## Applied Mathematics Seminar

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

Spring  2018

Title:  Existence of positive solutions with a prescribed singular set for fractional Yamabe Problem
Speaker: Weiwei Ao (Wuhan University, China)
Date and Time: Thursday, April 5, 3:00pm-4:00pm
Place: Funger 221?

Abstract. We consider the problem of the existence of positive solutions with prescribed singularity set of the fractional Yamabe problem. The singularity set can be isolated points or higher dimensional manifold with dimension less than $\frac{n-2\gamma}{2}$. This result is the analogous result for the classical Yamabe problem studied by Mazzeo and Pacard.

Title:  Sobolev regularity for first order Mean Field Games
Speaker: Jameson Graber (Baylor University)
Date and Time: Monday, March 26, 5:00pm-6:00pm
Place: Monroe 113

Abstract. In their seminal 2007 paper, Lasry and Lions proposed a system of PDE to model games with a continuum of infinitesimal interacting agents. Since then many authors have proved several results on the well-posedness and regularity of solutions, mostly employing techniques from the theory of parabolic PDE. More recently, a satisfactory definition of weak solution has been provided which allows us to study weak solutions of degenerate and first-order systems, for which the question of regularity can no longer be addressed using classical methods. In this talk, based on recent joint work with Alpar Meszaros, I will show how methods from the calculus of variations can be used to prove additional regularity of weak solutions to first-order mean field games. Additionally I will illustrate some connections to other work in optimal transport theory.

Title:  Signal transmission properties of unidirectional chains of phase oscillators.
Speaker: Stan Mintchev
Affiliation: National Science Foundation
Date and Time: Friday, April 6, 11:00am-12:00pm
Place: Phillips 640

Abstract. Phase oscillator ensembles exhibiting a preferred direction of coupling can mimic a spatially distributed communication line with effective mesoscale properties. In mathematical neuroscience, such networks provide a framework for understanding the generation and propagation of electrical impulses across brain tissues. Motivated by such applications, we consider signal transmission problems in this idealized setting. We focus on generation and stability of traveling wave solutions (TW) in feedforward chains of idealized neural oscillators featuring a pulse emission / Type-I phase response (PRC) interaction. Our prior investigations examined a smooth version of this model by way of a combination of numerical and analytical techniques: an iterative fixed point scheme was used to verify existence of TW, and these findings motivated an abstract hypothesis that turns out sufficient to establish global stability of the solution. We have since completed a full mathematically-rigorous study of a piecewise affine version of this model, establishing all of the observed phenomenology (both existence and stability of TW) analytically. In addition, we now have an understanding of how a robust TW solution in this setting may be generated with a variety of external forcing stimuli, including such that are structurally distinct from the consequent wave.

Title: Traveling Wave of Gray-Scott system, Existence, Multiplicity and Stability
Speaker: Yuanwei Qi
Affiliation: University of Central Florida
Date and Time: Friday, March 9, 3:00pm-4pm
Place: Rome Hall #204

Abstract. In this talk, I shall present some recent results on the existence and structure of  Traveling Wave solutions to an important model in Turing Pattern Formation: Gray-Scott System of Auto-Catalysis with and without feeding.

Title: The fractional Yamabe problem and the prescribed scalar curvature problem
Speaker: Seunghyeok Kim (Hanyang University, South Korea)
Date and time: Tuesday, January 23, 4-5pm
Place: Monroe 113

Abstract: In this talk, we consider two relatively new problems in conformal geometry, namely,  the fractional Yamabe problem and the prescribed fractional scalar curvature problem. They are geometric questions which concern the existence of a metric in a given conformal class  whose associated fractional scalar curvature is constant or a prescribed function. By conformal covariance of the fractional conformal Laplacian, they are reduced to finding positive solutions to nonlocal elliptic equations having nonlinear terms of critical growth. I will first define some geometric objects for which the problems make sense,  and then provide several geometric conditions which ensure the existence of a desired metric. The results for the fractional Yamabe problem were obtained through a collaboration with Monica Musso (PUC, Chile) and Juncheng Wei (UBC, Canada).

Archive