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.

October 24, 2017

Darien DeWolf (Saint Francis-Xavier)

*Restriction bicategories: two approaches*

**Abstract:**In this talk, I will introduce restriction bicategories: intuitively, a restriction bicategory is a bicategory B equipped with a family of functors r : B(A,B) --> B(A,A) which encode partiality in a way reminiscent of Cockett and Lack's restriction categories. Motivating this definition is the ``restriction bicategory'' of restriction bimodules.

Two approaches to defining such structures will be discussed:

(i) Cockett's approach has each restriction idempotent r(f) come with a monic
r(f) --> dom(f).

(ii) The approach taken in my thesis is more general in that it does not
require these monics.
Each approach has both merit and drawbacks, which will also be discussed.

October 31, 2017

Dorette Pronk (Dalhousie),

*On Suborbifolds*

**Abstract:**Just as the notion of orbifold has developed over the 60 years, so has the notion of suborbifold. It was introduced by Thurston in the late 70s, along with the first revision of the definition of orbifold. This original definition was geometrically elegant, but fails to encompass some of their examples that have since been introduced by a more topological/homotopical view of orbifolds. This has led to various more recent definitions of suborbifolds, all based on presenting orbifolds as groupoids. However, none of these definitions fully captures the properties and phenomena their authors intend to include. Therefore, we propose an alternate approach, based on atlases and modules rather than groupoid homomorphisms. We believe that this new definition will better describe and encompass the examples and geometric structure in the literature.

This is joint work with Laura Scull and Matteo Tommasini

November 14, 2017

Francisco Rios (Dalhousie),

*On Categorical Models of Intuitionistic Linear Logic*

**Abstract:**In the 1980s, Jean-Yves Girard introduced the substructural logic called linear logic as a refinement of classical and intuitionistic logic. Its emphasis on the role of formulas as resources has found many applications in fields as diverse as computer science, quantum physics, and linguistics. In this talk, I will introduce some of the most relevant categorical models of the intuitionistic fragment of linear logic, namely Lafont categories, Seely categories, and linear categories. Time permitting, I will show how these models are subsumed by the LNL (linear/non-linear) models introduced by Nick Benton in the 1990s.

November 21, 2017

Marzieh Bayeh (Dalhousie),

*A lambda calculus for quantum computation*

**Abstract:**"PhD Thesis of Benoit Valiron" is a part of what I am studying on quantum computation and "A lambda calculus for quantum computation" is a part of that thesis. So I am going to talk about part^2 of what I am studying on quantum computation. To make it part^3, introducing the terms, types, and typing rule for that lambda calculus is a part of this talk.

January 9, 2018, Bob Paré,

*Introduction to double categories*

**Abstract:**I will define double categories and their morphisms and give some examples. I will speculate on why I think they may be useful.

January 16, 2018, Bob Rosebrugh (Mount Allison),

*Symmetric lenses and cospans*

**Abstract:**A symmetric lens is state synchronization data and synchronization restoration operations between model domains. An asymmetric lens has only one-way data and operations. Some time ago we showed that spans of asymmetric lenses represent symmetric lenses. In the special case that we named (asymmetric) c-lenses the restoration operation satisfies a universal property. We have recently considered cospans of asymmetric lenses. They also generate symmetric lenses and we can now characterize the special case of symmetric lenses which so arise. When the cospan consists of c-lenses then the symmetric lens has a universal property.

(Joint work with Michael Johnson)

January 23, 2018, Bob Paré (Dalhousie),

*Kleisli double categories*

**Abstract:**I will explain some of the basic concepts of double category theory (companions, conjoints, tabulators, Cauchy completeness, products, etc.) and examine in detail how they play out in Kleisli double categories.

January 30, 2018, Michael Lambert (Dalhousie),

*Flat Category-Valued Pseudo-Functors*

**Abstract:**This talk will be a continuation of my talk from the end of last term. I will recall the notion of a flat set-valued functor on a small category. I will then show how a straightforward generalization of this definition for category-valued pseudo-functors recovers many of the expected properties of ordinary flatness.

February 6, 2018, Frank Fu (Dalhousie),

*An introduction to initial algebraic semantics for data types*

**Abstract:**The initial algebras of endofunctors can be used to model regular data types in the typed functional programming languages. In this talk, we will recall the standard approach by applying it to model the list data type. Then we will discuss its limitations and a well-known generalization. Finally we will show the limitations of the generalization and pose some questions.

February 13, 2018, Marzieh Bayeh (Dalhousie),

*Type inference for quantum lambda calculus*

**Abstract:**In this talk, first I will explain the problem of type inference for a typed lambda calculus. Then I will talk about the type inference for quantum lambda calculus.

February 27, 2018, Julien Ross (Dalhousie),

*Quantum magic games*

**Abstract:**In this talk, I will discuss two-player cooperative games known as quantum magic games. These games have the property that players sharing quantum resources can win the game with certainty whereas players sharing only classical resources cannot. During the talk, I will introduce quantum magic games and review known results.

March 6, 2018, Evangelia Aleiferi (Dalhousie),

*Involutive Structures on Cartesian double categories*

**Abstract:**This talk will be a continuation of my talk on Cartesian double categories from last year's @Cat seminar. We will recall the basic definitions on the subject and we will talk about involutions on a fibrant double category. We will also work towards the construction of an involution on any Cartesian and fibrant double category.

March 20, 2018, Xiaoning Bian (Dalhousie),

*Relations for Clifford+T operators on two qubits*

**Abstract:**In this talk, I will give a finite presentation for 2-qubit Clifford+T operators. First I will show you some background and state my question in a Symmetric Monoidal groupoid setting. Then I will introduce two tricks in finding group presentations.

1) if you have a norm form for group elements, then you might be able to find a finite presentation.

2) if you know a presentation for a group, then you are able to find a presentation for its subgroups of finite index.

These two tricks together with the Giles-Selinger algorithm (which works like Gaussian elimination) solves my question. Then I will show you some computer implementation details in applying trick 2) and in reducing 270 relations obtained from trick 2) into 40.

March 27, 2018, Fahimeh Bayeh (Dalhousie),

*A Categorical Model of Linear Logic*

**Abstract:**In this talk, first I will give an introduction of Linear Logic and Monoidal Categories. Then I will explain the categorical structure of Linear Logic.

April 3, 2018, Francisco Rios (Dalhousie),

*Typed Calculi, Constructive Logics, and Categorical Models: The Intuitionistic Linear Logic Case*

**Abstract:**For decades, classical and intuitionistic logics have served extensively in the design of programming languages and analysis of programs and specifications. However, the recent advancements in quantum computing, in general, and the likely arrival of scalable programmable quantum devices in the near future, in particular, have propelled the research and development of quantum programming languages--for which classical and intuitionistic logics are not sufficient. For these languages to be useful, they must, among other things, be able to handle quantum information as a non-duplicable resource. Linear logic, being a resource-sensitive formal system, provides an appropriate framework for the study of such languages. In this talk, I will discuss the remarkable relationship among typed linear term calculi (the bases for functional quantum programming languages), constructive logics, and categorical models, with an emphasis on those corresponding to the intuitionistic fragment of linear logic.

August 21, 2018

Simon Willerton (Sheffield),

*The magnitude of odd balls*

**Abstract:**Tom Leinster introduced the magnitude of finite metric spaces by formal analogy with his notion of Euler characteristic of finite categories. This can be thought of an 'effective number of points' of the metric space. This simple idea has turned out to have connections with all sorts of mathematics, including diversity measurement, Hausdorff dimension, categorification, semi-classical analysis and curvature measures. In this talk I'll give a fair amount of background and then focus on obtaining a formula for the magnitude of odd dimensional balls, explaining what it has to do with nineteenth century integrals and twentieth century enumerative combinatorics.