The tumor growth paradox refers to the observation that partially treated tumors might grow bigger than they were before treatment. The cancer stem cell hypothesis provides a model that can explain this behavior. Cancer stem cells are believed to be the organizing centers of a solid tumor. They are immortal and they populate the tumor mass through asymmetric division to produce differentiated cancer cells. If these differentiated cancer cells are killed (through treatment, for example), then space and resources become available for the stem cells to duplicate and, as a result, produce a larger tumor. I present a mathematical model which clearly supports this effect.

I will present the proofs of some recent results of Viale
and Weiss. Weiss introduced the notion of a slender function in his
dissertation: roughly, a function $M \mapsto F(M) \subset M$ (where
$M$ models a fragment of set theory) is slender iff for every
countable $Z \in M$, $Z \cap F(M) \in M$; i.e. $M$ can see countable
fragments of $F(M)$. Viale and Weiss proved that under the Proper
Forcing Axiom, for every regular $\theta \ge \omega_2$, there are
stationarily many $M \in P_{\omega_2}(H_{(2^\theta)^+})$ which
``catch'' $F(M \cap H_\theta)$ whenever $F$ is slender (i.e. whenever
$F$ is slender then there is some $X_F \in M$ such that $F(M \cap
H_\theta) = M \cap X_F$). The stationarity of this collection implies
many of the known consequences of PFA; e.g. failure of weak square at
every regular $\theta \ge \omega_2$; and separating internally
approachable sets from sets of uniform uncountable cofinality.

Minimization of functionals related to Euler's elastica energy has a wide range of applications in computer vision and image processing.
An issue is that a high order nonlinear partial differential equation (PDE) needs to be solved and the conventional algorithm usually takes high computational cost. In this talk, we propose a fast and efficient numerical algorithm to solve minimization problems related to the Euler's elastica energy and show applications to variational image denoising, image inpainting, and image zooming. We reformulate the minimization problem as a constrained minimization problem, followed by an operator splitting method and relaxation. The proposed constrained minimization problem is solved by using an augmented Lagrangian approach. Numerical tests on real and synthetic cases are supplied to demonstrate the efficiency of our method.

Many applications require the solution of time-dependent
partial differential equations (PDEs) on surfaces or more general
manifolds. Methods for treating such problems include surface
parameterization, methods on triangulated surfaces and embedding
techniques. This talk considers an embedding approach based on the
closest point representation of the surface which is very general with
respect to the underlying surface and PDE, yet is extremely simple.
Recent applications to high-order PDEs and Laplace-Beltrami
eigenmodes are given to illustrate the approach.

Many of todays environmental problems, such as air pollution and climate change, are closely related to surprisingly small changes in the composition of our atmosphere. A large variety of very sensitive experimental methods are used today to track these changes with the goal to monitor how human activity impacts the atmosphere and to provide information on which to base possible solutions. Among the many methods to study and monitor atmospheric composition, optical remote sensing has become one of the most widely used techniques. In the ultraviolet and visible spectral regions, where the sun intensity has its maximum and many artificial light sources exist, the method of choice to measure trace gases is Differential Optical Absorption Spectroscopy (DOAS). Examples of DOAS applications include atmospheric chemistry research, emission measurements from industrial facilities, monitoring of volcano activity, global air pollutant observations from space, etc.

DOAS is a method that relies on the measurements of narrow band trace gas absorption features in light originating from the sun, artificial light sources, or solar light scattered in the atmosphere. A number of challenges emerge from this approach. Trace gas absorption features are often present at the same wavelength range and need to be separated accurately from each other. Similarly, the spectral structure of the respective light and the impact of unwanted absorbers must be separated from the trace gas absorptions of interest. As the trace gas absorptions are often very weak, a number of instrumental effects have to be considered when deriving concentrations and their uncertainties. These challenges have lead to the development of numerical retrieval methods, which are at the heart of the DOAS method.

In this talk I will give a general introduction into the DOAS method and present some of its most significant applications. I will discuss the mathematical methods to retrieve trace concentrations from optical absorption measurements and point out the current limitations of the retrieval approach and thus the DOAS method in general.