**Spring 2015 Cascade**

We'll be hosting the Spring 2015 Cascade Topology seminar here in Victoria. Exact dates, speakers and such are yet to be decided. But together with the Applied Topology Workshop it will make for a relatively busy 2015 for topology in Victoria.

**Speaker:** Ryan Budney

**Title:** An action of the splicing operad on the functor calculus Taylor tower

**Abstract:** Recently two rather distinct objects have come about to study the homotopy-type of spaces of embeddings. The Taylor tower of Goodwillie-Weiss-Klein is a fundamentally cohomological object, giving a map out of embedding spaces to a space that is a homotopy-limit of cubical diagrams of configuration spaces. The splicing operad, on the other hand, is a fundamentally homological object, giving families of maps into the embedding space. I will describe an action of the splicing operad on the Taylor-tower itself, making the evaluation map from knots to the Taylor tower an equivariant map.

**Date:** TBA

**Time:** TBA

**Location:** TBA

If you wish to subscribe to the mailing list for the University of Victoria Topology Seminar, the interface is here.

**Speaker:** Ken Baker (U. Miami)

**Title:**Links of divides vs. L-space knots

**Abstract:** Yamada has shown that all known knots with Dehn surgeries to lens spaces have presentations as A'Campo's links of divides. Is this true for all knots with Dehn surgeries to L-spaces, the L-space knots? L-spaces are the Heegaard Floer generalization of lens spaces, and L-space knots are known to have many of the basic properties of links of divides. While the sets of links of divides and have a substantial intersection with the set of L-space knots, neither contains the other. We highlight their distinctions by examining their relationships with cabling and satellite operations.

**Date:**Friday, May 16th

**Location:** DSB C114

**Speaker:** Sam Churchill

**Title:**The Turaev Reconstruction Theorem

**Date:** Friday March 21st

**Time:**3pm-4pm

**Location:** DSB C128

**Speaker:** Robin Koytcheff

**Title:**Embedding calculus yields finite-type knot invariants

**Date:** Friday March 7th

**Time:**3pm-4pm

**Location:** DSB C128

**Speaker:** Ryan Budney

**Title:** Fibering 4-manifolds over the circle

**Date:** Friday February 21st

**Time:** 3pm-4pm

**Location:** DSB C128

**Speaker:** Sam Churchill

**Title:** 4-manifolds bounding 3-manifolds

**Date:** Friday October 4th

**Time:** 3pm-4pm

**Location:** CLE A211

**Speaker:** Joseph Cheng (UBC)

**Title:** Poincare duality for orbifolds in Morava K-theory

**Abstract:** It was showed by Greenlees and Sadofsky that the classifying spaces of ﬁnite groups are self-dual with respect to Morava K-theory K(n). Their duality map was constructed using a transfer map. I will describe the map and its generalization which would induce a K(n)-version of Poincare duality for classifying spaces of orbifolds. Some examples of K(n)-fundamental class and intersection product will be given. If time permits, I will explain the similarity of this duality map with that of the Spanier-Whitehead duality for manifolds from the point of view of diﬀerentiable stacks.

**Date:** Friday September 27th

**Time:** 3pm-4pm

**Location:** CLE A211

**Speaker:** Robin Koytcheff

**Title:** A coloured operad for string link infection

**Abstract:** This talk builds on the work of Budney on operads and knot spaces described in the past two seminars. In particular, we build on his operad for splicing of knots, which we will briefly review. Infection of knots or links by string links is a generalization of splicing from knots to links and is useful for studying concordance of knots. In joint work with Burke, we have constructed a coloured operad that encodes this infection operation. This work has motivated us to prove a prime decomposition for 2-component string links in joint work with Burke and Blair. This suggests the possibility of proving further decomposition theorems analogous to those of Budney; this last item is work in progress.

**Date:** Friday September 13th

**Time:** 3pm-4pm

**Location:** CLE A211

**Speaker:** Ryan Budney

**Title:** Spaces of embeddings. Part 1 of 2

**Abstract:** I will describe recent work on the homotopy-type of spaces of embeddings, primarily the space of embeddings of the circle in the 3-sphere. The talk will consist of an introduction to operads, and the basics of what is known about knots and embedding spaces. I will describe how Schubert's connect-sum decomposition can be elaborated into an action of the 2-cubes operad on knot spaces, and how that relates to the homotopy-type of knot spaces via geometrization. This inspired a further operad called the splicing operad, which has further insights into the topology of knot spaces, and deeper connections to geometrization. Towards the end of this 2-part talk I will discuss how there appears to be nice connections to the tools coming from the calculus of functors for describing embedding spaces.

**Date:** Friday August 23rd

**Time:** 3pm-4pm

**Location:** CLE A211

**Speaker:** Ryan Budney

**Title:** Spaces of embeddings. Part 2 of 2

**Abstract:** See previous talk.

**Date:** Friday August 30th

**Time:** 3pm-4pm

**Location:** CLE A211

** Speaker: ** Dale Rolfsen

** Title:** Orderable groups and topology

**Abstract:** I will discuss how the concept of orderability of a group can be applied to problems in topology, particularly in the theory of 3-dimensional manifolds.

**Date/Time: ** 3:30pm CLE A202 on April 2nd

** Speaker: ** Dale Rolfsen

** Title:** A topological view of orderable groups

**Abstract:** Algebra and topology are old friends. Many topological problems are solved by applying algebraic methods. But sometimes the relationship can work the other way. My talk will discuss how the topological viewpoint can be used to establish the basic facts regarding orderability of groups. These facts can be used, in turn, to show that certain groups of interest to topologists are orderable, for example knot groups and the group of PL homeomorphisms of a disk fixed on the boundary.

**Date/Time: ** 3:30pm CLE A207 on April 3rd

** Speaker: ** Ryan Budney

** Title:** Persistent homology applied to musical data

**Abstract:** This talk will describe various metric spaces that are useful for the analysis of musical data, such as spaces of chords and spaces of rhythms. Bill Sethares and I analyze readily-available musical data through the lens of "Persistent Homology". This talk will be a brief report on our results. Will take place in the U.Vic Stats Seminar.

**Date/Time: ** 2:30pm, Friday March 1st, 2013. DSB C128.

** Speaker: ** Jim McClure (Purdue)

** Title:** Poincare duality and sheaf theory

**Abstract:** The main theorem is that the Poincare duality isomorphism that is obtained from sheaf theory is the same as the classical isomorphism obtained from the cap product. This talk requires a minimal knowledge of sheaf theory (basically just a prior exposure to the definition of sheaf and to the derived category).

**Date/Time: ** 1pm Thursday September 27th.

**Location: ** HSD A264

**Speaker:** William Sethares (U. Wisconsin)

**Title:** Topology of Musical Data

**Abstract:** Techniques for discovering topological structures in large data sets are now becoming practical. This talk argues why the musical realm is a particularly promising arena in which to expect to find nontrivial topological features. The analysis is able to recover three important topological features in music: the circle of notes, the circle of fifths, and the rhythmic repetition of timelines, often pictured in the necklace notation.

**Date/Time:** March 8th, 3:30pm--4:30pm

**Location:** SSM A104

**Speaker**: Ryan Budney

**Title**: Some simple triangulations

**Abstract**: I'll describe the story of how Thurston observed some very simple triangulations of knot and link complements in the 3-sphere. This allowed for a relatively simple way to find hyperbolic structures on such manifolds, and was a key inspiration for the Geometrization Conjecture of 3-manifolds. Recently Ben Burton and I found an analogous triangulation for the complement of an embedded 2-sphere in the 4-sphere. While this does not lead to an amazing conjecture like Geometrization, it does lead to an interesting insight into things called Cappell-Shaneson knots, which are connected to the 4-dimensional Poincare conjecture. This is joint work with Ben Burton, and Jonathan Hillman.

**Date/Time**: September 28th, 3:30pm-4:30pm

**Location**: DSB C114

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)

All details will appear here.

**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: 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.

**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.

**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.