Georgia Institute of Technology College of Sciences

Inspired by the interval decomposition of persistence modules and the extended Newick format of phylogenetic networks, we show that, inside the larger category of partially ordered Reeb graphs, every Reeb graph with n leaves and first Betti number s, is equal to a coproduct of at most 2s trees with (n + s) leaves. An implication of this result, is that Reeb graphs are fixed parameter tractable when the parameter is the first Betti number. We propose partially ordered Reeb graphs as a natural framework for modeling time consistent phylogenetic networks.  We define a notion of interleaving distance on partially ordered Reeb graphs which is analogous to the notion of interleaving distance for ordinary Reeb graphs. This suggests using the interleaving distance as a novel metric for time consistent phylogenetic networks.

This presentation reviews different concepts of solution of a differential equation, in particular stressing the need to modify the classical theory when we want to deal with discontinuous systems.  We will review the concept of classical solution, and then of Caratheodory solution and Filippov solution, motivating with simple examples the need for these extensions.

We present a multiscale modeling and computational scheme for optical-
mechanical responses of nanostructures. The multi-physical nature of
the problem is a result of the interaction between the electromagnetic
(EM) field, the molecular motion, and the electronic excitation. To
balance accuracy and complexity, we adopt the semi-classical approach
that the EM field is described classically by the Maxwell equations,
and the charged particles follow the Schr ̈oidnger equations quantum
mechanically. To overcome the numerical challenge of solving the high
dimensional multi-component many- body Schr ̈odinger equations, we
further simplify the model with the Ehrenfest molecular dynamics to
determine the motion of the nuclei, and use the Time- Dependent
Current Density Functional Theory (TD-CDFT) to calculate the
excitation of the electrons. This leads to a system of coupled
equations that computes the electromagnetic field, the nuclear
positions, and the electronic current and charge densities
simultaneously. In the regime of linear responses, the resonant
frequencies initiating the out-of-equilibrium optical-mechanical
responses can be formulated as an eigenvalue problem. A
self-consistent multiscale method is designed to deal with the well
separated space scales. The isomerization of Azobenzene is presented as a numerical example.