2014--2015 Topology Seminar Archive

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The mailing list for the topology seminar is: Topology-talks@lists.uvic.ca

Upcoming Events

A joint applied-topology and spatial statistics meeting.  This is part of a PIMS CRG.  It will be two weeks long, with a gentle first week with some (hopefully!) quite accessible topics, targeted at grad students.

Old Seminar Archives, 2014--2015

Speaker: Helge Ruddat (U.Mainz)

Title: Skeleta of Affine Hypersurfaces

Abstract: Any smooth affine hypersurface Z of complex dimension n deformation retracts to a cell complex of real dimension n. Starting from the Newton polytope of the defining equation of Z, I will give an explicit combinatorial construction of a compact space S, comprised of n-dimensional components, which embeds into Z as a deformation retract. In particular, Z is homotopy equivalent to S. The construction uses toric degenerations, Nakayama-Ogus's work in log geometry and the Kato-Nakayama space. It is motivated by the homological mirror symmetry program. If time permits, I will explain the connections. This work is joint with Nicolò Sibilla, David Treumann and Eric Zaslow

Date/Time: Friday, September 26th at 1pm in HSD A270.

 

Speaker: Ryan Budney

Title: A little obstruction theory

Abstract: Obstruction theory is a topic that is almost as old as manifold theory itself.  I will outline how it began in the work of Whitney and Stiefel, and go on to describe orientations, spin structures and characteristic classes in this language.  I'll end with a brief description of my work on combinatorial spin (and spinc) structures on triangulated manifolds, and how one can encode these structures on computers. 

Date/Time: Friday October 3rd at 1pm in HSD A270.

 

Speaker: Tali Pinsky (UBC)

Title: Templates for Hecke triangles

Abstract: This talk will discuss the theory of templates for flows on three dimensional manifolds, and its application to study closed geodesics on Hecke triangles. By lifting a geodesic to the unit tangent bundle of a Hecke triangle, it becomes a knot in a three manifold. The template then allows us to investigate some topological invariants of these knots. Time permitting, we'll see a simple proof that every closed geodesic on a Hecke triangle (or in fact on any hyperbolic surface) is a prime knot.
 
Date/Time: Friday October 10th at 1pm in HSD A270.

 

Speaker: David Carchedi (UBC)

Title:Dg-manifolds as derived manifolds:

Abstract: Given two smooth maps of manifolds f:M  L and g:N  L, if they are not transverse, the fibered product M xL N may not exist, or may not have the correct cohomological properties. In the world of derived manifolds, such a fibered product always exists as a smooth object, regardless of transversality. In this talk we will describe recent progress of ours with D. Roytenberg on giving an accessible geometric model or derived manifolds using differential graded manifolds.
 

Date/Time: November 21st, 1pm HSD A270.

Speaker: Ryan Budney

Title: Early manifold theory

Abstract: I will describe the state of early manifold theory, primarily in regards to the development of 3-manifold theory.  This starts with the work of Poincare and ends with the discovery of the JSJ-decomposition of 3-manifolds.  Some of the major themes in this story-arc are questions such as: 1) To what extent does the fundamental group (and invariants from algebraic topology) distinguish 3-manifolds?  2) When are homotopy-equivalent 3-manifolds diffeomorphic?   Building on the work of people like Dehn, Papakryakopolous and Reidemeister, Waldhausen gave a complete answer to this question for a family of manifolds perhaps misleadingly called "sufficiently-large".  I will outline these developments leading up to the JSJ-decomposition. 

Date/Time: December 5th, 1pm. HSD A270

Speaker: Ryan Budney

Title: The 70's Revolution

Abstract: The JSJ-decomposiiton raised a glaring, unanswered quesiton at the heart of 3-manifold theory.  All 3-manifolds have a canonical decomposition into "atoroidal" manifolds, but very few atoroidal 3-manifolds were known!   Which manifolds are they?  A magic moment between Bob Riley and Bill Thurston at Warwick University set the stage for the answer, and what is now known as the Geometrization Conjecture.  This is "essentially" a classification of 3-manifolds, modulo the problem of effectively knowing how to navigate the diversity of hyperbolic 3-manifolds.  I will describe these developments and Thurston's partial proof of the Geometrization Conjecture.

Date/Time: Friday January 23rd, 1pm. CLE A302.

Speaker: Andrew Rechnitzer (UBC)

Title: Counting knots

Abstract: Recently a great deal of attention from biologists has been directed to understanding the role of knots in perhaps the most famous of long polymers - DNA. In order for our cells to replicate, they must somehow untangle the approximately two metres of DNA that is packed into each nucleus. Biologists have shown that DNA of various organisms is non-trivially knotted with certain topologies preferred over others. The aim of our work is to determine the "natural" distribution of different knot-types in random closed curves and compare that to the distributions observed in DNA.

Our tool to understand this distribution is a canonical model of long chain polymers - self-avoiding polygons (SAPs). These are embeddings of simple closed curves into a regular lattice. The exact computation of the number of polygons of length n and  fixed knot type K is extremely difficult - indeed the current best algorithms can barely touch the first knotted polygons. Instead of exact methods, in this talk I will describe an approximate enumeration method - which we call the GAS algorithm. This is a generalisation of the famous Rosenbluth method for simulating linear polymers. Using this algorithm we have  uncovered strong evidence that the limiting distribution of different knot-types is universal. Our data shows that a long closed curve is about 28 times more likely  to be a trefoil than a figure-eight, and that the natural distribution of knots is quite different from those found in DNA.

Coordinates: Thursday March 5th, 3:30pm.  Room COR A120.

Speaker: Tullia Dymarz (U.Wisconsin)

Title: Coarsely dense nets in amenable groups

Abstract: In 1998 Burago-Kleiner and McMullen constructed the first examples of coarsely dense and uniformly discrete subsets of Rn that are not biLipschitz equivalent to the standard lattice Zn. Similarly we find subsets inside the three dimensional solvable Lie group SOL that are not bilipschitz to any lattice in SOL. The techniques involve combining ideas from Burago-Kleiner with quasi-isometric rigidity results from geometric group theory. 

Date/Time: April 2nd.  Time 1pm.

Location: DSB C128

Speaker: Ben Williams (UBC)

Title: Topology and Azumaya Algebras

Abstract: Azumaya algebras are generalizations to commutative rings—and beyond---of Central Simple Algebras over fields. In this talk, I will explain how we can use classical homotopy theory to construct counterexamples to purely algebraic conjectures about these objects.

Date/Time: April 3rd.  Time 1pm.

Location: CLE A302

The 54th Cascade Topology Seminar, Spring 2015. 

Speaker: Andy Nicas (McMaster)

Title:  Large Scale Geometry and Topology

Abstract: The goal of large scale geometry and topology is to understand those features of an unbounded space that remain visible after measurements are taken at increasingly large scales.  A key notion in this subject is the "asymptotic dimension" of a metric space due to Gromov. The concept of "finite decomposition complexity" introduced by Guentner, Tessera and Yu is a generalization of asymptotic dimension that applies to many interesting spaces arising in geometry and topology.  I will discuss these ideas and their applications.

Time: Friday July 17th, 11am.

Location: CLE D125