1. CJM 2016 (vol 68 pp. 999)
 Izumi, Masaki; Morrison, Scott; Penneys, David

Quotients of $A_2 * T_2$
We study unitary quotients of the free product unitary pivotal
category $A_2*T_2$.
We show that such quotients are parametrized by an integer $n\geq
1$ and an $2n$th root of unity $\omega$.
We show that for $n=1,2,3$, there is exactly one quotient and
$\omega=1$.
For $4\leq n\leq 10$, we show that there are no such quotients.
Our methods also apply to quotients of $T_2*T_2$, where we have
a similar result.
The essence of our method is a consistency check on jellyfish
relations.
While we only treat the specific cases of $A_2 * T_2$ and $T_2
* T_2$, we anticipate that our technique can be extended to a
general method for proving nonexistence of planar algebras with
a specified principal graph.
During the preparation of this manuscript, we learnt of Liu's
independent result on composites of $A_3$ and $A_4$ subfactor
planar algebras
(arxiv:1308.5691).
In 1994, BischHaagerup showed that the principal graph of a
composite of $A_3$ and $A_4$ must fit into a certain family,
and Liu has classified all such subfactor planar algebras.
We explain the connection between the quotient categories and
the corresponding composite subfactor planar algebras.
As a corollary of Liu's result, there are no such quotient categories
for $n\geq 4$.
This is an abridged version of
arxiv:1308.5723.
Keywords:pivotal category, free product, quotient, subfactor, intermediate subfactor Category:46L37 

2. CJM 2015 (vol 67 pp. 990)
 Amini, Massoud; Elliott, George A.; Golestani, Nasser

The Category of Bratteli Diagrams
A category structure for Bratteli diagrams is proposed and a
functor from
the category of AF algebras to the category of Bratteli diagrams
is
constructed. Since isomorphism of Bratteli diagrams in this
category coincides
with Bratteli's notion of equivalence, we obtain in particular
a functorial formulation of Bratteli's
classification of AF algebras (and at the same time, of Glimm's
classification of UHF~algebras).
It is shown that the three approaches
to classification of AF~algebras, namely, through Bratteli diagrams,
Ktheory, and
abstract classifying categories, are essentially the same
from a categorical point of view.
Keywords:C$^{*}$algebra, category, functor, AF algebra, dimension group, Bratteli diagram Categories:46L05, 46L35, 46M15 

3. CJM 2014 (vol 67 pp. 28)
 Asadollahi, Javad; Hafezi, Rasool; Vahed, Razieh

Bounded Derived Categories of Infinite Quivers: Grothendieck Duality, Reflection Functor
We study bounded derived categories of the category of representations of infinite quivers over a ring $R$. In case $R$ is a commutative noetherian ring with a dualising complex, we investigate an equivalence similar to Grothendieck duality for these categories, while a notion of dualising complex does not apply to them. The quivers we consider are left, resp. right, rooted quivers that are either noetherian or their opposite are noetherian. We also consider reflection functor and generalize a result of Happel to noetherian rings of finite global dimension, instead of fields.
Keywords:derived category, Grothendieck duality, representation of quivers, reflection functor Categories:18E30, 16G20, 18E40, 16D90, 18A40 

4. CJM 2012 (vol 65 pp. 82)
 Félix, Yves; Halperin, Steve; Thomas, JeanClaude

The Ranks of the Homotopy Groups of a Finite Dimensional Complex
Let $X$ be an
$n$dimensional, finite, simply connected CW complex and set
$\alpha_X =\limsup_i \frac{\log\mbox{ rank}\, \pi_i(X)}{i}$. When
$0\lt \alpha_X\lt \infty$, we give upper and lower bound for $
\sum_{i=k+2}^{k+n} \textrm{rank}\, \pi_i(X) $ for $k$ sufficiently
large. We show also for any $r$ that $\alpha_X$ can be estimated
from the integers rk$\,\pi_i(X)$, $i\leq nr$ with an error bound
depending explicitly on $r$.
Keywords:homotopy groups, graded Lie algebra, exponential growth, LS category Categories:55P35, 55P62, , , , 17B70 
