The classical Cauchy integral is a fundamental object of complex analysis whose analytic properties are intimately related to the geometric properties of its supporting curve.

In this talk I will begin by reviewing the most relevant features of the classical Cauchy integral. I will then move on to the (surprisingly more involved) construction of the Cauchy integral for a hypersurface in
$\mathbb C^n$.

I will conclude by presenting new results joint with E. M. Stein concerning the regularity properties of this integral and their relations with the geometry of the hypersurface.

(Time permitting) I will discuss applications of these results to the Szeg\H o and Bergman projections (that is, the orthogonal projections of the Lebesgue space $L^2$ onto, respectively, the Hardy and Bergman spaces of holomorphic functions).

In this talk we present a survey of our results on quasi-periodic Jacobi operators whose diagonal and off-diagonal elements are generated from two analytic functions on the circle. Such operators arise as effective Hamiltonians describing the effects of external magnetic fields on a tight binding, infinite crystal layer. The main motivation of our investigations was extended Harper's model (EHM), whose description on both the level of spectral analysis, as well the Lyapunov exponent (LE) had posed an open problem even from the point of view of physics literature. Among the topics that will be addressed are: Singular components of spectral measures for ergodic Jacobi operators, Singular analytic cocycles and joint continuity of the Lyapunov exponent, Recovery of spectral data from rational frequency approximants, Almost constant cocycles and the complexified LE of EHM, Spectral theory of EHM.

A systematic understanding of traffic dynamics on road networks is crucial for many transportation studies and can help to develop more efficient ramp metering, evacuation, signal control, and other management and control strategies. In this study, we present a theory of multi-commodity network traffic flow based on the Lighthill-Whitham-Richards (LWR) model. In particular, we attempt to analyze kinematic waves of the Riemann problem for a general junction with multiple upstream and downstream links. In this theory, kinematic waves on a link can be determined by its initial condition and prevailing stationary state. In addition to stationary states, a flimsy interior state can develop next to the junction on a link. In order to pick out unique, physical solutions, we introduce two types of entropy conditions in supply-demand space such that (i) speeds of kinematic waves should be negative on upstream links and positive on downstream links, and (ii) fair merging and First-In-First-Out diverging rules are used to prescribe fluxes from interior states. We prove that, for given initial upstream demands, turning proportions, and downstream supplies, there exists a unique critical demand level satisfying the entropy conditions. It follows that stationary states and kinematic waves on all links exist and are unique, since they are uniquely determined by the critical demand level. For a simple model of urban or freeway intersections with four upstream and four downstream links, we demonstrate that theoretical solutions are consistent with numerical ones from a multicommodity Cell Transmission Model. In a sense, the proposed theory can be considered as the continuous version of the multi-commodity Cell Transmission Model with fair merging and First-In-First-Out diverging rules. Finally we discuss future research topics along this line.