We study some problems relating to polynomials evaluated either at random Gaussian or random Bernoulli inputs.  We present a structure theorem for degree-d polynomials with Gaussian inputs. In particular, if p is a given degree-d polynomial, then p can be written in terms of some bounded number of other polynomials q_1,...,q_m so that the joint probability density function of q_1(G),...,q_m(G) is close to being bounded.  This says essentially that any abnormalities in the distribution of p(G) can be explained by the way in which p decomposes into the q_i.  We then present some applications of this result.

First-passage percolation is a model of a random metric on a infinite network. It deals with a collection of points which can be reached within a given time from a fixed starting point, when the network of roads is given, but the passage times of the road are random. It was introduced back in the 60's but most of its fundamental questions are still open. In this talk, we will overview some recent advances in this model focusing on the existence, fluctuation and geometry of its geodesics. Based on joint works with M. Damron and J. Hanson.

In an inverse boundary value problem one is interesting in determining the internal properties of a medium by making measurements on the boundary of the medium. In mathematical terms, one wishes to recover the coefficients of a partial differential equation inside the medium from the knowledge of the Cauchy data of solutions on the boundary. These problems have numerous applications, ranging from medical imaging to exploration geophysics. We shall discuss some recent progress in the analysis of inverse boundary problems, starting with the celebrated Calderon problem, and point out how the methods of microlocal and harmonic analysis can be brought to bear on these problems.  In particular, inverse problems with rough coefficients and with measurements performed only on a portion of the boundary will be addressed.

The ability of eukaryotic cells to polarize is essential for their division, differentiation into distinct tissues, and migration. During polarization various polarity proteins segregate to form a distinct front and rear. To understand a mechanism for polarization we consider a simplified PDE model describing the interchange of a polarity protein,  between an active membrane-bound form and an inactive cytosolic form. An initial transient signal results in a traveling front of activation that stops at some point in the domain, representing segregation of the cell into front and back. Using phase plane methods and numerical continuation we analyze the transition from a spatially heterogeneous (pinned wave) to a spatially homogeneous steady state as the ratio of the diffusion coefficients of the two forms and the total amount of material in the domain is varied. We discover a second spatially heterogeneous solution that acts as a threshold for polarity establishment, and give biological interpretation for this phenomenon.

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.