Radiotherapy is the most common form of treatment for lung cancer patients.
Thoracic specialists develop radiation treatment plans based on CT (computed
tomography)
scans of the lungs. These treatment plans specify the amount of radiation to be
delivered to the patient, as well as which areas of the lungs to irradiate.

Using mathematical and computational tools, my collaborators and
I have developed a software package which can be used to improve the radiation
treatment
planning process. The software is based on a well studied problem in image processing
referred to as "image registration".

In this talk, I will discuss the relationship between image registration and
radiation treatment planning, as well as the types of math and computational tools
required to do research in this field.

Partially motivated by the observation that the curl-curl operator
behaves differently when it is applied to the divergence-free field and
the gradient field in the Hodge decomposition of a vector field, we
introduce the reduced time-harmonic Maxwell (RTHM) equations which solve the divergence-free component of the solution to the time-harmonic Maxwell equations. Three numerical schemes are formulated for solving the RTHM system. Two of them use the classical nonconforming finite element approximations, and the other is based on the interior penalty type discontinuous Galerkin methods. To weakly impose the divergence-free condition satisfied by the solutions, the schemes either work with the locally divergence-free trial spaces, or contain a weighted divergence term in the formula. With the properly chosen stabilizing terms, the optimal error estimates are established on graded meshes. These schemes and the error estimate results are further extended for solving reduced curl-curl problems.

The discrete operators in these schemes naturally define three Maxwell
eigensolvers which are free of spurious eigenmodes and do not contain any
to-be-tuned parameter. The analysis for these solvers is closely related
to the reduced curl-curl problems and their numerical approximations. Not like those Maxwell eigensolvers based on the full curl-curl problems, the compactness of the involved operator and the uniform error estimates for the source problems greatly simplify the analysis of our proposed
eigensolvers. Numerical examples for both Maxwell source and eigenproblems will be presented to demonstrate the performance of the
proposed schemes as well as their relative advantages.

n the talk I will study the spectral properties of a class of
SturmLiouville-type operators on the real line where the derivatives are replaced by a q-difference operator which has been introduced in the context of orthogonal polynomials. Using the relation of this operator to a direct integral of doubly-infinite Jacobi matrices, one can construct examples for isolated pure point, dense pure point, purely absolutely continuous and purely singular continuous spectrum. I will show that the last two spectral types are generic for analytic coefficients and for a class of positive, uniformly continuous coefficients, respectively. A key ingredient in the proof is the so- called Wonderland theorem.
The talk is based on joint work with Malcolm Brown and Karl
Michael Schmidt.

I will present a criterion for the validity of the central limit theorem
for a class of dependent random variables and then I will discuss some applications of
it on random, boundary homogenization problems of nonlinear PDEs such nonlinear
parabolic ones and Navier walls.