1. CJM Online first
 Eilers, Søren; Restorff, Gunnar; Ruiz, Efren; Sørensen, Adam P. W.

Geometric classification of graph C*algebras over finite graphs
We address the classification problem for graph $C^*$algebras of
finite graphs (finitely many edges and vertices), containing
the class of CuntzKrieger algebras as a
prominent special case. Contrasting earlier work, we do not assume
that the graphs satisfy the standard condition (K), so that the
graph
$C^*$algebras may come with uncountably many ideals.
We find that in this generality, stable isomorphism of graph
$C^*$algebras does not coincide with the geometric notion of Cuntz
move equivalence. However, adding a modest condition on the
graphs, the two notions are proved to be mutually equivalent and
equivalent to the $C^*$algebras having isomorphic $K$theories. This
proves in turn that under this condition, the graph
$C^*$algebras are in fact classifiable by $K$theory, providing in
particular complete classification when the $C^*$algebras in question
are either of real rank zero or type I/postliminal. The key ingredient
in obtaining these results is a characterization of Cuntz move
equivalence using the adjacency matrices of the graphs.
Our results are applied to discuss the classification problem
for the quantum lens spaces defined by Hong and SzymaÅski,
and to complete the classification of graph $C^*$algebras associated to
all simple graphs with four vertices or less.
Keywords:graph $C^*$algebra, geometric classification, $K$theory, flow equivalence Categories:46L35, 46L80, 46L55, 37B10 

2. CJM 2015 (vol 68 pp. 241)
 Allermann, Lars; Hampe, Simon; Rau, Johannes

On Rational Equivalence in Tropical Geometry
This article discusses the concept of rational equivalence
in tropical
geometry
(and replaces an older and imperfect version).
We give the basic definitions in the context of tropical varieties
without boundary points and prove some basic properties.
We then compute the ``bounded'' Chow groups of $\mathbb{R}^n$ by showing
that they are isomorphic
to the group of fan cycles. The main step in the proof is of
independent interest:
We show that every tropical cycle in $\mathbb{R}^n$ is a sum of (translated)
fan cycles. This also
proves that the intersection ring of tropical cycles is generated
in codimension 1 (by hypersurfaces).
Keywords:tropical geometry, rational equivalence Category:14T05 

3. CJM 2014 (vol 67 pp. 369)
 Graham, Robert; Pichot, Mikael

A Free Product Formula for the Sofic Dimension
It is proved that if $G=G_1*_{G_3}G_2$ is free product of probability
measure preserving $s$regular ergodic discrete groupoids amalgamated
over an amenable subgroupoid $G_3$, then the sofic dimension $s(G)$
satisfies the equality
\[
s(G)=\mathfrak{h}(G_1^0)s(G_1)+\mathfrak{h}(G_2^0)s(G_2)\mathfrak{h}(G_3^0)s(G_3)
\]
where $\mathfrak{h}$ is the normalized Haar measure on $G$.
Keywords:sofic groups, dynamical systems, orbit equivalence, free entropy Category:20E06 

4. CJM 2014 (vol 67 pp. 184)
 McReynolds, D. B.

Geometric Spectra and Commensurability
The work of Reid, ChinburgHamiltonLongReid,
PrasadRapinchuk, and the author with Reid have demonstrated that
geodesics or totally geodesic submanifolds can sometimes be used to
determine the commensurability class of an arithmetic manifold. The
main results of this article show that generalizations of these
results to other arithmetic manifolds will require a wide range of
data. Specifically, we prove that certain incommensurable arithmetic
manifolds arising from the semisimple Lie groups of the form
$(\operatorname{SL}(d,\mathbf{R}))^r \times
(\operatorname{SL}(d,\mathbf{C}))^s$ have the same commensurability
classes of totally geodesic submanifolds coming from a fixed
field. This construction is algebraic and shows the failure of
determining, in general, a central simple algebra from subalgebras
over a fixed field. This, in turn, can be viewed in terms of forms of
$\operatorname{SL}_d$ and the failure of determining the form via certain classes of
algebraic subgroups.
Keywords:arithmetic groups, Brauer groups, arithmetic equivalence, locally symmetric manifolds Category:20G25 

5. CJM 2013 (vol 67 pp. 132)
 Clouâtre, Raphaël

Unitary Equivalence and Similarity to Jordan Models for Weak Contractions of Class $C_0$
We obtain results on the unitary equivalence of weak contractions of
class $C_0$ to their Jordan models under an assumption on their
commutants. In particular, our work addresses the case of arbitrary
finite multiplicity. The main tool is the
theory of boundary representations due to Arveson. We also
generalize and improve previously known results concerning unitary
equivalence and similarity to Jordan models when the minimal function
is a Blaschke product.
Keywords:weak contractions, operators of class $C_0$, Jordan model, unitary equivalence Categories:47A45, 47L55 

6. CJM 2003 (vol 55 pp. 42)
 Benanti, Francesca; Di Vincenzo, Onofrio M.; Nardozza, Vincenzo

$*$Subvarieties of the Variety Generated by $\bigl( M_2(\mathbb{K}),t \bigr)$
Let $\mathbb{K}$ be a field of characteristic zero, and $*=t$ the
transpose involution for the matrix algebra $M_2 (\mathbb{K})$. Let
$\mathfrak{U}$ be a proper subvariety of the variety of algebras with
involution generated by $\bigl( M_2 (\mathbb{K}),* \bigr)$. We define
two sequences of algebras with involution $\mathcal{R}_p$,
$\mathcal{S}_q$, where $p,q \in \mathbb{N}$. Then we show that
$T_* (\mathfrak{U})$ and $T_* (\mathcal{R}_p \oplus \mathcal{S}_q)$
are $*$asymptotically equivalent for suitable $p,q$.
Keywords:algebras with involution, asymptotic equivalence Categories:16R10, 16W10, 16R50 

7. CJM 1997 (vol 49 pp. 1281)
 Sottile, Frank

Pieri's formula via explicit rational equivalence
Pieri's formula describes the intersection product of a Schubert
cycle by a special Schubert cycle on a Grassmannian.
We present a new geometric proof,
exhibiting an explicit chain of rational equivalences
from a suitable sum of distinct Schubert cycles
to the intersection of a Schubert cycle with a special
Schubert cycle. The geometry of these rational equivalences
indicates a link to a combinatorial proof of Pieri's formula using
Schensted insertion.
Keywords:Pieri's formula, rational equivalence, Grassmannian, Schensted insertion Categories:14M15, 05E10 
