We will describe some recent applications of resolution of singularities methods to questions of interest in analysis. In particular, we will describe a recent local resolution of singularities algorithm of the speaker for real-analytic functions. This algorithm is elementary and self-contained, and makes extensive use of Newton polyhedra and local coordinate systems. Some applications will be given. These include applications to oscillatory integrals, asymptotic expansions for sublevel set volumes, and the determination of the supremum of the positive e for which |f|^{- e} is locally integrable. Here f denotes a real-analytic function.

Determining the long-term behavior of large biochemical models has proved to be a remarkably difficult problem. Yet these models exhibit several characteristics that might make them amenable to study under the right perspective. One possible approach (first suggested by Sontag and
Angeli) is their decomposition in terms of so-called monotone systems, which can be thought of as systems with exclusively positive feedback.

In this talk I discuss some general properties of monotone dynamical systems, including recent results regarding their generic convergence
towards an equilibrium. Then I will discuss the use of monotone systems to model biochemical behaviors such as switches and oscillations under
time delays.

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.

A result of Clemens and Griffiths says that a smooth cubic threefold can be recovered from its intermediate Jacobian. In this talk I will discuss the possible degenerations of these abelian varieties, and give a description of the compactification of the moduli space of cubic threefolds obtained in this way. The relation between this compactification and those constructed in the work of Allcock-Carlson-Toledo and Looijenga-Swierstra will also be considered, and is similar in spirit to the relation between the various compactifications of the moduli spaces of low genus curves. This is joint work with Radu Laza.