Title: Observability for Evolution Equations on Simplices
Abstract: In this talk, I will present some observability results for wave, Schrodinger, and heat equations on Euclidean simplices. A simplex has Lipschitz boundary, so standard results for smooth domains do not apply. However, the affine nature of a simplex allows the use of simple integration by parts arguments to prove asymptotic boundary observability for the wave equation from any face of the simplex. Similar results hold for the Schrodinger equation. Using a transmutation method, the observability for the wave equation provides an estimate on the cost of observability for solutions to the heat equation. This is joint work with subsets of Evan Stafford, Ziqing Lu, Sarah Carpenter, and Ameer Qaqish.

Title: Sharp critical thresholds for a class of nonlocal traffic flow models
Abstract: I will introduce a class of macroscopic traffic flow models with nonlocal look-ahead inter- actions. The global regularity of solutions depend on the initial data. I will then present sharp critical threshold conditions that distinguish the initial data into a trichotomy: Subcritical initial conditions lead to global smooth solutions, while two types of supercritical initial conditions lead to two kinds of finite time shock formations. The existence of non-trivial subcritical initial data indicates that the nonlocal look-ahead interactions can help avoid shock formations, and hence prevent the creation of traffic jams. This is joint work with my advisor, Dr. Changhui Tan Key words: Macroscopic traffic flow, partial differential equations, nonlocal conservation law, continuity equations, phase plane analysis, fluid dynamics, critical threshold

Title: Metric entropy for scalar conservation laws
Abstract: Inspired by a question posed by Lax in 2002, the study of metric entropy for nonlinear partial differential equations has received increasing attention in recent years. This talk demonstrates methods to obtain sharp upper and lower bounds on the metric entropy for a class of bounded total generalized variation functions taking values in a general totally bounded metric space. Thereafter we use this result to establish metric entropy estimates for the set of entropy admissible weak solutions to a scalar conservation law with weakly genuinely nonlinear flux. Estimates of this type could provide a measure of the order of resolution of a numerical method required to solve the equation.

Title: A quantitative analysis on total number of shocks for scalar conservation laws
Abstract: In this talk, I will present a quantitative version of the transversality theorem. More precisely, given a continuous function \(g \in C([0,1]^d,\mathbb{R}^m)\) and a smooth manifold \(W \subset \mathbb{R}^m\) of dimension p, one establishes a quantitative estimate on the (d + p − m)-dimensional Hausdorff measure of the set \(Z^g_W ={x \in [0,1]^d : g(x) \in W}\) over \(\varepsilon\)-perturbations in \(C^0\). The obtained result is applied to bound the total number of shock curves in weak entropy solutions to scalar conservation laws with uniformly convex fluxes in one space dimension.