We develop, in an axiomatic fashion, mathematical equations that describe the neutrophil dynamics after a chemotherapy treatment with various G-CSF support protocols (the majority of the white blood cells are neutrophils. G- CSF is a factor that may be injected to help patients produce more neutrophils). The resulting formulas are capable of better tailoring G-CSF treatments for a specific patient in a robust manner and are based upon easily obtained clinical prognostic variables. Our results clarify and revise the current American Society of Clinical Oncology recommendations for G-CSF administration in neutropenia (dangerous drop in the neutrophil's level) following intensive chemotherapy regimens: a) We identify a group of patients that need new treatment strategies - we clearly show that G-CSF alone will not cure the neutropenia in such patients. b) We identify a second group of patients for which a change in the current standard protocol, to a protocol which is clinically available, may be critical for the success of the treatment of neutropenia by G-CSF injections.
In view of this potential curative effect, we propose to compare between the standard and the sustained G-CSF regimens within a framework of a prospective randomized clinical trial.

One approach to investigate the structure of an algebraic variety X is to study the geometry of curves, especially the rational curves, that X contains. This approach relies on classical geometric ideas and strives to understand the intrinsic geometry of varieties. It is nowadays understood that if X contains many rational curves, then their geometry determines X to a large degree.

After Shigefumi Mori showed in his landmark works that many interesting varieties contain rational curves, their systematic study became a standard tool in algebraic geometry. The spectrum of application is diverse and covers long-standing problems such as deformation rigidity, stability of the tangent bundle, classification problems, and generalizations of the Shafarevich hyperbolicity conjecture.

The talk concentrates on examples and basic properties of minimal degree rational curves on projective varieties. Some of the more advanced applications will be discussed.

In this talk, I will introduce two spatial problems in theoretical ecology together with their mathematical solutions.

The first part of the talk concerns competition between plants for sunlight. In it, I use a mechanistic Kolmogorov-type competition model to connect plant population vertical leaf profiles (or VLPs) to the asymptotic behavior of the resulting dynamical system. For different VLPs, conditions can be obtained for either competitive exclusion to occur or stable coexistence at one or more equilibrium points.

The second part of the talk concerns the spatial spread of infectious diseases. Here, I use a family of SI-type models to examine the ability of a disease, such as rabies, to invade or persist in a spatially heterogeneous habitat. I will discuss properties of the disease-free equilibrium and the behavior of the endemic equilibrium as the mobility of healthy individuals becomes very small relative to that of infecteds. The family of disease models consists variously of systems of difference equations (which I will emphasize), ODEs, and reaction-diffusion equations.

Using the Landau-Brazovskii model, a new numerical implementation is developed to investigate the phase behavior of the diblock copolymer system. Though the method is based on the Fourier expansion of order parameter, a priori symmetric information is not required, and more significantly, the period structure can be adjusted automatically during the iteration as well. The method enables us to calculate the phase diagram, discover new meta-stable phases, validate the epitaxial relation in the phase transition process, and find the inefficiency of the Landau-Brazovskii model for some situations.

We will also introduce a new numerical method to study the nucleation in ordered phases. Nucleation is the decay of a metastable state via the thermally activated formation and subsequent growth of droplets of the equilibrium phase. We will consider the nucleation in diblock copolymer melts, whose equilibrium phases are well understood. We apply a new numerical method, called the string method, to compute the minimum energy path (MEP). Then from the MEP, we find the size and shape of the critical droplet and the free-energy barrier to nucleation. This method is generally useful for other systems, such as nucleation of liquid crystal based on Landau-de Gennes theory and binary alloy system by phase field method etc.