# Applied Mathematics and Analysis Seminar

As part of the Duke University Department of Mathematics, the Program in Applied Mathematics hosts this ongoing series of seminars. The presentations cover a broad range of topics including numerical analysis, ordinary and partial differential equations, nonlinear systems, scientific computing, dynamical systems theory, mathematical biology, pattern formation, and complex physical systems.

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As a convenience, some selected seminars and presentations can be viewed live via the web. Further, we have video archives of past talks, which are also publicly available for you to view at any time.

Upcoming Seminars:
• Tuesday, November 3, 2020, 3:15pm, Physics 119, Applied Math And Analysis Seminar
Uniqueness of the 2D incompressible Euler equation on corner domains
Siddhant Agrawal (U. Mass Amherst)

We consider the 2D incompressible Euler equation on a corner domain with an angle between π/2 and π. In this setup, the uniqueness of solutions in the Yudovich class is not known in general due to the fact that the velocity is very far from being Lipschitz. In this work we prove that if the initial vorticity is non-negative and supported on one side of the angle bisector of the domain, then solutions in the Yudovich class are unique. This is the first result which proves uniqueness when the velocity is far from Lipschitz and the initial vorticity is nontrivial around the boundary. This is joint work with Andrea Nahmod.

• Tuesday, November 10, 2020, 3:15pm, Physics 119, Applied Math And Analysis Seminar
TBA
Dominique Maldague (MIT)

• Tuesday, November 17, 2020, 3:15pm, Physics 119, Applied Math And Analysis Seminar
Fractional diffusion limit of a linear kinetic transport equation in a bounded domain
Pedro Aceves-Sanchez (NC State)

In recent years, the study of evolution equations featuring a fractional Laplacian has received much attention due to the fact that they have been successfully applied into the modelling of a wide variety of phenomena, ranging from biology, physics to finance. The stochastic process behind fractional operators is linked, in the whole space, to an $\alpha$-stable processes as opposed to the Laplacian operator which is linked to a Brownian stochastic process. In addition, evolution equations involving fractional Laplacians offer new interesting and very challenging mathematical problems. There are several equivalent definitions of the fractional Laplacian in the whole domain. However, in a bounded domain there are several options depending on the stochastic process considered. In this talk we shall present results on the rigorous passage from a velocity jumping stochastic process in a bounded domain to a macroscopic evolution equation featuring a fractional Laplace operator. More precisely, we shall consider the long-time/small mean-free path asymptotic behaviour of the solutions of a re-scaled linear kinetic transport equation in a smooth bounded domain.

• Tuesday, November 24, 2020, 3:15pm, Zoom link, Applied Math And Analysis Seminar
TBA
Yimin Zhong (Duke University)

• Tuesday, December 1, 2020, 3:15pm, Zoom link, Applied Math And Analysis Seminar
Localized edge modes in subwavelength resonator arrays
Erik Hiltunen (ETH)

A topological insulator is a material which is insulating in the bulk but conducting along its edge. Intriguingly, the existence of conducting edge modes stems not from the structure of the edge, but from the insulating bulk. The goal of this talk is to generalize these concepts to the setting of high-contrast subwavelength resonators. The main mathematical challenges comes from the discontinuous material parameters, which we solve using layer potential techniques. We set out by computing the well-studied topological number called the Zak phase of the material, thereby proving that it can support a topologically nontrivial frequency band gap. We then perturb the periodic structure by introducing a dislocation. This way, mid-gap frequencies (corresponding to edge modes), enter the band gap and converge to a single frequency as the dislocation becomes arbitrarily large. Finally, we show that edge modes can alternatively appear by introducing gain and loss, without altering the periodic geometry.

All seminars take place on Mondays at 4:30 pm in Room 119 Physics Building unless otherwise noted.

Tea and refreshments are served before the seminars at 4:00 pm in Physics 101.

Related Seminars

Past speakers in the Duke Applied Mathematics seminars (1997+)