Singularities occur in Ricci flow because of curvature blowup. For dimensional reasons, when approaching a singularity, one expects the curvature to blow up like the inverse of the time to the singularity. If this does not happen, the singularity is said to be type II. The first example of a type II singularity, studied by Daskalopoulos-Del Pino-Hamilton-Sesum, occurs on a noncompact surface which is the result of capping off a hyperbolic cusp. The analysis in the surface case uses isothermal coordinates. It is not immediately clear whether it extends to higher dimensions. We look at the Ricci flow on finite-volume metrics that live on the complement of a divisor in a compact Khler manifold. We compute the blowup time in terms of cohomological data and give sufficient conditions for a type II singularity to emerge. This is joint work with Zhou Zhang.

In this expository talk we relate the spectral properties of a discrete
Schr"odinger operator on a d-dimentional lattice to its dynamical
features. Dynamical quantities of interest include Fourier transforms of
spectral measures, time averaged moments of the position operator, as well
as time-averaged observables for a compact operator. The RAGE theorem in
its various formulations predicts the asymptotic behaviour of these
quantities for any state in the continuous subspace of the Hilbert space:
observables for a compact operator decay to zero, whereas the moments of
the position operator asymptotically diverge. In order to quantify this
decay/divergence, we present a decomposition of the spectral measure with
respect to Hausdorff measures of dimension $\alpha \in [0,1]$. This
decomposition due to Rogers and Taylor generalizes the classical
decomposition of the spectral measure w.r.t. Lebesgue measure into pure
point and continous component. Whereas for $\alpha = 1$ it recasts the
classical result, for $\alpha < 1$ one obtains a decomposition different
to the classical one. For each Hausdorff dimension, the spectral measure
then splits in an $\alpha$-continuous and an $\alpha$-singular component.
$\alpha$-continuous measures are shown to be limits of uniformly $\alpha$
H"older continuous (U$\alpha$H) measures w.r.t. to a suitable topology.
For U$\alpha$H spectral measures lower and upper bounds for various
dynamical quantities are available.

references:
Y. Last, Quantum dynamics and decompositions of singular continuous
spectra, J. Funct. Anal 142, 406-445 (1996).
W. Kirsch: An invitation to random Schr"odinger operators,
arXiv:0709.3707v1[math-ph].
G. Teschl: Mathematical Methods in Quantum mechanics with application to
Schroedinger operators, Graduate Studies in Mathematics, Amer. Math. Soc.,
Providence, 2008. (to appear).

Understanding the seasonal/periodic reoccurrence of influenza will be
very helpful in designing successful vaccine programs and introducing
public health interventions. However, the reasons for seasonal/periodic
influenza epidemics are still not clear even though various explanations
have been proposed. In this talk, we present an age-structured type evolutionary epidemiological model of influenza A drift, in which the susceptible class is continually replenished because the pathogen changes genetically and immunologically from one epidemic to the next, causing previously immune hosts to become susceptible. Applying our recent established center manifold theory for semilinear equations with non-dense domain, we show that Hopf bifurcation occurs in the model. This
demonstrates that the age-structured type evolutionary epidemiological model of influenza A drift has an intrinsic tendency to oscillate due to the evolutionary and/or immunological changes of the influenza viruses.
(based on joint work with Pierre Magal).