Expand all Collapse all | Results 1 - 15 of 15 |
1. 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 |
2. CJM Online first
Non-tame Mice from Tame Failures of the Unique Branch Hypothesis In this paper, we show that the failure of the unique branch
hypothesis (UBH) for tame trees
implies that in some homogenous generic extension of $V$ there is a
transitive model $M$ containing $Ord \cup \mathbb{R}$ such that
$M\vDash \mathsf{AD}^+ + \Theta \gt \theta_0$. In particular, this
implies the existence (in $V$) of a non-tame mouse. The results of
this paper significantly extend J. R. Steel's earlier results
for tame trees.
Keywords:mouse, inner model theory, descriptive set theory, hod mouse, core model induction, UBH Categories:03E15, 03E45, 03E60 |
3. CJM 2013 (vol 65 pp. 1125)
On the Global Structure of Special Cycles on Unitary Shimura Varieties In this paper, we study the reduced loci of special cycles on local
models of the Shimura variety for $\operatorname{GU}(1,n-1)$. Those special cycles are defined by Kudla and Rapoport. We explicitly compute the irreducible components of the reduced locus of a single special cycle, as well as of an arbitrary intersection of special cycles, and their intersection behaviour in terms of Bruhat-Tits
theory. Furthermore, as an application of our results, we prove the connectedness of arbitrary intersections of special cycles, as conjectured by Kudla and Rapoport.
Keywords:Shimura varieties, local models, special cycles Category:14G35 |
4. CJM 2013 (vol 66 pp. 387)
Composition of Inner Functions We study the image of the model subspace $K_\theta$ under the
composition operator $C_\varphi$, where $\varphi$ and $\theta$ are
inner functions, and find the smallest model subspace which contains
the linear manifold $C_\varphi K_\theta$. Then we characterize the
case when $C_\varphi$ maps $K_\theta$ into itself. This case leads to
the study of the inner functions $\varphi$ and $\psi$ such that the
composition $\psi\circ\varphi$ is a divisor of $\psi$ in the family of
inner functions.
Keywords:composition operators, inner functions, Blaschke products, model subspaces Categories:30D55, 30D05, 47B33 |
5. CJM 2012 (vol 65 pp. 1020)
Monotone Hurwitz Numbers in Genus Zero Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers related to the expansion of complete symmetric functions in the Jucys-Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone join-cut equation, a partial differential equation with initial conditions that characterizes the generating function for monotone Hurwitz numbers in arbitrary genus. The second is our main result, in which we give an explicit formula for monotone Hurwitz numbers in genus zero.
Keywords:Hurwitz numbers, matrix models, enumerative geometry Categories:05A15, 14E20, 15B52 |
6. CJM 2009 (vol 61 pp. 1325)
Uniqueness of Shalika Models Let $\BF_q$ be a finite field of $q$ elements, $\CF$ a $p$-adic field,
and $D$ a quaternion division algebra over $\CF$. This paper proves
uniqueness of Shalika models for $\GL_{2n}(\BF_q) $ and $\GL_{2n}(D)$,
and re-obtains uniqueness of Shalika models for $\GL_{2n}(\CF)$ for
any $n\in \BN$.
Keywords:Shalika models, linear models, uniqueness, multiplicity free Category:22E50 |
7. CJM 2009 (vol 61 pp. 222)
Klyachko Models for General Linear Groups of Rank 5 over a $p$-Adic Field This paper shows the existence and uniqueness of Klyachko models for
irreducible unitary representations of $\GL_5(\CF)$, where $\CF$ is
a $p$-adic field. It is an extension of the work of Heumos and Rallis on $\GL_4(\CF)$.
Keywords:Klyachko models, Whittaker-symplectic model Category:22E50 |
8. CJM 2008 (vol 60 pp. 734)
Genus 2 Curves with Quaternionic Multiplication We explicitly construct the canonical rational models of Shimura
curves, both analytically in terms of modular forms and
algebraically in terms of coefficients of genus 2 curves, in the
cases of quaternion algebras of discriminant 6 and 10. This emulates
the classical construction in the elliptic curve case. We also give
families of genus 2 QM curves, whose Jacobians are the corresponding
abelian surfaces on the Shimura curve, and with coefficients that
are modular forms of weight 12. We apply these results to show
that our $j$-functions are supported exactly at those primes where
the genus 2 curve does not admit potentially good reduction, and
construct fields where this potentially good reduction is attained.
Finally, using $j$, we construct the fields of moduli and definition
for some moduli problems associated to the Atkin--Lehner group
actions.
Keywords:Shimura curve, canonical model, quaternionic multiplication, modular form, field of moduli Categories:11G18, 14G35 |
9. CJM 2006 (vol 58 pp. 1095)
A Casselman--Shalika Formula for the Shalika Model of $\operatorname{GL}_n$ The Casselman--Shalika method is a way to compute explicit
formulas for periods of irreducible unramified representations of
$p$-adic groups that are associated to unique models (i.e.,
multiplicity-free induced representations). We apply this method
to the case of the Shalika model of $GL_n$, which is known to
distinguish lifts from odd orthogonal groups. In the course of our
proof, we further develop a variant of the method, that was
introduced by Y. Hironaka, and in effect reduce many such problems
to straightforward calculations on the group.
Keywords:Casselman--Shalika, periods, Shalika model, spherical functions, Gelfand pairs Categories:22E50, 11F70, 11F85 |
10. CJM 2004 (vol 56 pp. 1290)
Equivariant Formality for Actions of Torus Groups This paper contains a comparison of several
definitions of equivariant formality for actions of torus groups. We
develop and prove some relations between the definitions. Focusing on
the case of the circle group, we use $S^1$-equivariant minimal models
to give a number of examples of $S^1$-spaces illustrating the
properties of the various definitions.
Keywords:Equivariant homotopy, circle action, minimal model,, rationalization, formality Categories:55P91, 55P62, 55R35, 55S45 |
11. CJM 2003 (vol 55 pp. 1264)
Admissible Majorants for Model Subspaces of $H^2$, Part II: Fast Winding of the Generating Inner Function |
Admissible Majorants for Model Subspaces of $H^2$, Part II: Fast Winding of the Generating Inner Function This paper is a continuation of \cite{HM02I}. We consider the model
subspaces $K_\Theta=H^2\ominus\Theta H^2$ of the Hardy space $H^2$
generated by an inner function $\Theta$ in the upper half plane. Our
main object is the class of admissible majorants for $K_\Theta$,
denoted by $\Adm \Theta$ and consisting of all functions $\omega$
defined on $\mathbb{R}$ such that there exists an $f \ne 0$, $f \in
K_\Theta$ satisfying $|f(x)|\leq\omega(x)$ almost everywhere on
$\mathbb{R}$. Firstly, using some simple Hilbert transform techniques,
we obtain a general multiplier theorem applicable to any $K_\Theta$
generated by a meromorphic inner function. In contrast with
\cite{HM02I}, we consider the generating functions $\Theta$ such that
the unit vector $\Theta(x)$ winds up fast as $x$ grows from $-\infty$
to $\infty$. In particular, we consider $\Theta=B$ where $B$ is a
Blaschke product with ``horizontal'' zeros, {\it i.e.}, almost
uniformly distributed in a strip parallel to and separated from $\mathbb{R}$.
It is shown, among other things, that for any such $B$, any even
$\omega$ decreasing on $(0,\infty)$ with a finite logarithmic integral
is in $\Adm B$ (unlike the ``vertical'' case treated in \cite{HM02I}),
thus generalizing (with a new proof) a classical result related to
$\Adm\exp(i\sigma z)$, $\sigma>0$. Some oscillating $\omega$'s in
$\Adm B$ are also described. Our theme is related to the
Beurling-Malliavin multiplier theorem devoted to $\Adm\exp(i\sigma z)$,
$\sigma>0$, and to de~Branges' space $\mathcal{H}(E)$.
Keywords:Hardy space, inner function, shift operator, model, subspace, Hilbert transform, admissible majorant Categories:30D55, 47A15 |
12. CJM 2003 (vol 55 pp. 1231)
Admissible Majorants for Model Subspaces of $H^2$, Part I: Slow Winding of the Generating Inner Function |
Admissible Majorants for Model Subspaces of $H^2$, Part I: Slow Winding of the Generating Inner Function A model subspace $K_\Theta$ of the Hardy space $H^2 = H^2
(\mathbb{C}_+)$ for the upper half plane $\mathbb{C}_+$ is
$H^2(\mathbb{C}_+) \ominus \Theta H^2(\mathbb{C}_+)$ where $\Theta$
is an inner function in $\mathbb{C}_+$. A function $\omega \colon
\mathbb{R}\mapsto[0,\infty)$ is called {\it an admissible
majorant\/} for $K_\Theta$ if there exists an $f \in K_\Theta$, $f
\not\equiv 0$, $|f(x)|\leq \omega(x)$ almost everywhere on
$\mathbb{R}$. For some (mainly meromorphic) $\Theta$'s some parts
of $\Adm\Theta$ (the set of all admissible majorants for
$K_\Theta$) are explicitly described. These descriptions depend on
the rate of growth of $\arg \Theta$ along $\mathbb{R}$. This paper
is about slowly growing arguments (slower than $x$). Our results
exhibit the dependence of $\Adm B$ on the geometry of the zeros of
the Blaschke product $B$. A complete description of $\Adm B$ is
obtained for $B$'s with purely imaginary (``vertical'') zeros. We
show that in this case a unique minimal admissible majorant exists.
Keywords:Hardy space, inner function, shift operator, model, subspace, Hilbert transform, admissible majorant Categories:30D55, 47A15 |
13. CJM 1999 (vol 51 pp. 3)
On a Conjecture of Goresky, Kottwitz and MacPherson We settle a conjecture of Goresky, Kottwitz and MacPherson related
to Koszul duality, \ie, to the correspondence between differential
graded modules over the exterior algebra and those over the
symmetric algebra.
Keywords:Koszul duality, Hirsch-Brown model Categories:13D25, 18E30, 18G35, 55U15 |
14. CJM 1998 (vol 50 pp. 1119)
Ward's solitons II: exact solutions In a previous paper, we gave a correspondence between certain exact
solutions to a \((2+1)\)-dimensional integrable Chiral Model and
holomorphic bundles on a compact surface. In this paper, we use
algebraic geometry to derive a closed-form expression for those
solutions and show by way of examples how the algebraic data which
parametrise the solution space dictates the behaviour of the
solutions.
Dans un article pr\'{e}c\'{e}dent, nous avons d\'{e}montr\'{e} que
les solutions d'un mod\`{e}le chiral int\'{e}grable en dimension \(
(2+1) \) correspondent aux fibr\'{e}s vectoriels holomorphes sur
une surface compacte. Ici, nous employons la g\'{e}om\'{e}trie
alg\'{e}brique dans une construction explicite des solutions. Nous
donnons une formule matricielle et illustrons avec trois exemples
la signification des invariants alg\'{e}briques pour le
comportement physique des solutions.
Keywords:integrable system, chiral field, sigma model, soliton, monad, uniton, harmonic map Category:35Q51 |
15. CJM 1997 (vol 49 pp. 855)
Rational Classification of simple function space components for flag manifolds. Let $M(X,Y)$ denote the space of all continous functions
between $X$ and $Y$ and $M_f(X,Y)$ the path component
corresponding to a given map $f: X\rightarrow Y.$ When $X$ and
$Y$ are classical flag manifolds, we prove the components of
$M(X,Y)$ corresponding to ``simple'' maps $f$ are classified
up to rational homotopy type by the dimension of the kernel of
$f$ in degree two
cohomology. In fact, these components are themselves all products
of flag manifolds and odd spheres.
Keywords:Rational homotopy theory, Sullivan-Haefliger model. Categories:55P62, 55P15, 58D99. |