Topology Seminar 2008--2012 Archive

The Department of Mathematics and Statistics at the University of Victoria will host the Spring 2011 Cascade Topology Seminar. The conference will be on the April 15th weekend of 2011. If flying in, one should plan to arrive at latest on the evening of April 15th. Talks will begin at 10am on April 16th. Activities should be over by 1pm on April 17th.

The speaker list is:

* Ian Agol (Berkeley)

* Yi Liu (Berkeley)

* Tom Church (Chicago)

* Jesse Johnson (Oklahoma State)

* Johanna Mangahas (Brown)

* Alexandra Pettet (Michigan / UBC)

* Jing Tao (Utah)

Speaker: Noah Kieserman (Bowdoin)

Title: Infinitesimally multiplicative structures on Lie algebroids

Abstract: I will give a survey of known results on multiplicative structures on Lie groupoids, with a particular interest in how to view the corresponding infinitesimal data. Poisson manifolds and Dirac structures are important examples.

Date/Time: March 24th, 4:30pm-5:30pm

Location: DSB C126

Speaker: Victor Turchin (Kansas State)

Title: Everything the audience wants to know about the Vassiliev Spectral Sequence

Date/Time: March 22nd, 2pm--5pm

Location: Hickman 116

Speaker: Daniel Moskovich

Title: First steps in coloured knot theory.

Abstract: A colouring is one useful structure with which one might wish to equip a knot. The theory of coloured knots parallels ordinary knot theory to some extent. There are coloured analogues to Seifert matrices, crossing changes, knot polynomials, and a lot more. We will take some first steps in coloured knot theory with a Dehn surgery theoretic approach.

Speaker: Samson Black

Title: A new state-sum formula for the Alexander polynomial

Abstract: Markov's theorem gives conditions under which a function defined on braids descends to a link invariant, upon closing up the ends of the braid. Thus, representations of the braid group, suitably rescaled, are effective tools in knot theory. Perhaps the nicest representations factor through the Iwahori-Hecke algebra (of type A), and the corresponding invariants enjoy a skein relation on the link diagrams. I will present a new diagram calculus for obtaining the Alexander polynomial, beginning with a braid diagram and obtaining various combinatorial states and summing their weights. If we look "under the hood," then we find a version of Young's seminormal representations adapted to the Hecke algebra, and some character formulas due to Ocneanu. Time permitting, I will discuss a particular quotient of the Hecke algebra on which these combinatorics are based.

Speaker: Garret Flowers

Title: Star cocircularities of knots

Abstract: The type-2 Vassiliev invariant of knots has many descriptions, such as the z^2-coefficient of the Conway polynomial. This talk will provide a (real) algebraic-geometric interpretation of the invariant, as a certain count of round circles intersecting the knot in precisely five points. I will also discuss other algebraic-geometric counts on knots and their potential for giving knot invariants.

Speaker: Robin Koytcheff (Stanford)

Title: A homotopy-theoretic view of Bott-Taubes integrals and knot spaces

Abstract: Bott and Taubes constructed knot invariants by considering a bundle over the space of knots and performing integration along the fiber. This method was subsequently used to construct real cohomology classes in spaces of knots in R^n, n > 3. Replacing integration of differential forms by a Pontrjagin-Thom construction, I have constructed cohomology classes with arbitrary coefficients. Motivated by work of Budney and F. Cohen on the homology of the space of long knots in R^3, I have also proven a product formula for these classes with respect to connect-sum. If time permits, I will outline work in progress towards explicit calculations using cosimplicial models for knot spaces coming from the Goodwillie-Weiss embedding calculus.

Date/Time: Thursday March 25th, 4:30pm.

Location: David Strong Building C128

Speaker: Alexandra Pettet (U. Michigan)

Title: Dynamics of Out(F): twisting out fully irreducible automorphisms.

Abstract: The outer automorphism group Out(F) of a free group F of finite rank shares many properties with the mapping class group of a surface, however the techniques for studying these groups are generally quite different. Analogues of the pseudo-Anosov elements of the mapping class group are the so-called fully irreducible automorphisms, which exhibit north-south dynamics on Culler-Vogtmann's Outer Space. We will explain a method for constructing these automorphisms and suggest why this construction should be useful. This is joint work with Matt Clay (University of Oklahoma).

Date/Time: 4pm Friday February 27th.

Location: Cornett A128

Speaker: Ben Burton (RMIT)

Title: A guided tour through the census of minimal 3-manifold triangulations

Abstract: A minimal triangulation is a method of building a 3-manifold using the smallest possible number of tetrahedra. In this talk we get our hands dirty and examine the combinatorial structures of minimal triangulations for several families of closed 3-manifolds, using computational census data as our guide. Along the way we will see both theoretical and computational examples of why it is useful to understand these structures, and we will close with a smattering of results and open questions regarding minimal triangulations in general.

Date/Time: Friday November 7th at 2:30pm.

Location: David Strong Building C130.

Speaker: Ryan Budney

Title: Smooth embeddings of 3-manifolds in the 4-sphere

Abstract: These talks will describe what is known about smooth embeddings of 3-manifolds in the 4-sphere.

Date/Time: Friday October 24th and 31st at 2:30pm.
Location: David Strong Building C130.

Speaker: Ekaterina Yurasovskaya (UBC)

Title: Homotopy string links over surfaces

Abstract: In his 1947 work "Theory of Braids" Emil Artin asked whether the braid group remained unchanged when one considered classes of braids under link-homotopy, allowing each strand of a braid to pass through itself but not through other strands. The problem remained open for a long time until in her 1974 paper "Homotopy of Braids - in answer to a Question of E. Artin", Deborah Goldsmith described a subgroup of isotopically non-trivial braids that became trivial under the relation of link-homotopy. In a seminal paper "Classification of links up to link-homotopy"(1990) Nathan Habegger and Xiao-Song Lin re-introduced Goldsmith's quotient of the pure braid group as a group of homotopy string links, which they used as a fundamental tool to accomplish classification.

We generalize Artin's question to string links over orientable surface M and show that under link-homotopy surface string links form a group, which is isomorphic to a quotient of the surface pure braid group PBn(M). Our work explores the geometric and visual beauty of the subject as we compute a presentation of the group of homotopy string links in terms of generators and relations.

Date/Time: Friday September 26th at 2:30pm.

Location: David Strong Building C130.

Speaker: Chan-Ho Suh

Title: Recent perspectives on normal surface theory (part 2 of 2)

Abstract: I explain some modern versions of the classic normal surface theory. In particular, I will briefly explain Tollefson's Q-theory and Rubinstein's algorithm to recognize the 3-sphere using almost normal surfaces. Some non-algorithmic applications of normal surface theory will also be discussed, particularly bounding the number of Reidemeister moves to unknot or split a link. If time permits, I will also explain a diagrammatic form of normal surface theory that is particularly apt for normal surface theoretic problems in knot complements.

Date/Time: Friday September 19th at 2:30pm.

Location: David Strong Building C130.

Speaker: Chan-Ho Suh

Title: Recent perspectives on normal surface theory (part 1 of 2)

Abstract: I will cover the basics of normal surface theory, a fundamental tool in studying 3-manifolds. Since its inception by Wolfgang Haken, the theory has seen many extensions and refinements. I will also explain some of these, with an eye toward explaining new developments.

Date/Time: Friday September 12th at 2:30pm.

Location: David Strong Building C130.

Speaker: Ryan Budney

Title: Enumerative geometry of knots.

Abstract: This talk will be about how one can construct an invariant of knots by counting an algebraic-geometric coincidence number: the (signed) number of straight lines that intersect the knot in precisely four points. This gives a "type two" invariant of knots in R^3, and can also be thought of as a description of the first non-trivial homotopy group of the space of knots in higher-dimensional Euclidean space.

Date/Time: Friday August 15th at 2pm.

Location: David Strong Building C130.

Speaker: Ryan Budney

Title: Knot spinning.

Abstract: Consider a knot to be a smooth embedding of the j-sphere in the n-sphere. n-j is the co-dimension of the knot. Spinning is an inductive procedure that starts with an embedding of a j-sphere in the n-sphere, and produces an embedding of the (j+1)-sphere in the (n+1)-sphere. I will describe two results: if the co-dimension is larger than 2, every knot is deform-spun. When the co-dimension is 2, not every knot is deform-spun. The first result comes from an observation on the boundary-map in a pseudo-isotopy fibration and the h-cobordism theorem. The second observation is due to an Alexander polynomial obstruction.

Date/Time: Friday July 25th at 2pm.

Location: David Strong Building C130.

Speaker: Ryan Budney

Title: Abelian covering spaces and their uses II.

Abstract: This is a continuation of last week's talk. Items to include are: signature invariants of 3-manifolds such as the Milnor signature invariants and Tristram-Levigne invariants, and how they provide obstructions to 3-manifolds embedding in the 4-sphere. I will also talk about Poincare Duality for the Burau and Lawrence-Krammer representations.

Date/Time: Friday July 11th at 2pm.
Location: David Strong Building C130.

Speaker: Ryan Budney

Title: Abelian covering spaces and their uses.

Abstract: I'll describe some basic uses of abelian covering spaces, such as the construction of the Alexander polynomial, the signature of a knot, the homotopy and diffeomorphism classification of lens spaces, the Burau and Lawrence-Krammer representations of braid groups and their basic properties. Most likely this talk will continue on July 11th with a detailed description of how Poincare duality works for the Lawrence-Krammer representation.

Date/Time: Friday July 4th at 2pm.

Location: David Strong Building C130.


2008 marks the beginning of the University of Victoria Topology Seminar.

First speaker: Melissa Macasieb (UBC)

Title: Character varieties of a family of 2-bridge knot complements.

Abstract: To every hyperbolic 3-manifold M with nonempty boundary, one can associate a pair of related algebraic varieties X(M) and Y(M) called the character varieties of M. These varieties carry much topological information about M, but are in general difficult to compute. In this talk, I will discuss how properties of these varieties reflect the topology of M in the case M is a hyperbolic knot complement. I will also show how to obtain explicit equations for the the character varieties associated to a family of hyperbolic two-bridge knots K(m,n) and discuss consequences of these results related to the existence of integral points on these curves. This is joint work with Kate Petersen and Ronald van Luijk.

Date/Time: Friday April 4th, 3:30pm to 4:30pm

Location: U.Vic David Strong Building room C128

If you decide to drive, be aware Google Maps is sending you to Campus Security -- this is where you can acquire a parking permit. To find the Mathematics Department, go to the 4th floor of the Social Sciences and Mathematics building (SSM). See the Campus Maps for details.