1. CJM 2013 (vol 66 pp. 303)
 Elekes, Márton; Steprāns, Juris

Haar Null Sets and the Consistent Reflection of Nonmeagreness
A subset $X$ of a Polish group $G$ is called Haar null if there exists
a Borel set $B \supset X$ and Borel probability measure $\mu$ on $G$ such that
$\mu(gBh)=0$ for every $g,h \in G$.
We prove that there exist a set $X \subset \mathbb R$ that is not Lebesgue null and a
Borel probability measure $\mu$ such that $\mu(X + t) = 0$ for every $t \in
\mathbb R$.
This answers a question from David Fremlin's problem list by showing
that one cannot simplify the definition of a Haar null set by leaving out the
Borel set $B$. (The answer was already known assuming the Continuum
Hypothesis.)
This result motivates the following Baire category analogue. It is consistent
with $ZFC$ that there exist an abelian Polish group $G$ and a Cantor
set $C \subset G$ such that for every nonmeagre set $X \subset G$ there exists a $t
\in G$ such that $C \cap (X + t)$ is relatively nonmeagre in $C$. This
essentially generalises results of BartoszyÅski and BurkeMiller.
Keywords:Haar null, Christensen, nonlocally compact Polish group, packing dimension, Problem FC on Fremlin's list, forcing, generic real Categories:28C10, 03E35, 03E17, , , , , 22C05, 28A78 

2. CJM 2011 (vol 63 pp. 755)
 Chu, Kenneth C. K.

On the Geometry of the Moduli Space of Real Binary Octics
The moduli space of smooth real binary octics has five connected
components. They parametrize the real binary octics whose defining
equations have $0,\dots,4$ complexconjugate pairs of roots
respectively. We show that each of these five components has a real
hyperbolic structure in the sense that each is isomorphic as a
realanalytic manifold to the quotient of an open dense subset of
$5$dimensional real hyperbolic space $\mathbb{RH}^5$ by the action of an
arithmetic subgroup of $\operatorname{Isom}(\mathbb{RH}^5)$. These subgroups are
commensurable to discrete hyperbolic reflection groups, and the
Vinberg diagrams of the latter are computed.
Keywords:real binary octics, moduli space, complex hyperbolic geometry, Vinberg algorithm Categories:32G13, 32G20, 14D05, 14D20 

3. CJM 2011 (vol 63 pp. 1083)
 Kaletha, Tasho

Decomposition of Splitting Invariants in Split Real Groups
For a maximal torus in a quasisplit semisimple simplyconnected group over a local field of characteristic $0$,
Langlands and Shelstad constructed a
cohomological invariant called the splitting invariant, which is an important
component of their endoscopic transfer factors. We study this invariant in the
case of a split real group and prove a
decomposition theorem which expresses this invariant for a general torus as a product of the corresponding
invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants
between different tori in the given real group.
Keywords:endoscopy, real lie group, splitting invariant, transfer factor Categories:11F70, 22E47, 11S37, 11F72, 17B22 

4. CJM 2010 (vol 62 pp. 1058)
5. CJM 2009 (vol 62 pp. 109)
 Li, ChiKwong; Poon, YiuTung

Sum of Hermitian Matrices with Given Eigenvalues: Inertia, Rank, and Multiple Eigenvalues
Let $A$ and $B$ be $n\times n$ complex Hermitian (or real symmetric) matrices
with eigenvalues $a_1 \ge \dots \ge a_n$ and $b_1 \ge \dots \ge b_n$.
All possible inertia values, ranks, and multiple eigenvalues
of $A + B$ are determined. Extension of the results to the sum of $k$ matrices
with $k > 2$ and connections of the results to other subjects such
as algebraic combinatorics are also discussed.
Keywords:complex Hermitian matrices, real symmetric matrices, inertia, rank, multiple eigenvalues Categories:15A42, 15A57 

6. CJM 2009 (vol 62 pp. 646)
 Rupp, R.; Sasane, A.

Reducibility in A_{R}(K), C_{R}(K), and A(K)
Let $K$ denote a compact real symmetric subset of $\mathbb{C}$ and let
$A_{\mathbb R}(K)$ denote the real Banach algebra of all real
symmetric continuous functions on $K$ that are analytic in the
interior $K^\circ$ of $K$, endowed with the supremum norm. We
characterize all unimodular pairs $(f,g)$ in $A_{\mathbb R}(K)^2$
which are reducible.
In addition, for an arbitrary compact $K$ in $\mathbb C$, we give a
new proof (not relying on Banach algebra theory or elementary stable
rank techniques) of the fact that the Bass stable rank of $A(K)$ is
$1$.
Finally, we also characterize all compact real symmetric sets $K$ such
that $A_{\mathbb R}(K)$, respectively $C_{\mathbb R}(K)$, has Bass
stable rank $1$.
Keywords:real Banach algebras, Bass stable rank, topological stable rank, reducibility Categories:46J15, 19B10, 30H05, 93D15 

7. CJM 2008 (vol 60 pp. 958)
 Chen, Yichao

A Note on a Conjecture of S. Stahl
S. Stahl (Canad. J. Math. \textbf{49}(1997), no. 3, 617640)
conjectured that the zeros of genus polynomial are real.
L. Liu and Y. Wang disproved this conjecture on the basis
of Example 6.7. In this note, it is pointed out
that there is an error in this example and a new generating matrix
and initial vector are provided.
Keywords:genus polynomial, zeros, real Categories:05C10, 05A15, 30C15, 26C10 

8. CJM 2006 (vol 58 pp. 529)
 Dijkstra, Jan J.; Mill, Jan van

On the Group of Homeomorphisms of the Real Line That Map the Pseudoboundary Onto Itself
In this paper we primarily consider two natural subgroups of the autohomeomorphism group of the real
line $\R$, endowed with the compactopen topology. First, we prove that the subgroup of
homeomorphisms that map the set of rational numbers $\Q$ onto itself
is homeomorphic to the infinite power of $\Q$ with
the product topology. Secondly, the group consisting of homeomorphisms that map the pseudoboundary
onto itself is shown to be homeomorphic to the hyperspace of nonempty compact subsets of $\Q$ with
the Vietoris topology. We obtain similar results for the Cantor set but we also prove that these
results do not extend to $\R^n$ for $n\ge 2$, by linking the groups in question with Erd\H os
space.
Keywords:homeomorphism group, real line, countable dense set, pseudoboundary, Erd\H{o}s space, hyperspace Category:57S05 

9. CJM 2001 (vol 53 pp. 592)
 Perera, Francesc

Ideal Structure of Multiplier Algebras of Simple $C^*$algebras With Real Rank Zero
We give a description of the monoid of Murrayvon Neumann equivalence
classes of projections for multiplier algebras of a wide class of
$\sigma$unital simple $C^\ast$algebras $A$ with real rank zero and stable
rank one. The lattice of ideals of this monoid, which is known to be
crucial for understanding the ideal structure of the multiplier
algebra $\mul$, is therefore analyzed. In important cases it is shown
that, if $A$ has finite scale then the quotient of $\mul$ modulo any
closed ideal $I$ that properly contains $A$ has stable rank one. The
intricacy of the ideal structure of $\mul$ is reflected in the fact
that $\mul$ can have uncountably many different quotients, each one
having uncountably many closed ideals forming a chain with respect to
inclusion.
Keywords:$C^\ast$algebra, multiplier algebra, real rank zero, stable rank, refinement monoid Categories:46L05, 46L80, 06F05 

10. CJM 2000 (vol 52 pp. 920)
 Evans, W. D.; Opic, B.

Real Interpolation with Logarithmic Functors and Reiteration
We present ``reiteration theorems'' with limiting values
$\theta=0$ and $\theta = 1$ for a real interpolation method
involving brokenlogarithmic functors. The resulting spaces
lie outside of the original scale of spaces and to describe them
new interpolation functors are introduced. For an ordered couple
of (quasi) Banach spaces similar results were presented without
proofs by Doktorskii in [D].
Keywords:real interpolation, brokenlogarithmic functors, reiteration, weighted inequalities Categories:46B70, 26D10, 46E30 

11. CJM 1999 (vol 51 pp. 1240)
12. CJM 1998 (vol 50 pp. 719)
 Göbel, Rüdiger; Shelah, Saharon

Indecomposable almost free modulesthe local case
Let $R$ be a countable, principal ideal domain which is not a field and
$A$ be a countable $R$algebra which is free as an $R$module. Then we
will construct an $\aleph_1$free $R$module $G$ of rank $\aleph_1$
with endomorphism algebra End$_RG = A$. Clearly the result does not
hold for fields. Recall that an $R$module is $\aleph_1$free if all
its countable submodules are free, a condition closely related to
Pontryagin's theorem. This result has many consequences, depending on
the algebra $A$ in use. For instance, if we choose $A = R$, then
clearly $G$ is an indecomposable `almost free' module. The existence of
such modules was unknown for rings with only finitely many primes like
$R = \hbox{\Bbbvii Z}_{(p)}$, the integers localized at some prime $p$. The result
complements a classical realization theorem of Corner's showing that
any such algebra is an endomorphism algebra of some torsionfree,
reduced $R$module $G$ of countable rank. Its proof is based on new
combinatorialalgebraic techniques related with what we call {\it rigid
treeelements\/} coming from a module generated over a forest of trees.
Keywords:indecomposable modules of local rings, $\aleph_1$free modules of rank $\aleph_1$, realizing rings as endomorphism rings Categories:20K20, 20K26, 20K30, 13C10 

13. CJM 1997 (vol 49 pp. 963)
 Lin, Huaxin

Homomorphisms from $C(X)$ into $C^*$algebras
Let $A$ be a simple $C^*$algebra
with real rank zero, stable rank one and weakly
unperforated $K_0(A)$ of countable rank. We show that
a monomorphism $\phi\colon C(S^2) \to A$ can be approximated
pointwise by homomorphisms from $C(S^2)$ into $A$ with
finite dimensional range if and only if certain index
vanishes. In particular, we show that every homomorphism
$\phi$ from $C(S^2)$ into a UHFalgebra can be approximated
pointwise by homomorphisms from $C(S^2)$ into the UHFalgebra
with finite dimensional range. As an application, we show
that if $A$ is a simple $C^*$algebra of real rank zero
and is an inductive limit of matrices over $C(S^2)$ then
$A$ is an AFalgebra. Similar results for tori are also
obtained. Classification of ${\bf Hom}\bigl(C(X),A\bigr)$
for lower dimensional spaces is also studied.
Keywords:Homomorphism of $C(S^2)$, approximation, real, rank zero, classification Categories:46L05, 46L80, 46L35 
