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.

Serre proved in 1972 that the image of the adelic Galois representation
associated to an elliptic curve E without complex multiplication has open
image; moreover, he also proved that for an elliptic curve over Q the
index of the image is always divisible by 2 (and in particular never
surjective). More recently, Greicius in his thesis gave criteria for
surjectivity and gave an explicit example of an elliptic curve E over a
number field K with surjective adelic representation. Soon after, Zywina,
building on earlier work of Duke, Jones, and others, proved that the
adelic image `random' elliptic curve is maximal.

In this talk I will explain recent joint work with David Zywina in which
we generalize these theorems and prove that a random abelian variety in a
family with big monodromy has maximal image of Galois. I'll explain what
big monodromy and maximal mean an explain the analytic and geometric
techniques used in previous work and the new geometric ideas -- in
particular, Nori's method of semistable approximation-- needed to
generalized to higher dimension.