Details about the Atlantic Category Theory (ATCAT) Seminar for the 2019 summer term can be found below. Information about current seminars can be found here.
Curran McConnell (Dalhousie University)
Spaces of Trees: Investigating Metric Spaces Used in
Phylogenetics
In order to quantitatively describe "trees of life" representing evolutionary history, biologists have developed a theory of tree metrics. I will discuss some findings about the spaces that arise from these metrics, and how I think these findings might be useful for approaching combinatorial problems related to trees. I will relate the findings to an application using persistent homology to determine whether a lateral gene transfer event has occurred in evolutionary history.
Pablo Andres-Martinez (University of Edinburgh)
The Input-Output Behaviour of Quantum Walks: a
Traced Category
In 2008, Andrew Childs proved that quantum walks are universal for quantum computation, and since then multiple quantum algorithms have been designed using this model. However, in order to run an algorithm in a quantum computer, you must describe it as a circuit, and currently there's no automatic translator (compiler) from quantum walks to circuits. To come up with a translator, we first need to define a mathematical formalism where the behaviour (semantics) of any walk is easily described and manipulated. In this talk, I will sketch the first steps towards such a formalism, defining a suitable traced monoidal category and making use of standard tools from control theory. In the end, I will compare this result to a similar formalism that Hines and Scott proposed in 2012 for single-photon optics.
Jeff Egger
Real and Complex Numbers in a Topos (Part
I)
In this talk, I will give an unconventional account of conventional notions of "extended positive real numbers" in a topos, by which I mean the interval [0,infinity]. Part 2 will continue with notions of (arbitrary) real and complex numbers, not all of which seem to have been considered before.
Jeff Egger
Real and Complex Numbers in a Topos (Part
II)
This is a continuation of the May 14th talk.
Barry Gardner (University of Tasmania)
Generalized Monoid Rings and Other Ring
Extensions
We get a skew polynomial ring over a ring A by defining a skew multiplication by a condition
xa = b+cx
and requiring associativity. Then associating with each a the corresponding, c ,b defines, respectively, an endomorphism f and an (f, id) derivation d of A. Conversely for any such f, d we get an associative multiplication by defining
xa = d(a) + f(a)x.
We generalize this to monoid rings, using a family of self-maps of the coefficient ring to define a new multiplication for which coefficients and monoid elements need not commute. In this context what is usually called a skew monoid ring corresponds to a skew polynomial ring for which the derivation is the zero map.
The connections between our generalized monoid rings and some other types of ring extensions – nornalizing extensions and subnormalizing extensions – will also be discussed.
Jean-Simon Lemay (University of Oxford)
The Missing Link Between Hopf Monads and
*‑Autonomous Monads
A Hopf monad on a monoidal closed category is a monad which lifts the monoidal closed structure to its category of algebras and the forgetful functor preserves this structure strictly. On the other hand, a *‑autonomous monad on a *‑autonomous category lifts the *‑autonomous structure to its category of algebras and the forgetful functor preserves this structure strictly. Since a *‑autonomous category is a monoidal closed category, every *‑autonomous monad is a Hopf monad. However not every Hopf monad is a *‑autonomous monad.
In this talk, I will explain that a Hopf monad is a *‑autonomous monad precisely when the dualizing object is an algebra. The proof is based on the characterization of *‑autonomous monads are linearly distributive categories with negation.
This talk is based on this paper, which is joint work with Masahito Hasegawa.