying the finite sheeted covering spaces of a 3-manifold $M$. A well studied question in this setting is the following: given a $\pi_1$-injective immersion $f$ from a manifold $N$ into $M$, when can $f$ be lifted to a non-trivial finite sheeted covering space of $M$? I will describe how the inverse limit of the collection of all finite sheeted covering spaces of $M$, the universal solenoid over $M$, can be used to study this question. Topologically this solenoid is a Cantor set bundle $E\to M$, and recent work of Wilton-Zalesskii can be used to show, modulo the virtual fibered conjecture for Haken hyperbolic 3-manifolds recently announced by Wise, that this bundle has trivial cohomology over finite coefficients groups for any compact 3-manifold $M$. I will then show how these results can be used to resolve Grothendieck's problem for the fundamental groups of many compact 3-manifolds, and how this in turn can be used as a tool to give a positive answer to the lifting problem stated above under suitable hypotheses.
April 15th	Jozef Przytycki	George Washington University	Homology of distributive magmas	While homology theory of associative structures, such as groups and rings, has been extensively studied in the past beginning with the work of Hopf, Eilenberg, and Hochschild, the non-associative structures, such as quandles, were neglected until recently. The distributive structures have been studied for a long time and even C.S. Peirce in 1880 emphasized the importance of (right) self-distributivity in algebraic structures. However, homology for such universal algebras was introduced only fifteen years ago by Fenn, Rourke and Sanderson. I will develop this theory in the historical context and describe relations to topology and similarity with some structures in logic. I will also speculate how to define homology for Yang-Baxter operators and how to relate our work to Khovanov homology and categorification. We use here the fact that Yang Baxter equation can be thought of as a generalization of self-distributivity.
April 22nd	Owen Baker	Cornell University	The Jacobian Map on Outer Space	Culler-Vogtmann Outer Space X_n is a space on which the outer automorphism group of a rank n free group acts nicely. I will define the Jacobian map that associates a positive definite quadratic form to each point of Culler-Vogtmann Outer Space. I will show how this Jacobian map can be used to study the homology of the kernel of the natural map from Out(F_3) to GL_3(Z).
April 29th	Gregory Schneider	University at Buffalo	(Dissertation Defense)