Applied Mathematics Seminar

Department of Mathematics, Phillips Hall, 801 22nd Street NW, Washington, DC 20052.

Spring 2020

"no current seminars"

Fall 2019

Title: Periodic Minimizers of a Ternary Non-Local Isoperimetric Problem
Speaker:Chong Wang
Affiliation: McMaster University
Date and Time:Tuesday, October 8, 3:30pm-4:30pm
Place: Phillips 730

Abstract:  We study a two-dimensional ternary inhibitory system derived as a sharp-interface limit of the Nakazawa-Ohta density functional theory of triblock copolymers. This free energy functional combines an interface energy favoring micro-domain growth with a Coulomb-type long range interaction energy which prevents micro-domains from unlimited spreading. Here we consider a limit in which two species are vanishingly small, but interactions are correspondingly large to maintain a nontrivial limit. In this limit two energy levels are distinguished:  the highest order limit encodes information on the geometry of local structures as a two-component isoperimetric problem, while the second level describes the spatial distribution of components in global minimizers.  We provide a sharp rigorous derivation of the asymptotic limit, both for minimizers and in the context of Gamma-convergence.  Geometrical descriptions of limit configurations are derived; among other results, we will show that, quite unexpectedly, coexistence of single and double bubbles can arise. The main difficulties are hidden in the optimal solution of two-component isoperimetric problem: compared to binary systems, not only it lacks an explicit formula, but, more crucially, it can be neither concave nor convex on parts of its domain.

Title: Boundary operator associated to $\sigma_k$ curvature
Speaker: Yi Wang
Affiliation: Johns Hopkins University
Date and Time:Tuesday, September 24, 3:30pm-4:30pm
Place: Phillips 730

Abstract:  On a Riemannian manifold $(M, g)$, the $\sigma_k$ curvature is the $k$-th elementary symmetric function of the eigenvalues of the Schouten tensor $A_g$. It is known that the prescribing $\sigma_k$ curvature equation on a closed manifold without boundary is variational if k=1, 2 or $g$ is locally conformally flat; indeed, this problem can be studied by means of the energy $\int \sigma_k(A_g) dv_g$. We construct a natural boundary functional which, when added to this energy, yields as its critical points solutions of prescribing $\sigma_k$ curvature equations with general non-vanishing boundary data. Moreover, we prove that the new energy satisfies the Dirichlet principle. If time permits, I will also discuss applications of our methods. This is joint work with Jeffrey Case.

Spring 2019

Title: The Mathematical Structure of Diabetes on a Slow Manifold
Speaker: Dr. Joon Ha
Affiliation: NIH and GWU
Date and Time:Monday, March 4, 4:00pm-5:00pm
Place: Rome 730

Abstract: Type 2 diabetes (T2D) is a chronic disorder in glucose homeostasis, caused by both genetic and environmental factors. A common form of pathophysiology of the disease is the failure of insulin-secreting pancreatic β-cells to increase levels of insulin, demanded mainly by obesity and aging. Such increased insulin levels are utilized to maintain a normal range of blood glucose concentration. Blood glucose sharply rises at the onset of the disease, as clinically observed, suggesting that there exists a threshold in glucose homeostasis. Using a model of T2D pathogenesis, we found that the threshold is not solely determined by glucose concentration, but corresponds to the crossing of a separatrix in the plane of insulin sensitivity and beta-cell mass, which is the phase plane of the slow subsystem of the model. The product of sensitivity and mass corresponds to the disposition index (DI), which is frequently used in clinical trials to quantify diabetes risk. This finding highlights that a dynamic interaction of insulin secretion and sensitivity plays a major role in progression to diabetes. Furthermore, the threshold formed in the DI plane addresses well how both genetic and environmental factors contribute to developing diabetes. These insights will aid in applying the mathematical model of progression to diabetes to clinical studies of personalized medicine.

Title: Nonparametric inference of interaction laws in systems of agents from trajectory data
Speaker: Ming Zhong
Affiliation: Johns Hopkins University
Date and Time:Monday, February 25, 3:30pm-4:30pm
Place: Rome 730

Abstract. Inferring the laws of interaction between particles and agents in complex dynamical systems from observational data is a fundamental challenge in a wide variety of disciplines. We propose a non-parametric statistical learning approach to estimate the governing laws of distance-based interactions, with no reference or assumption about their analytical form, from data consisting trajectories of interacting agents. We demonstrate the effectiveness of our learning approach both by providing theoretical guarantees, and by testing the approach on a variety of prototypical systems in various disciplines. These systems include homogeneous and heterogeneous agents systems, ranging from particle systems in fundamental physics to agent-based systems modeling opinion dynamics under the social influence, prey-predator dynamics, flocking and swarming, and phototaxis in cell dynamics.

Title: Primal-Dual Weak Galerkin Finite Element Methods for PDEs
Speaker: Chunmei Wang
Affiliation: Texas Tech University
Date and Time:Monday, January 14, 11:00am-12:00pm(RESCHEDULED FOR TUESDAY JANUARY 15, 2019 FROM 11AM-12PM DUE TO SNOW DAY ON MONDAY)
Place: Rome 771

Abstract. Weak Galerkin (WG) finite element method is a numerical technique for PDEs where the differential operators in the variational form are reconstructed/approximated by using a framework that mimics the theory of distributions for piecewise polynomials. The usual regularity of the approximating functions is compensated by carefully-designed stabilizers. The fundamental difference between WG methods and other existing finite element methods is the use of weak derivatives and weak continuities in the design of numerical schemes based on conventional weak forms for the underlying PDE problems. Due to its great structural flexibility, WG methods are well suited to a wide class of PDEs by providing the needed stability and accuracy in approximations. The speaker will present a recent development of WG, called "Primal-Dual Weak Galerkin (PD-WG)", for problems for which the usual numerical methods are difficult to apply. The essential idea of PD-WG is to interpret the numerical solutions as a constrained minimization of some functionals with constraints that mimic the weak formulation of the PDEs by using weak derivatives. The resulting Euler-Lagrange equation offers a symmetric scheme involving both the primal variable and the dual variable (Lagrange multiplier). PD-WG method is applicable to several challenging problems for which existing methods may have difficulty in applying; these problems include the second order elliptic equations in nondivergence form, Fokker-Planck equation, and elliptic Cauchy problems. An abstract framework for PD-WG will be presented and discussed for its potential in other scientific applications.

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