July 11, 2017
Soumen Sarkar (Indian Institute of Technology Madras, Chennai), On integral cohomology of toric orbifolds
Abstract: There are several advantages to study topological spaces whose integral cohomology groups $H^{*}(X, \mzthbb{Z})$ are torsion-free and concentrated in even degrees; for instance, their complex $K$-theory and complex cobordism groups can be deduced with little bit extra effort. We call such spaces 'even'. In this talk, I will identify certain families of toric orbifolds which are even, and then I will compute the structure of their cohomology ring.

This is a joint work with Tony Bahri and Jongbaek Song.



September 12, 2017
Kohei Kishida (Dalhousie), Non-Locality, Contextuality, and Topology
Abstract: Non-locality and contextuality are among the most paradoxical properties of quantum physics contradicting the intuitions behind classical physics. In addition to their foundational significance, non-locality is fundamental to quantum information, and recent studies suggest contextuality may constitute a key computational resource of quantum computation. This has motivated inquiries into higher-level, structural expressions of non-locality and contextuality that are independent of the concrete formalism of quantum mechanics. One approach uses the mathematical tool of sheaf theory, and has yielded the insight that non-locality and contextuality are topological in nature.

In this talk, I first review several ideas as well as formal expressions of non-locality, and extract from them the topological formalism for quantum measurement scenarios and a characterization of non-locality in this formalism. In fact, as we show, the same characterization captures contextuality as well (so that non-locality amounts to a special case of contextuality). We will then illustrate the power of this higher-level, unifying formalism: On the one hand, it leads to several new methods of contextuality argument. On the other hand, it shows contextuality to be a ubiquitous phenomenon that can be found in various other disciplines.

This is joint work with Samson Abramsky, Rui Soares Barbosa, Ray Lal and Shane Mansfield.


September 19, 2017
Geoff Cruttwell (Mount Allison), Calculus via functors and natural transformations
Abstract: In this talk, we focus on the "category of multivariable calculus": the category whose objects are open subsets of R^n's and whose maps are smooth maps between these spaces. We show that this category has a variety of endofunctors and natural transformations whose existence is essentially equivalent to various important properties of the derivative (such as the chain rule and the symmetry of mixed partial derivatives).

The intent of the talk is two-fold: (1) to demonstrate a different way to work with the ideas of multivariable calculus and differential geometry than is normally presented in courses on these subjects, and (2) to give an introduction to tangent categories at a concrete level, both for those new to the subject and for those who need a refresher.



September 26, 2017
Frank Fu (Dalhousie), Encoding Data in Lambda Calculus: An Introduction
Abstract: Lambda calculus is a formalism introduced by Alonzo Church in the 1930s for his research on the foundations of mathematics. It is now widely used as a theoretical foundation for the functional programming languages (e.g. Haskell, OCaml, Lisp). I will first give a short introduction to lambda calculus, then I will discuss how to encode natural numbers using the encoding schemes invented by Alonzo Church, Dana Scott and Michel Parigot. Although we will mostly focus on numbers, these encoding schemes also works for more general data structures such as lists and trees. If time permits, I will talk about the type theoretical aspects of these encodings.

October 3, 2017
Geoff Cruttwell (Mount Allison), Geometric spaces in a tangent category
Abstract: A connection on the tangent bundle of a smooth manifold arises frequently in differential geometry, and gives as a consequence associated notions of curvature, parallel transport, and geodesics. In this talk, we look at how to define such connections in the abstract setting of a tangent category, calling an object equipped with such a connection a "geometric object".

We look at three different aspects of this theory of such objects: (a) how one can define maps between such geometric objects, giving a new tangent category, (2) what it means for such connections to be "flat" and "torsion-free", (3) how to characterize flat torsion-free connections as "connection-preserving connections". The last result appears to be new in differential geometry.

This is joint work with Rick Blute and Rory Lucyshyn-Wright.


October 10, 2017
Marzieh Bayeh (Dalhousie), Higher Topological Complexity
Abstract:This is an introductory talk on higher topological complexity. Topological complexity of the configuration space of a mechanical system was introduced by M. Farber in 2003 to estimate the complexity of a motion planning algorithm. Generalizing this concept, higher topological complexity was introduced by Y. Rudyak in 2010. This invariant may be used to find the sequential motion planning algorithms.

October 17, 2017
Geoff Cruttwell (Mount Allison), The rich structure of affine geometric spaces
Abstract: Continuing the previous talk on geometric spaces in a tangent category, we focus on the sub-tangent category of affine geomtric spaces: those objects equipped with a flat torsion-free connection. Following ideas of Jubin, we show that this category has an astonishing variety of monads and comonads on it, with many distributive laws relating these monads and comonads.

This is joint work with Rick Blute and Rory Lucyshyn-Wright.