AARMS-PIMS Summer School in
Differential Equations and Numerical Analysis
July 6 - 31st, 2015, Dalhousie University, Halifax

All classes take place in DUNN building, room 117. Any computer labs will be held in DUNN room 301. Class schedule:
8:30: course 4 (Stynes)
10:30: course 1 (Carretero/Bertozzi)
12:30-1:30: lunch break
1:30: course 2 (Ward/Wei)
3:30: course 3 (Celledoni/Owren)

Students are housed at Howe Hall (Dalhousie residence). From the airport, take shuttle or taxi or bus there. Please see here (public facebook page) for schedule, arrival/housing information, and any other questions. I will be using that page in the future for further information such as social events etc.

This summer school is intended for graduate and senior undergraduate students, as well as young researchers. Each participant is expected to register for two of the four courses. Each course runs for four weeks and consists of five ninety-minute lectures per week. These are graduate courses approved by Dalhousie and we will facilitate transfer credit to the extent possible.

Please click here to apply. The courses are free for students and we will provide local accomodations. We have student residences reserved for students outside of Halifax attending the summer school.

In addition to the summer school, AARMS is organizing three associated workshops in differential equations that the school participans are welcome to attend:

Organizers: Theodore Kolokolnikov and Hermann Brunner.

Summer school posters: black-and white version | color version

Course descriptions
Course 1: Waves and patterns in nonlinear systems
Instructors: Andrea Bertozzi and Ricardo Carrettero
This topics course on waves and pattern formation covers a variety of settings. Topics include:

Course 2: Topics in Reaction-Diffusion Systems: Theory and Applications
Instructors: Michael Ward and Juncheng Wei
This course is intended to expose students to analytical aspects for the study of spatial-temporal patterns in reaction-diffusion (RD) systems. This course consists of two parts: In Part I, we briefly discuss Turing stability analysis, weakly nonlinear theory, and bifurcations of spatially homogeneous solutions to RD systems. For various specific systems with an extreme diffusivity ratio of the diffusing species, we then illustrate and analyze asymptotically different classes of localized patterns, including transition layers and spot patterns. The goal is to develop a facility in the use of systematic asymptotic and singular perturbation methodologies for analyzing various types of localized patterns. Part II is devoted to a rigorous treatment of localized solutions. First we will introduce the method of finite dimensional Liapunov-Schmidt method for scalar equations and then we will apply it to construct various localized patterns for reaction-diffusion systems (spikes, layers, etc). Next we study the stability of these localized patterns. For large eigenvalues we introduce the method of Nonlocal Eigenvalue Problem (NLEP). For small eigenvalues, we return to the method of finite dimensional reduction. Finally if time permits we will introduce the infinite dimensional reduction method.

Course 3: Structure-preserving discretization of differential equations
Instructors: Elena Celledoni and Brynjulf Owren
Since the early 1990s the study of structure-preserving discretization and geometric integration have emerged as important new fields in the numerical approximation of ordinary and (time-dependent) partial differential equations. Here, concepts like Lie group integrators play a key role in the design of time integrators for PDEs that preserve, e.g., first integrals. This course, presented by two of the leading contributors to the field, will provide a thorough introduction to the basic concepts underlying structure-preserving discretization and then lead the students to the present state of the art. (A representative reference: S. Christiansen, H.Z. Munthe-Kaas and B. Owren, Topics in structure-preserving discretization, Acta Numerica 20 (2011), 1-119.)

Course 4: Numerical analysis of singularly perturbed ODEs and PDES
Instructor: Martin Stynes
Boundary-value problems for singularly perturbed ordinary differential equations and partial differential equations of convection-diffusion type, and their time-dependent versions arise in many physical applications. Since their solutions usually possess sharp boundary (or interior) layers, one is interested in designing robust numerical schemes whose convergence properties are independent of the singular perturbation parameter 0 < epsilon << 1. Here, layer-adapted (spatial or temporal) meshes (for example Shishkin meshes) usually form a key part of such methods. This course is presented by one of the world leaders in this field; it will provide a thorough introduction to the numerical analysis of singularly perturbed differential equations and to its current state of the art. (A representative reference: G.-G. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly perturbed Differential Equations (2nd edition), Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 2008.)

Teacher biographies

Andrea Bertozzi is Professor of Mathematics at UCLA and currently serves as Director of the Program in Computational and Applied Mathematics. Since 2012 she is the Betsy Wood Knapp Chair for Innovation and Creativity at UCLA. She received AB, MA, and PhD degrees in Mathematics from Princeton University. Her current research interests include image in painting, image segmentation, cooperative control of robotic vehicles, swarming, crime modeling, and fluid interfaces. Prof. Bertozzi has served as a plenary/distinguished lecturer for both SIAM and AMS and is associate editor for the SIAM journals Multiscale Modelling and Simulation and SIAM Journal on Mathematical Analysis. She also serves on the editorial boards for 8 journals. Her past honors include a Sloan Foundation Research Fellowship and the Presidential Career Award for Scientists and Engineers from the Office of Naval Research. She is a member of the American Academy of Arts and Sciences and a Fellow of the American Mathematical Society and the Society for Industrial and Applied Mathematics. In addition to serving on the MSRI board, she is Chair of the Science Board for the Institute for Computational and Experimental Research in Mathematics at Brown University and serves on the board for the Banff International Research Station.

Ricardo Carretero is a professor of Applied Mathematics at San Diego State University (SDSU). He obtained his PhD at Queen Mary University in London in 1997. He took postdoctoral positions at University College London and at Simon Fraser University before joining SDSU in 2002. His research spans numerous topics at the intersection of physics and nonlinear complex systems. His current research interests lie in the formation, stability and dynamics of localized structures in nonlinear media. Specifically, he is interested in the nonlinear evolution of solitons, vortices and vortex rings in nonlinear Schrodinger-type systems with special emphasis in understanding these nonlinear structures in Bose-Einstein condensates. He has published over 100 papers and three journal issues and, according to google scholar, he has an h-index of 30 with more than 2700 citations.

Michael Ward is a Professor of Mathematics at the University of British Columbia. His research focuses on the development and application of singular perturbation methodology to analyze partial differential equations in various applications, including localization phenomena in reaction-diffusion systems, nonlinear eigenvalue problems, narrow escape phenomena, and low Reynolds number fluid flow. He received his Ph.D in Applied Mathematics at Caltech in 1988, and was held postdoctoral fellowships as a Szego Instructor at Stanford from 1988-1991 and at the Courant Institute from 1991-1993 before joining UBC in 1993. He is the recipient of several awards including, the Andre-Aisensdat Prize awarded by the CRM (1995), a junior Killam Research Prize (1995), the Coxeter James prize awarded by the CMS (1998), a Steacie Fellowship from NSERC (1998-1999), and he was a Christensen Fellow at St. Catherines College, Oxford, in 1998. He is the recipient of the senior CAIMS Research Prize in 2011. He was an invited plenary speaker at the ICIAM in Hamburg in 1995. He is the co-editor in chief of the European Journal of Applied Math. According to Google scholar citations, his H-index is 30 and he has over 2500 citations.

Juncheng Wei holds a Tier I CRC chair in nonlinear partial differential equations in the Mathematics department at UBC. He is one of the leading international experts in nonlinear partial differential equations, concentration phenomena, and related applications, having published over 300 research papers since his PhD in 1994 from the University of Minnesota. His awards include the NSERC Discovery accelerator (2013); the Morningside medal of the International Congress of Chinese Mathematicians in 2010, a First Class Award of Natural Science 2010 as awarded by the Ministry of Education of China, a Croucher Senior Fellowship 2005-2006, and a Research excellence prize awarded from the Chinese University of Hong KongUHK in 2010. In addition, his Annals paper on the De Giorgi conjecture, joint with M. Del Pino and M. Kowalczyk, was the topic of M. Del Pino's invited lecture at the prestigious International Congress of Mathematics (ICM) in India in 2010. Wei is an invited ICM speaker for 2014. He has over 3000 citations in the Web of Science, and is included in the Thompson ISIHighlyCited.com 2010 list.

Brynjulf Owren has been a professor in the Department of Mathematical Sciences at the Norwegian University of Science and Technology (NTNU) in Trondheim since 1994. He received his PhD in Numerical Analysis from NTH in 1990 under the supervision of Prof. S.P. Norsett. His research areas include the numerical analysis of ordinary and partial differential equations, integration methods for differential equations on manifolds (geometric integration), and mathematical modelling. He is the author or co-author of some 25 papers in leading research journals, and he was president of the Norwegian Mathematical Society from 2007 to 2011.

Elena Celledoni is a professor in the Department of Mathematical Sciences at NTNU. Following the award of her PhD in Computational Mathematics from the University of Padova (Italy) she spent postdoctoral years at the University of Cambridge (1997-1998) and the University of Berkeley (1998-1999). She joined NTNU in 1999. Her research is concerned with the numerical analysis of differential equations, Lie group integrators for differential equations, Hamiltonian systems, and energy-preserving Runge-Kutta methods. Since 1997 she has published 24 papers in leading research journals.

Martin Stynes is Professor Emeritus in the Department of Mathematics at the National University of Ireland and currently a member of the Beijing Computational Science Research Center. He is also a member of Irish Research Group on Singularly Perturbed Differential Equations, and he was president of the UK/Republic of Ireland Section of SIAM (2003-2005). His research has focused on the design and the analysis of numerical methods for singularly perturbed (ordinary and partial) differential equations, and this is reflected in his publication list of more than 100 papers (including the survey article on "Steady-state convection-diffusion problems" in Acta Numerica (2005), pp. 445-508). He is also the co-author of two books (Numerical Treatment of Partial Differential Equations, Springer-Verlag, 2007, and Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer-Verlag, 2008 (2nd edition)).

Bluenose Workshop: July 11-12

Organizers: Paul Muir, Wayne Enright, and Morven Gentleman. Overview | Link to the workshop website

Workshop on Pattern Formation: July 18-19

Organizers: Theodore Kolokolnikov, Ricardo Carretero and Michael Ward. Summer school students are encouraged to attend. Click here for information.

Workshop on Domain Decomposition Methods for PDE's: Aug 4-8

Organizer: Ronald Haynes, Herman Brunner and Paul Muir. Overview | Link to the workshop website