Topology Seminar, Fall 2003

Topology Seminar

Fall 2003

The topology seminar will usually meet on Tuesdays 2-3:30 pm, in Room 117 Goldsmith Hall. Occasionally a talk may have to be scheduled at a different time.

Schedule

Date Speaker/Title

September 24 Murray Elder (Tufts University)
Pattern theory for some Cayley graphs of groups

Abstract: In this talk I will introduce a simple idea called a "pattern" to try to understand the geometry of the Cayley graph of certain groups. Patterns were defined in the hope of proving that a certain example is or is not "automatic", and I will describe the progress made towards this goal. .

September 30 Danny Ruberman
Rohlin's invariant and gauge theory

Abstract: In 1985, Andrew Casson showed how to lift the Z/2-valued Rohlin invariant of homology 3-spheres to an integer now known as Casson's invariant. I will discuss some 4-dimensional analogs, and their implications. .
October 8, 5 PM
Room: 117 Shelly Harvey (MIT)
Title: Invariants of 3-manifolds from Non-commutative Algebra

Abstract: We will discuss some non-commutative generalizations of the Alexander polynomial of a 3-manifold and some of their applications to 3- and 4-manifolds. In particular, we will define two sequences of integral invariants r_n and d_n and discuss their behavior. We show that they give computable algebraic obstructions to a 3-manifolds fibering over S^1, a 3-manifold being a Seifert fibered space and a 4-manifold of the form X x S^1 admitting a symplectic structure. We will also present the result that, for 3-manifolds, the d_n are non-increasing while the r_n are non-decreasing (as a function of n).
October 14 Jerry Levine
Signature-type invariants of knots and links and applications to concordance

Abstract: I will review the use of signatures associated to unitary representations of the knot group to study and classify knot concordance, and then discuss attempts to do similar things for links. .
October 21 Indira Chatterji (Cornell University)
Wall spaces and CAT(0) cube complexes.

Abstract: .
October 28 Mario Bourgoin
Classifying immersed curves (Including non-orientable surfaces)

Abstract: In 1991, Carter used signed Gauss codes to classify generic immersed curves in closed oriented surfaces. We introduce a new method that uses Gauss codes to classify such curves also in non-orientable surfaces. Then, something fun happens: the classification has a mod~2 cohomology-theoretic interpretation, and allows us to include a classification of two-colorable immersions. And this last classification makes a large class of immersions stand out that leads us to introduce a new form of virtual knots that includes Kauffman's 1996 construction. .
Spring 2003 schedule

Date	Speaker/Title
September 24	Murray Elder (Tufts University) Pattern theory for some Cayley graphs of groups
Abstract: In this talk I will introduce a simple idea called a "pattern" to try to understand the geometry of the Cayley graph of certain groups. Patterns were defined in the hope of proving that a certain example is or is not "automatic", and I will describe the progress made towards this goal. .
September 30	Danny Ruberman Rohlin's invariant and gauge theory
Abstract: In 1985, Andrew Casson showed how to lift the Z/2-valued Rohlin invariant of homology 3-spheres to an integer now known as Casson's invariant. I will discuss some 4-dimensional analogs, and their implications. .
October 8, 5 PM Room: 117	Shelly Harvey (MIT) Title: Invariants of 3-manifolds from Non-commutative Algebra
Abstract: We will discuss some non-commutative generalizations of the Alexander polynomial of a 3-manifold and some of their applications to 3- and 4-manifolds. In particular, we will define two sequences of integral invariants r_n and d_n and discuss their behavior. We show that they give computable algebraic obstructions to a 3-manifolds fibering over S^1, a 3-manifold being a Seifert fibered space and a 4-manifold of the form X x S^1 admitting a symplectic structure. We will also present the result that, for 3-manifolds, the d_n are non-increasing while the r_n are non-decreasing (as a function of n).
October 14	Jerry Levine Signature-type invariants of knots and links and applications to concordance
Abstract: I will review the use of signatures associated to unitary representations of the knot group to study and classify knot concordance, and then discuss attempts to do similar things for links. .
October 21	Indira Chatterji (Cornell University) Wall spaces and CAT(0) cube complexes.
Abstract: .
October 28	Mario Bourgoin Classifying immersed curves (Including non-orientable surfaces)
Abstract: In 1991, Carter used signed Gauss codes to classify generic immersed curves in closed oriented surfaces. We introduce a new method that uses Gauss codes to classify such curves also in non-orientable surfaces. Then, something fun happens: the classification has a mod~2 cohomology-theoretic interpretation, and allows us to include a classification of two-colorable immersions. And this last classification makes a large class of immersions stand out that leads us to introduce a new form of virtual knots that includes Kauffman's 1996 construction. .