Georgia Institute of Technology College of Sciences

We study a class of linear delay-differential equations, with a singledelay, of the form$$\dot x(t) = -a(t) x(t-1).\eqno(*)$$Such equations occur as linearizations of the nonlinear delay equation$\dot x(t) = -f(x(t-1))$ around certain solutions (often around periodicsolutions), and are key for understanding the stability of such solutions.Such nonlinear equations occur in a variety of scientific models, anddespite their simple appearance, can lead to a rather difficultmathematical analysis.We develop an associated linear theory to equation (*) by taking the$m$-fold wedge product (in the infinite dimensional sense of tensorproducts) of the dynamical system generated by (*). Remarkably, in the caseof a ``signed feedback'' where $(-1)^m a(t) > 0$ for some integer $m$, theassociated linear system is given by an operator which is positive withrespect to a certain cone in a Banach space. This leads to very detailedinformation about stability properties of (*), in particular, informationabout characteristic multipliers.

They may flow like fluids but under constraints of mechanical energies from their crystal aspects. As a result, they exhibit very rich phenomena that grant them tremendous applications in modern technology. Based on works of Oseen, Z\"ocher, Frank and others, a continuum theory (not most general but satisfactory to a great extent) for liquid-crystals was formulated by Ericksen and Leslie in 1960s. We will first give a brief introduction to this classical theory and then focus on various important special settings in both static and dynamic cases. These special flows are rather simple for classical fluids but are quite nonlinear for liquid-crystals. We are able to apply abstract theory of nonlinear dynamical systems upon revealing specific structures of the problems at hands.

An optimal transport path may be viewed as a geodesic in the space of probability measures under a suitable family of metrics. This geodesic may exhibit a tree-shaped branching structure in many applications such as trees, blood vessels, draining and irrigation systems. Here, we extend the study of ramified optimal transportation between probability measures from Euclidean spaces to a geodesic metric space. We investigate the existence as well as the behavior of optimal transport paths under various properties of the metric such as completeness, doubling, or curvature upper boundedness. We also introduce the transport dimension of a probability measure on a complete geodesic metric space, and show that the transport dimension of a probability measure is bounded above by the Minkowski dimension and below by the Hausdorff dimension of the measure. Moreover, we introduce a metric, called "the dimensional distance", on the space of probability measures. This metric gives a geometric meaning to the transport dimension: with respect to this metric, the transport dimension of a probability measure equals to the distance from it to any finite atomic probability measure.