Topology

Disconnecting the G_2 Moduli Space

Author: 
Johannes Nordstrom
Date: 
Tue, Jul 7, 2015
Location: 
PIMS, University of British Columbia
Conference: 
PIMS Symposium on the Geometry and Topology of Manifolds
Abstract: 
Little is currently known about the global properties of the $G_2$ moduli space of a closed 7-manifold, ie the space of Riemannian metrics with holonomy $G_2$ modulo diffeomorphisms. A holonomy $G_2$ metric has an associated $G_2$-structure, and I will define a Z/48 valued homotopy invariant of a $G_2$-structure in terms of the signature and Euler characteristic of a Spin(7)-coboundary. I will describe examples of manifolds with holonomy $G_2$ metrics where the invariant is amenable to computation in terms of eta invariants, and which are candidates for having a disconnected moduli space. This is joint work in progress with Diarmuid Crowley and Sebastian Goette.
Notes: 
PDF icon nordstromPIMS2015.pdf
Class: 
Scientific
Subject: 
Mathematics
Topology

Universal torsion, L^2-invariants, polytopes and the Thurston norm

Author: 
Wolfgang Lück
Date: 
Thu, Jul 2, 2015
Location: 
PIMS, University of British Columbia
Conference: 
PIMS Symposium on the Geometry and Topology of Manifolds
Abstract: 
We introduce universal torsion which is defined for $L^2$-acyclic manifolds with torsion free fundamental group and takes values in certain $K_1$-groups of a skew field associated to the integral group ring. It encompasses well-know invariants such as the Alexander polynomial and $L^2$-torsion. We discuss also twisted $L^2$-torsion and higher order Alexander polynomials which can also be derived from the universal invariant and assign certain polytopes to the universal torsion. This gives especially in dimension 3 interesting invariants which recover for instance the Thurston norm.
Notes: 
PDF icon Lueck_vancouver_thurston_norm1507.pdf
Class: 
Scientific
Subject: 
Mathematics
Topology

Introduction to the Farrell-Jones Conjecture

Author: 
Wolfgang Lück
Date: 
Tue, Jul 7, 2015
Location: 
PIMS, University of British Columbia
Conference: 
PIMS Symposium on the Geometry and Topology of Manifolds
Abstract: 

The Farrell-Jones Conjecture identifies the algebraic K- and L-groups for group rings with certain equivariant homology groups. We will give some details of its formulation, its status and indicate some ideas of proofs for certain classes of groups. We will try to convince the audience about its significance by considering special cases and presenting the surprizing large range of its applications to prominent problems in topology, geometry, and group theory.

Notes: 
PDF icon Lueck_vancouver_Farrell-Jones1507.pdf
Class: 
Scientific
Subject: 
Mathematics
Topology

Decision problems, curvature and topology

Author: 
Martin Birdson
Date: 
Tue, Jul 7, 2015
Location: 
PIMS, University of British Columbia
Conference: 
PIMS Symposium on the Geometry and Topology of Manifolds
Abstract: 

I shall discuss a range of problems in which groups mediate between topological/geometric constructions and algorithmic problems elsewhere in mathematics, with impact in both directions. I shall begin with a discussion of sphere recognition in different dimensions. I'll explain why there is no algorithm that can determine if a compact homology sphere of dimension 5 or more has a non-trivial finite-sheeted covering. I'll sketch how ideas coming from the study of CAT(0) cube complexes were used by Henry Wilton and me to settle isomorphism problems for profinite groups, and to settle a conjecture in combinatorics concerning the extension problem for sets of partial permutations.

Notes: 
PDF icon bridsonPIMS2015up copy.pdf
Class: 
Scientific
Subject: 
Mathematics
Differential Geometry and Geometric Analysis
Topology

Khovanov homology

Author: 
Robin Koytcheff
Janis Lazovskis
Date: 
Thu, Jul 10, 2014
Location: 
PIMS, University of British Columbia
Conference: 
2014 West Coast Algebraic Topology Summer School
Abstract: 

Lecture notes on Khovanov homology.

Notes: 
PDF icon wcatss3-3KhovanovHomology.pdf
Class: 
Scientific
Subject: 
Mathematics
Topology

Duality Notes

Author: 
Dominic Culver
Mitchell Faulk
Date: 
Mon, Jul 7, 2014
Location: 
PIMS, University of British Columbia
Conference: 
West Coast Algebraic Topology Summer School
Abstract: 

These are notes for the Duality talk presented as part of the West Coast Algebraic Toplogy Summer School.

Notes: 
PDF icon Duality Notes.pdf
Class: 
Scientific
Subject: 
Mathematics
Topology

Verlinde Algebra

Author: 
Lee Cohn
Date: 
Fri, Jul 11, 2014
Location: 
PIMS, University of British Columbia
Conference: 
West Coast Algebraic Topology Summer School
Abstract: 

Lecture Notes on Verlinde Algebra

Notes: 
PDF icon VerlindeAlgebra1.pdf
Class: 
Scientific
Subject: 
Mathematics
Topology

WCATSS Problem Set 5

Author: 
WCATSS
Date: 
Fri, Jul 11, 2014
Location: 
PIMS, University of British Columbia
Conference: 
West Coast Algebraic Topology Summer School
Abstract: 

This is the third problem of the West Coast Algebraic Topology Summer School (WCATSS)

Notes: 
PDF icon ProblemSet5.pdf
Class: 
Scientific
Subject: 
Mathematics
Topology

WCATSS Problem Set 4

Author: 
WCATSS
Date: 
Thu, Jul 10, 2014
Location: 
PIMS, University of British Columbia
Conference: 
West Coast Algebraic Topology Summer School
Abstract: 

This is the third problem of the West Coast Algebraic Topology Summer School (WCATSS)

Notes: 
PDF icon ProblemSet4.pdf
Class: 
Scientific
Subject: 
Mathematics
Topology

WCATSS Problem Set 3

Author: 
WCATSS
Date: 
Wed, Jul 9, 2014
Location: 
PIMS, University of British Columbia
Conference: 
West Coast Algebraic Topology Summer School
Abstract: 

This is the third problem of the West Coast Algebraic Topology Summer School (WCATSS)

Notes: 
PDF icon ProblemSet3 (2).pdf
Class: 
Scientific
Subject: 
Mathematics
Topology

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