1. CJM 2016 (vol 69 pp. 1036)
 Carlen, Eric; Maggi, Francesco

Stability for the BrunnMinkowski and Riesz Rearrangement Inequalities, with Applications to Gaussian Concentration and Finite Range Nonlocal Isoperimetry
We provide a simple, general argument to obtain improvements
of concentrationtype inequalities starting from improvements
of their corresponding isoperimetrictype inequalities. We apply
this argument to obtain robust improvements of the BrunnMinkowski
inequality (for Minkowski sums between generic sets and convex
sets) and of the Gaussian concentration inequality. The former
inequality is then used to obtain a robust improvement of the
Riesz rearrangement inequality under certain natural conditions.
These conditions are compatible with the applications to a finiterange
nonlocal isoperimetric problem arising in statistical mechanics.
Keywords:BrunnMinkowski inequality, Riesz rearrangement, Gaussian Concentration, GatesPenroseLebowitz energy Category:49N99 

2. CJM 2016 (vol 69 pp. 453)
 Marquis, Timothée; Neeb, KarlHermann

Isomorphisms of Twisted Hilbert Loop Algebras
The closest infinite dimensional relatives of compact Lie algebras are HilbertLie algebras, i.e. real Hilbert spaces with a Lie
algebra
structure for which the scalar product is invariant.
Locally affine Lie algebras (LALAs)
correspond to double extensions of (twisted) loop algebras
over simple HilbertLie algebras $\mathfrak{k}$, also called
affinisations of $\mathfrak{k}$.
They possess a root space decomposition
whose corresponding root system is a locally affine root system
of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$,
$D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some
infinite set $J$. To each of these types corresponds a ``minimal"
affinisation of some simple HilbertLie algebra $\mathfrak{k}$,
which we call standard.
In this paper, we give for each affinisation $\mathfrak{g}$ of
a simple HilbertLie algebra $\mathfrak{k}$ an explicit isomorphism
from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from
the classification
of locally affine root systems, but
for representation theoretic purposes it is crucial to obtain
it explicitly
as a deformation between two twists which is compatible
with the root decompositions.
We illustrate this by applying our isomorphism theorem to the
study of positive energy highest weight representations of $\mathfrak{g}$.
In subsequent work, the present paper will be used to obtain
a complete classification
of the positive energy highest weight representations of affinisations
of $\mathfrak{k}$.
Keywords:locally affine Lie algebra, HilbertLie algebra, positive energy representation Categories:17B65, 17B70, 17B22, 17B10 

3. CJM 2014 (vol 66 pp. 1110)
 Li, Dong; Xu, Guixiang; Zhang, Xiaoyi

On the Dispersive Estimate for the Dirichlet SchrÃ¶dinger Propagator and Applications to Energy Critical NLS
We consider the obstacle problem for the SchrÃ¶dinger evolution
in the exterior of the unit ball with Dirichlet boundary condition. Under
the radial symmetry we compute explicitly the fundamental solution
for the linear Dirichlet SchrÃ¶dinger
propagator $e^{it\Delta_D}$
and give a robust algorithm to prove sharp $L^1 \rightarrow
L^{\infty}$ dispersive estimates. We showcase the analysis in
dimensions $n=5,7$. As an application, we obtain global
wellposedness and scattering for defocusing energycritical NLS on
$\Omega=\mathbb{R}^n\backslash \overline{B(0,1)}$ with Dirichlet boundary
condition and radial data in these dimensions.
Keywords:Dirichlet SchrÃ¶dinger propagator, dispersive estimate, Dirichlet boundary condition, scattering theory, energy critical Categories:35P25, 35Q55, 47J35 

4. CJM 2013 (vol 66 pp. 1413)
 Zhang, Xi; Zhang, Xiangwen

Generalized KÃ¤hlerEinstein Metrics and Energy Functionals
In this paper, we consider a generalized
KÃ¤hlerEinstein equation on KÃ¤hler manifold $M$. Using the
twisted $\mathcal K$energy introduced by Song and Tian, we show
that the existence of generalized KÃ¤hlerEinstein metrics with
semipositive twisting $(1, 1)$form $\theta $ is also closely
related to the properness of the twisted $\mathcal K$energy
functional. Under the condition that the twisting form $\theta $ is
strictly positive at a point or $M$ admits no nontrivial Hamiltonian
holomorphic vector field, we prove that the existence of generalized
KÃ¤hlerEinstein metric implies a MoserTrudinger type inequality.
Keywords:complex MongeAmpÃ¨re equation, energy functional, generalized KÃ¤hlerEinstein metric, MoserTrudinger type inequality Categories:53C55, 32W20 

5. CJM 2011 (vol 64 pp. 24)
 Borodachov, S. V.

Lower Order Terms of the Discrete Minimal Riesz Energy on Smooth Closed Curves
We consider the problem of minimizing the energy of $N$ points
repelling each other on curves in $\mathbb{R}^d$ with the potential
$xy^{s}$, $s\geq 1$, where $\, \cdot\, $ is
the Euclidean norm. For a sufficiently smooth, simple, closed,
regular curve, we find the next order term in the asymptotics of the
minimal $s$energy. On our way, we also prove that at
least for $s\geq 2$, the minimal pairwise distance in optimal configurations
asymptotically equals $L/N$, $N\to\infty$, where $L$ is the length
of the curve.
Keywords:minimal discrete Riesz energy, lower order term, power law potential, separation radius Categories:31C20, 65D17 

6. CJM 2008 (vol 60 pp. 457)
 Teplyaev, Alexander

Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure
We define sets with finitely ramified cell structure, which are
generalizations of postcrit8cally finite selfsimilar
sets introduced by Kigami and of fractafolds introduced by Strichartz. In general,
we do not assume even local selfsimilarity, and allow countably many cells
connected at each junction point.
In particular, we consider postcritically infinite fractals.
We prove that if Kigami's resistance form
satisfies certain assumptions, then there exists a weak Riemannian metric
such that the energy can be expressed as the integral of the norm squared
of a weak gradient with respect to an energy measure.
Furthermore, we prove that if such a set can be homeomorphically represented
in harmonic coordinates, then for smooth functions the weak gradient can be
replaced by the usual gradient.
We also prove a simple formula for the energy measure Laplacian in harmonic
coordinates.
Keywords:fractals, selfsimilarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric Categories:28A80, 31C25, 53B99, 58J65, 60J60, 60G18 

7. CJM 2004 (vol 56 pp. 529)
 MartínezFinkelshtein, A.; Maymeskul, V.; Rakhmanov, E. A.; Saff, E. B.

Asymptotics for Minimal Discrete Riesz Energy on Curves in $\R^d$
We consider the $s$energy
$$
E(\ZZ_n;s)=\sum_{i \neq j} K(\z_{i,n}z_{j,n}\;s)
$$
for point sets $\ZZ_n=\{ z_{k,n}:k=0,\dots,n\}$ on certain compact sets
$\Ga$ in $\R^d$ having finite onedimensional Hausdorff measure, where
$$
K(t;s)=
\begin{cases}
t^{s} ,& \mbox{if } s>0, \\
\ln t, & \mbox{if } s=0,
\end{cases}
$$
is the Riesz kernel. Asymptotics for the minimum $s$energy and the
distribution of minimizing sequences of points is studied. In
particular, we prove that, for $s\geq 1$, the minimizing nodes for a
rectifiable Jordan curve $\Ga$ distribute asymptotically uniformly with
respect to arclength as $n\to\infty$.
Keywords:Riesz energy, Minimal discrete energy,, Rectifiable curves, Bestpacking on curves Categories:52A40, 31C20 
