Over the past almost three decades dynamical systems have played a central role in spectral analysis of quasiperiodic Hamiltonians as well as certain quasiperiodic models in statistical mechanics (most notably: the Ising model, both quantum and classical). There are many ways of introducing quasiperiodicity into a model. We shall concentrate on the widely studied Fibonacci case (which is a prototypical example of so-called substitution systems on two letters with certain desirable properties). In this case a particular geometric scheme, arising from a certain smooth three-dimensional dynamical system associated to the quasiperiodic model in question (the so-called Fibonacci trace map) has been established. Our aim is to present a general dynamical/geometric framework and to demonstrate how information about the model in question (spectral properties for Hamiltonians, and Lee-Yang zeros distribution for classical Ising models) can be obtained from the aforementioned dynamical system and the geometry of certain dynamically invariant sets. In this first in a series of two (or three) talks, we'll briefly recall how dynamical systems are associated to Schroedinger and Jacobi operators, as well as classical Ising models. We'll establish notation, ask main questions and in general prepare the ground for a somewhat more general (in terms of geometry and dynamical systems) discussion for next time.

We complete the construction of the Martin-Solovay tree

Continuous logic is a relatively new logic better equipped for studying the model theory of structures based on complete metric spaces. There are continuous analogs of virtually every notion and theorem from classical model theory, often with equalities replaced by approximations. However, most of the work done in continuous logic has centered around sophisticated topics concerning stability and its generalizations. In this talk, I will discuss the more basic notion of definability in metric structures. More specifically, I will consider the question of which functions are definable in Urysohn's metric space. Urysohn's metric space is the unique (up to isometry) Polish space that is universal and ultrahomogeneous. In many ways, Urysohn's metric space is to continuous logic as the the infinite set is to classical logic. However, we will see that the task of understanding the definable functions in Urysohn's metric space involves some interesting topological considerations.

Differential and Integral Equations are powerful tools to model and analyze biological problems. In this talk, two different biological applications will be discussed: one is in biomedical images and the other is in cellular biology.

The basic medical science research and clinical diagnosis and treatment have strongly benefited from the development of various noninvasive biomedical imaging techniques and modeling, e.g. magnetic resonance imaging (MRI) and computed tomography (CT). We introduce integro-differential models to the morphology and connectome study of human brains from brain images, as well as the shape analysis of ciliary muscles from human eyes.

In the application of cellular biology, we investigate the cell differentiation model of T cells. T cells of the immune system, upon maturation, differentiate into either Th1 or Th2 cells that have different functions. The decision to which cell type to differentiate depends on the concentrations of transcription factors T-bet and GATA-3. We study a population density model of the T cells and show that, under some conditions on the parameters of the system of integro-differential equations, various T cells differentiation scenarios occur.