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1. CJM Online first
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 |
2. CJM Online first
Geometric Spectra and Commensurability The work of Reid, Chinburg-Hamilton-Long-Reid,
Prasad-Rapinchuk, 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 |
3. CJM Online first
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 |
4. CJM 2003 (vol 55 pp. 42)
$*$-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 |
5. CJM 1997 (vol 49 pp. 1281)
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 |