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Mathematics

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.

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.

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.

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.

Nassif Ghoussoub Receives Honorary Degree from the University of Victoria

Speaker: 
Nassif Ghoussoub
Date: 
Fri, Jun 12, 2015
Location: 
University of Victoria
Abstract: 
Dr. Nassif Ghoussoub received the honorary degree of DSc at UVic's convocation ceremonies on June 12, 2015. Ghoussoub is a world-leading mathematician who has played a critical role in building Canadian networks for the support of education and research in mathematical sciences. He is the founder of the Pacific Institute for Mathematical Sciences (PIMS), the Banff International Research Station (BIRS) and a co-founder of Mathematics of Information Technology and Complex Systems (MITACS). PIMS has transformed mathematics among research universities in western Canada by encouraging regional research initiatives. BIRS has gained an international reputation as a premier venue for mathematics conferences. MITACS, an offshoot of PIMS, promotes collaboration among mathematicians working in industry and academia. Ghoussoub, a math professor at the University of British Columbia, is a world leader in the field of partial differential equations.

A topological look at the vector (cross) product in three dimensions

Speaker: 
Peter Zvengrowski
Date: 
Sat, May 9, 2015
Location: 
PIMS, University of Lethbridge
Conference: 
Alberta Mathematics Dialog
Abstract: 
The vector product (or cross product) of two vectors in 3-dimensional real space $\mathbb{R}^3$ is a standard item covered in most every text in calculus, advanced calculus, and vector calculus, as well as in many physics and linear algebra texts. Most of these texts add a remark (or “warning”) that this vector product is available only in 3-dimensional space. In this talk we shall start with some of the early history, in the nineteenth century, of the vector product, and in particular its relation to quaternions. Then we shall show that in fact the 3-dimensional vector product is notthe only one, indeed the Swiss mathematician Beno Eckmann (a frequent visitor to Alberta) discovered a vector product in 7-dimensional space in 1942. Further- more, by about 1960 deep advances in topology implied that there were no further vector products in any other dimension. We shall also, following Eckmann, talk about the generalization to r-fold vector products for $r\geq 1$ (the familiar vector product is a 2-fold vector product), and give the complete results for which dimensions n and for which $r$ these can exist. In the above work it is clear that the spheres $S^3$, $S^7$ play a special role (as well as their “little cousin” $S^1$). In the last part of the talk we will briefly discuss how these special spheres also play a major part in the recent solution of the Kervaire conjecture by Hill, Hopkins, and Ravenel, as well as their relation to the author’s own research on the span of smooth manifolds.

It’s All in the Follow Through – what research in math education says ... and doesn’t say

Speaker: 
Rob Craigen
Date: 
Sat, May 9, 2015
Location: 
PIMS, University of Lethbridge
Conference: 
Alberta Mathematics Dialog
Abstract: 
We’ll be examining a few classic cases of how educational research has been handled that explain a lot about how we got where we are in public school math education today.

Robustness of Design: A Survey

Speaker: 
Doug Wiens
Date: 
Fri, May 8, 2015
Location: 
PIMS, University of Lethbridge
Conference: 
Alberta Mathematics Dialog
Abstract: 

When an experiment is conducted for purposes which include fitting a particular model to the data, then the ’optimal’ experimental design is highly dependent upon the model assumptions - linearity of the response function, independence and homoscedasticity of the errors, etc. When these assumptions are violated the design can be far from optimal, and so a more robust approach is called for. We should seek a design which behaves reasonably well over a large class of plausible models. I will review the progress which has been made on such problems, in a variety of experimental and modelling scenarios - prediction, extrapolation, discrimination, survey sampling, dose-response, etc

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