The question of global regularity remains open for many fundamental models of fluid dynamics.  In two dimensions, solutions to the incompressible Euler equations have been known to be globally regular since the 1930s, although their derivatives can grow double-exponentially with time.  On the other hand, this question has not yet been resolved for the more singular surface quasi-geostrophic (SQG) equation, which is used in atmospheric models.  The latter state of affairs is also true for the modified SQG equations, a family of PDE which interpolate between these two models.

I will present two results about the patch dynamics version of these equations on the half-plane.  The first is global-in-time regularity for the Euler patch model, even if the patches initially touch the boundary of the half-plane.  The second is local-in-time regularity for those modified SQG patch equations that are only slightly more singular than Euler, but also existence of their solutions which blow up in finite time. The latter appears to be the first rigorous proof of finite time blow-up in this type of fluid dynamics models.

Mathematics plays a central role in many recent technological advances. The speaker will describe his experience at a math institute that promotes connections between math and other disciplines. The impact of these interdisciplinary interactions will be demonstrated in three examples: Compression of very large datasets for medical imaging; machine learning as a tool for finding new materials for batteries; and mathematical modeling and computer simulation that enable predictive policing.

In this presentation, a class of high-order weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving nonlin- ear hyperbolic conservation law systems is presented. The construction of HWENO schemes is based on a finite volume formulation, Hermite interpolation, and nonlinearly stable Runge- Kutta methods. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first derivative values are evolved in time and used in the reconstruction, while only the function values are evolved and used in the original WENO schemes. Comparing with the original WENO schemes of Liu et al. [J. Comput. Phys. 115 (1994) 200] and Jiang and Shu [J. Comput. Phys. 126 (1996) 202], one major advantage of HWENO schemes is its compactness in the reconstruction. For example, five points are needed in the stencil for a fifth-order WENO (WENO5) reconstruction, while only three points are needed for a fifth-order HWENO (HWENO5) reconstruction in one dimensional case. Numerical results are presented for both one and two dimensional cases to show the efficiency of the schemes. 

We will talk about a paper by A. Moreira and J.C. Yoccoz, where they proved a conjecture by Palis according to which the arithmetic sums of generic pairs of regular Cantor sets on the line either has zero Lebesgue measure or contains an interval.

In this expository talk we will discuss aspects of spectral theory of the complex Laplacian, revolving around
the notion of positivity.  We will discuss geometric/potential theoretic characterizations
for positivity of the complex Neumann Laplacian and explain some applications of the theory in complex geometry.