# Applied Partial Differential Equations: Modeling, Analysis, and Computation

### Overview

The scientific focus of this CRG will be to study nonlinear partial differential equations (PDEs) with particular emphasis on problems involving pattern formation, defined in the broadest sense. Specific topics in this area include the study of reaction-diffusion patterns with applications to the biological and social sciences, the study of nonlocal PDE and ODE models of swarming or collective behavior and problems involving concentration or localization behavior including singularity formation and the interplay between geometry and PDEs. The mathematical methodologies needed to analyze problems in this area include stability and bifurcation theory, asymptotic and singular perturbation analysis, stochastic methods, dynamical systems, symmetry methods, differential geometry and PDE analysis. From a numerical viewpoint, several areas of focus that have direct applications to pattern formation problems are the development of algorithms for computing PDEs on surfaces, the design of fast multipole methods based on integral equation formulations, the development of numerical methodologies for rigorous computing and numerical methods for the computation of sharp interfaces using front tracking or level set methods. As evidenced above, there is a strong and diverse collective of researchers in Western and Atlantic Canada involved in either the analysis or computation of applied partial differential equations with relevance to pattern formation problems.

There are three primary goals of this CRG. The first goal is to create new mathematical and numerical methodologies for the analysis of pattern formation problems in diverse applications, through the fostering of new and lasting collaborations between the rather diverse membership of the CRG and with international researchers. This will be done through hosting both short term (two-day-long CRG Summits) and longer term (5 day workshop) events, and from the shared mentorship and training of postdoctoral fellows and graduate students with CRG members in Western and Atlantic (AARMS) Canada. The second main goal of this CRG is with regards to the cross-fertilization of mathematical and numerical techniques and methodologies to scientific communities outside mathematics where pattern formation problems arise in concrete applications. This outreach component of the CRG is primarily related to Theme I and is described below. The third goal of the CRG is to offer a high level training for graduate students and postdoctoral fellows in the analytical and numerical methodologies needed to study and model pattern formation problems in a variety of areas of application. This training will be done through a month-long AARMS summer school in 2015, a two-day graduate student summit in 2016, and a three-day short course in 2016 on stability theory.

### Research Interests

- Collective Dynamics in Biology and Social Sciences
- Concentration Phenomena and PDE's on Surfaces
- Stability, Bifurcation, and Rigorous Computing

### Organizers

- Thomas Hillen (U. Alberta)
- Theodore Kolokolnikov (Dalhousie)
- Steven Ruuth (SFU)
- Michael Ward (UBC)
- Juncheng Wei (UBC)

### Planned Activities:

**2015 Activities:**

- July 6- 31, 2015: AARM-PIMS Summer School in Differential Equations and Numerical Analysis - at Dalhousie University

**2016 Activities**:

- June 13- 17, 2015: Workshop on Non Local Variational PDEs - at UBC
- May 2016: Distinguished Visitor, Henri Berestycki- at UAlberta and UBC
- August 2016: Intermediate CRG Summit- at BIRS

**2017 Activities**:

- May 25- 28: Graduate Summit in Mathematical Biology and Applied PDE in Jasper, Alberta
- June 11-15: Workshop on Numerical Methods for PDEs on Surfaces in Vancouver, BC
- July: Workshop on Collective Dynamics in PDE Models of Biological, Physical and Social Interactions at Dalhousie University, NS

**CRG PostDoctorate Fellows:**

- Justin Tzou at UBC
- Ariana Bianchi at UAlberta