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Invited Paper Session Abstracts - Frontiers in Differential Equations and Applications

MAA Invited Paper Session

Frontiers in Differential Equations and Applications

Please note: all sessions are listed in Eastern Daylight Time (EDT = UTC-4:00)

Friday, August 4, 8:00 a.m. - 10:20 a.m., Room 118/119

This session seeks to explore problems and techniques at the interface of differential equations and applied mathematics. Topics of interest include multiscale analysis, inverse problems, numerical methods, and spectral analysis, with applications in materials science, fluid dynamics, mathematical biology, and medical imaging. Our main goal is to showcase new research in applications of differential equations and to promote collaboration between established and early- career mathematicians.

Robert Viator Jr., Swarthmore College
Chee Han Tan, Wake Forest University

Maximal Total Population of Species in a Diffusive Logistic Model

8:00 a.m. - 8:20 a.m.
Chiu-Yen Kao, Claremont Mckenna College

We investigate the maximization of the total population of a single species which is governed by a stationary diffusive logistic equation with a fixed amount of resources. For large diffusivity, qualitative properties of the maximizers like symmetry will be addressed. Our results are in line with previous findings which assert that for large diffusion, concentrated resources are favorable for maximizing the total population. Then, an optimality condition for the maximizer is derived based upon rearrangement theory. We develop an efficient numerical algorithm applicable to domains with different geometries in order to compute the maximizer. It is established that the algorithm is convergent. Our numerical simulations give a real insight into the qualitative properties of the maximizer and also lead us to some conjectures about the maximizer.

Modeling Microtubule Assembly and Polarity in Neurons

8:30 a.m. - 8:50 a.m.
Anna C. Nelson, Duke University

The microtubule cytoskeleton is responsible for sustained, long-range intracellular transport of cellular cargo in neurons. However, microtubules must also be dynamic and rearrange their orientation, or polarity, in response to injuries. While mechanisms that control the minus-end out microtubule orientation in Drosophila dendrites have been identified experimentally, it is unknown how these mechanisms maintain both dynamic rearrangement and sustained, long-term function. To better understand these mechanisms, we introduce a spatially-explicit mathematical model of dendritic microtubule growth dynamics using parameters informed by experimental data. We explore several hypotheses of microtubule growth using both a stochastic model and a continuous ordinary differential equation model. Using fluorescence microscopy experimental data, we validate mechanisms such as limited tubulin availability and catastrophe events that depend on microtubule length. By incorporating biological data, our modeling framework can uncover the impact of various mechanisms on the collective dynamics and polarity of microtubules in Drosophila dendrites.

Homogenization of Nonlinear Deformable Dielectrics

9:00 a.m. - 9:20 a.m.
Thuyen Dang, University of Chicago

In this talk, I will present the rigorous periodic homogenization for a weakly coupled electro-elastic system of a nonlinear electrostatic equation with an elastic equation enriched with electrostriction. Such coupling is employed to describe deformable (elastic) dielectrics. It is shown that the effective response of the system consists of a homogeneous deformable dielectric described by a nonlinear coupled system of PDEs whose coefficients depend on the coefficients of the original heterogeneous material and geometry of the composite and periodicity of the original microstructure. A classical corrector result for the homogenization of monotone operators is improved, and two Lp–gradient estimates for elastic systems with discontinuous coefficients are also shown.

Computation of Free Boundary Minimal Surfaces via Extremal Steklov Eigenvalue

9:30 a.m. - 9:50 a.m.
Braxton Osting, University of Utah

Recently A. Fraser and R. Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e., a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally. In this talk, I’ll discuss recent work on numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, Σ, with genus γ and b boundary components, we maximize σj (Σ, g) L(∂Σ, g) over a class of smooth metrics, g, where σj(Σ,g) is the j-th nonzero Steklov eigenvalue and L(∂Σ,g) is the length of ∂Σ. Our numerical method involves (i) using conformal uniformization of multiply connected domains to avoid explicit parameterization for the class of metrics, (ii) accurately solving a boundary-weighted Steklov eigenvalue problem in multi-connected domains, and (iii) developing gradient-based optimization methods for this non-smooth eigenvalue optimization problem. The corresponding eigenfunctions generate a free boundary minimal surface, which we describe. This is joint work with Chiu-Yen Kao and Èdouard Oudet.

Analyticity of Steklov Eigenvalues of Nearly-Hyperspherical Domains in Rd+1

10:00 a.m. - 10:20 a.m.
Chee Han Tan, Wake Forest University

We consider the Dirichlet-to-Neumann operator (DNO) on nearly-hyperspherical domains in dimension greater than 3. Treating such domains as perturbations of the ball, we prove the analytic dependence of the DNO on the shape perturbation pa- rameter for fixed perturbation functions. Consequently, we conclude that the Steklov eigenvalues are analytic in the shape perturbation parameter as well. This is joint work with Robert Viator.