April 1 – 30, 2019 » Topological and Rigorous Computational Methods for High Dimensional Dynamics

Organizers: Jean-Philippe Lessard (McGill), Konstantin Mischaikow (Rutgers), Jan Bouwe van den Berg (VU Amsterdam)

The focus of this program is on identifying explicit dynamical structures in nonlinear systems that are high dimensional, poorly resolved, or both. In these problems, computational mathematics is often the only feasible way forward.

The first featured workshop explores the computational challenges of rigorously identifying and extracting fundamental dynamical features such as equilibria, periodic orbits, connecting orbits and invariant manifolds in infinite-dimensional dynamical systems. The second featured workshop investigates the development of computational algebraic topological tools for studying multiparameter, nonlinear systems where the nonlinearities are poorly defined.

The main aim is to identify, characterize, and predict nonlinear dynamics from high-dimensional time series data sets. Each workshop is preceded by a hands-on tutorial aimed at graduate students, postdocs and early- to mid-career mathematicians.

May 1 – 31, 2019 » Data Assimilation: Theory, Algorithms, and Applications

Organizers: Tony Humphries (McGill), Sebastian Reich (Potsdam & Reading), Andrew Stuart (Caltech)

The seamless integration of large data sets into computational models provides one of the central challenges for the mathematical sciences of the 21st century. When the computational model is based on dynamical systems and the data is time ordered, the process of combining data and models is called data assimilation. Historically, the field has been primarily developed by practitioners within the geophysical sciences; however, it has enormous potential in many more subject areas.

This month-long thematic activity is aimed at developing the underpinning mathematical theory of data assimilation, the process of combining data with dynamical systems to learn hidden states and unknown parameters. The activities will be guided and informed by applications coming from the physical, biomedical, social and cognitive sciences. Methodologies based around particle filtering, ensemble Kalman filtering, optimization and Bayesian inverse problems will underpin the program. Long-term visitors in all of these fields will be present, and a number of short-term visitors will attend around the four workshops devoted to underpinning methodologies, geophysical applications, biomedical applications and applications from the social and cognitive sciences.

October 1 – 31, 2019 » Mixed Integer Nonlinear Programming: Theory and Computation

Organizers: Andrea Lodi (Polytechnique Montréal), Bruce Shepherd (McGill)

Mixed integer nonlinear programming (MINLP) is concerned with finding optimal solutions to mathematical optimization models that combine both discrete and nonlinear elements. Models with this flavor are arising in important applications in many domains, notably chemical engineering, energy, and transportation. Moreover, the well-developed frameworks for discrete and continuous optimization are not sufficient in themselves to attack this new class of problems. The underlying mathematical complexity is not well understood due to the interaction of non-convexities arising from both the discrete and nonlinear components. In particular, there remain theoretical, algorithmic and computational challenges before MINLP can enjoy a success similar to, say, smooth optimization or integer programming. These challenges are at the core of the activities of the “Mixed Integer Nonlinear Programming: Theory and Computation” thematic month at the CRM.

November 1 – 30, 2019 » Mathematical Physiology—Better Health Through Mathematics

Organizers: Jacques Bélair (Montréal), Fahima Nekka (Montréal), John Milton (Claremont)

The proposed activities will deal with the use of mathematical analysis of disease to help develop and deliver new therapies.  It will focus on past successes and new directions with an emphasis on translating theoretical insights into deliverables at the bedside.  We will gather mathematicians, statisticians, benchtop researchers, physicians and students, together with representatives from industry and computer science to discuss the role of mathematics in the detection and treatment of human illness. Workshop activities will include:

Workshops

Dynamical Disease—From the Blackboard to the Bedside

The rapid development of wearable devices, cell phone apps and cloud computing has the promise of providing the continuous monitoring of key physiological variables such as heart rate and body temperature of every individual in a population at risk. Implantable electronic devices and nanotechnologies make it possible to restore physiological functions lost by disease and to treat medical emergencies when they arise. Thus it is possible to develop a personalized medicine in which patients at risk can be identified and even treated before their health deteriorates. Thus the goals of mathematical physiology include the development of mathematical models (1) to uncover disease mechanisms, (2) to develop therapeutic strategies, and (3) to identify the dynamical changes in those physiological variables which can be easily monitored that warn of impending illness.  This workshop will focus on past successes of modeling dynamical diseases, address new modeling directions, and deal with practical aspects of translating theoretical insights into accepted diagnostics and therapy.

Dynamic Approaches to Disease Treatment

Neurons, skeletal and cardiac muscle cells, and certain endocrine cells are examples of excitable cells.  Over the last 70 years the mathematical analysis of excitability has provided fundamentally important insights into, for example, the genesis of cardiac arrhythmias and epileptic seizures.   These insights, in turn, have led to a growing area of medicine in which implantable electronic devices are used to treat medical emergencies when they arise, control pain, and replace functions lost by disease including movement to those who have lost the ability, such as amputees and patients with Parkinson’s disease and, most recently, an artificial pancreas to treat patients with diabetes. Advances in computational capabilities have made possible physiologically “realistic” representations of parts of, and in some cases, entire organs. This workshop addresses the crucial modeling question of determining at what level of detail, for a given organ or system, a mathematical model can be considered “adequate”.

Facilitating mHealth Implementation of Dynamical Approaches

The healthcare system is experiencing a paradigm shift in delivering its services, evolving from a reactive “one-size-fits-all” structure to a patient-centrist model focusing on individualized medicine. However, the dream of providing personalized healthcare to every individual on the planet requires that mathematicians obtain solutions to a number of practical problems.  These issues include, but are not limited to, the identification of the important physiologically accessible parameters to monitor, the development of efficient data mining techniques to detect abnormality and statistical analytic techniques to identify artifacts and determine levels of significance that would motivate medical intervention.   This workshop draws on the experience garnered from drug development protocols that incorporate of data gathered at the level of individuals.   Systems level mathematical insights are provided in the form of pharmacometrics-based decision support tools which bring together validated scientific components, available technical options, considerations of regulatory aspects, and achievement of efficient commercialization.   This workshop is aimed to raise awareness among applied mathematicians and computer scientists to emerging opportunities for the development of mobile applications targeting medical and health care, and discusses the regulatory aspects that should be part of the development process.