Details about the Atlantic Category Theory (ATCAT) Seminar for the 2020 winter term can be found below. Information about past seminars can be found here.
Toby Kenney (Dalhousie University)
Euclidean Convexity Spaces
A convexity space is a set equipped with a chosen family of "convex" subsets which are closed under intersections and disjoint unions. In this talk, we ask how much classical geometry can be developed in this framework. We begin be developing the axioms of standard Euclidean geometry.
Lisa Nicklasson (Stockholm University)
Ideals with Linear Powers
An ideal I of a polynomial ring has a linear resolution if all the maps in a minimal free resolution of the ideal are represented by matrices with linear entries. The ideal is said to have linear powers if all its powers I^k have a linear resolution. One family of ideals with linear powers are monomial ideals defined by discrete polymatroids. Another example, also coming from combinatorics, are edge ideals of a certain type of graphs. In this talk we will study these examples in more detail, along with some more general properties of ideals with linear powers. Especially we will discuss what can be said about their Betti numbers and their Rees algebras.
Matthew Amy (Dalhousie University)
Path Integrals and Symbolic Quantum Computing
In the 1940's Richard Feynman developed the path integral formulation of quantum mechanics, in which the amplitude of a quantum state was described as a sum over the amplitudes of paths leading to that state. In this talk I will review discrete analogs of Feynman's path integral for qubit quantum mechanics, and using these path integrals reframe previous results in quantum complexity theory from the perspective of evaluating certain exponential sums. I will then discuss applications to symbolic execution and circuit synthesis via the rewriting of exponential sums.
Jeff Egger
An Element-Free Description of the Mix Structure on
Ban
The category of finite-dimensional Banach spaces and linear contractions is a symmetric star-autonomous category which is not compact closed; in other words, "the" tensor product is not self-dual; which is to say, there are really two monoidal structures, the "usual" one (which is closed) and its dual (which is co-closed), linked by linear distributions; but they have the same unit, so they are also linked by a mix map.
The category of all Banach spaces and linear contractions is not star-autonomous, but both of the monoidal structures extend to the larger category, and continue to be linked by linear distributions and mix maps. I will try to explain why as abstractly as possible, using only the fact that Ban is a closed monoidal structure with a well-behaved factorisation system. (In particular, I will not use the fact that Ban_fd is star-autonomous.)
In principle, this means that similar mix structures might be induced on other categories, and indeed I do know of some, though none outside of functional analysis.
Martin Szyld (Dalhousie University)
Sigma limits in 2-categories and
applications
In (classical, 1-dimensional) category theory, the notions of limit and cone come together. A limit is "just" the universal cone for a diagram. This is no longer the case for higher dimensional categories, where not every limit is conical, and thus the more complicated notion of weight had to be introduced to deal with limits in their correct generality. However, some very basic category theory concepts and constructions depend, in one way or another, on conical limits and on the intuition that comes with them (think for example of filtered categories and filtered (co)limits).
I want to use my first (of hopefully many) seminar talk here to introduce sigma limits. Sigma limits form a notion of 2-dimensional limit which is as general as weighted limits, but at the same time they satisfy: every weighted sigma limit is conical. I will define these limits, show this fact, and show how, using just conical diagrams, we can find a concept of filtered category and filtered (co)limit for dimension 2. Time permitting, I will also mention several results we got so far using sigma limits and present possible future research directions.
Toby Kenney (Dalhousie University)
Euclidean Topological Convexity Spaces
In my previous talk, I showed how we can axiomatise Euclidean geometry in the framework of topological convexity spaces. This time, I will start again from a different perspective, which I hope will provide a simpler axiomatisation and offer more insights into topological convexity spaces in general.
Sara Faridi (Dalhousie University)
Breaking of Homology and Resolutions of Monomial
Ideals
The free resolutions of ideals generated by monomials in a polynomial ring can be studied via properties of topological objects such as simplicial complexes, and more generally CW complexes. This talk will be an introduction to this area, and we will also report on joint work with Mayada Shahada on how an open question about degrees of syzygies of monomial ideals can be considered as a question of breaking homological cycles into smaller ones.
Sarah Li (Dalhousie University)
Xiaoning Bian (Dalhousie University)