Mathematics Colloquium
Mario Gomez, John Hoffman and SeongHee Jeong
FSU
Titles:
1. Vietoris-Rips complexes of totally split-decomposable spaces
2. Square Functions Controlling Smoothness
3. Optimal control problem for coupled stationary flow and transport equations
Date: Friday, November 1, 2024
Place and Time: Room 101, Love Building, 3:05-3:55 pm
Abstracts. 1. Split-metric decompositions are an important tool in the theory of phylogenetics, particularly because of the link between the tight span and the class of totally decomposable spaces, i.e. a generalization of metric trees whose decomposition does not have a “prime” component. The connection with tight spans has been studied at least since the introduction of split-metric decompositions by Bandelt and Dress in 1992 and culminated with the characterization of the polytopal structure of the tight span of a totally decomposable metric by Huber, Koolen, and Moulton in 2018. We use this connection, along with recent results on the Vietoris-Rips complex of the circle and the connection between tight spans and Vietoris-Rips complexes, to characterize the homotopy type of the Vietoris-Rips complex of a large class of totally decomposable spaces.
2. We characterize the space $I_{\alpha}(BMO)$, the image of $BMO$ under the Riesz potential of order $\alpha$, for $0<\alpha<2$, in terms of Carleson measures which measure multiscale approximation by constants (in the case $0<\alpha<1$) or linear functions (in the case $1 \leq \alpha <2$). We discuss the relevance of these characterizations to higher-order rectifiability.
3. This work presents an optimal control problem constrained by coupled flow and transport equations. We explore two different cases: one with no pointwise constraints and the other with pointwise control constraints. The optimality conditions and convergence analysis are obtained for both cases, and they are numerically solved using a finite element method. The primal-dual active set algorithm is used for the pointwise control constraint case. Numerical examples are provided to validate the theoretical results.