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1. CJM 2013 (vol 66 pp. 303)
Haar Null Sets and the Consistent Reflection of Non-meagreness 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 non-meagre set $X \subset G$ there exists a $t
\in G$ such that $C \cap (X + t)$ is relatively non-meagre in $C$. This
essentially generalises results of BartoszyÅski and Burke-Miller.
Keywords:Haar null, Christensen, non-locally compact Polish group, packing dimension, Problem FC on Fremlin's list, forcing, generic real Categories:28C10, 03E35, 03E17, , , , , 22C05, 28A78 |
2. CJM 2012 (vol 65 pp. 961)
A Hilbert Scheme in Computer Vision Multiview geometry is the study of
two-dimensional images of three-dimensional scenes, a foundational subject in computer vision.
We determine a universal GrÃ¶bner basis for the multiview ideal of $n$ generic cameras.
As the cameras move, the multiview varieties vary in a family of dimension $11n-15$.
This family is the distinguished component of a multigraded Hilbert scheme
with a unique Borel-fixed point.
We present a combinatorial study
of ideals lying on that Hilbert scheme.
Keywords:multigraded Hilbert Scheme, computer vision, monomial ideal, Groebner basis, generic initial ideal Categories:14N, 14Q, 68 |
3. CJM 2011 (vol 63 pp. 1107)
Genericity of Representations of p-Adic $Sp_{2n}$ and Local Langlands Parameters Let $G$ be the $F$-rational points of the symplectic group $Sp_{2n}$,
where $F$ is a non-Archimedean local field
of characteristic
$0$. Cogdell, Kim, Piatetski-Shapiro, and Shahidi
constructed local Langlands functorial lifting from irreducible
generic representations of $G$ to irreducible representations of
$GL_{2n+1}(F)$.
Jiang and Soudry constructed the descent map from irreducible
supercuspidal representations of $GL_{2n+1}(F)$ to those of $G$,
showing that the local Langlands functorial lifting from the
irreducible supercuspidal generic representations is surjective. In
this paper, based on above results, using the same descent method of
studying $SO_{2n+1}$ as Jiang and Soudry, we will show the rest
of local Langlands functorial lifting is also surjective, and for any
local Langlands parameter $\phi \in \Phi(G)$, we construct a
representation $\sigma$ such that $\phi$ and $\sigma$ have the same
twisted local factors. As one application, we prove the $G$-case of a
conjecture of
Gross-Prasad and Rallis, that is, a local Langlands parameter $\phi
\in \Phi(G)$ is generic, i.e., the representation attached to
$\phi$ is generic, if and only if the adjoint $L$-function of $\phi$
is holomorphic at $s=1$. As another application, we prove for each
Arthur parameter $\psi$, and the corresponding local Langlands
parameter
$\phi_{\psi}$, the representation attached to $\phi_{\psi}$
is generic if and only if $\phi_{\psi}$ is tempered.
Keywords:generic representations, local Langlands parameters Categories:22E50, 11S37 |
4. CJM 2004 (vol 56 pp. 825)
Differentiability Properties of Optimal Value Functions Differentiability properties of optimal value functions associated with
perturbed optimization problems require strong assumptions. We consider such
a set of assumptions which does not use compactness hypothesis but which
involves a kind of coherence property. Moreover, a strict differentiability
property is obtained by using techniques of Ekeland and Lebourg and a result
of Preiss. Such a strengthening is required in order to obtain genericity
results.
Keywords:differentiability, generic, marginal, performance function, subdifferential Categories:26B05, 65K10, 54C60, 90C26, 90C48 |