July 1 – 31, 2019 » Expansions, Lie Algebras and Invariants

Organizers: Anton Alekseev (Genève), Dror Bar-Natan (Toronto), Roland van der Veen (Leiden)

Our workshop will bring together a number of experts working on “expansions” and a number of experts working on “invariants” in the hope that the two groups will learn from each other and influence each other. “Expansions” are solutions of a certain type of intricate equations within graded spaces often associated with free Lie algebras; they include Drinfel’d associators, solutions of the Kashiwara–Vergne equations, solutions of various deformation quantization problems, and more. By “invariants” we refer to quantum-algebra-inspired invariants of various objects within low-dimensional topology; these are often associated with various semi-simple Lie algebras. The two subjects were born together in the early days of quantum group theory, but have to a large extent evolved separately. We believe there is much to gain by bringing the two together again.

September 1 – 8, 2019 » Workshop on New Trends in Polynomial Differential Systems

Organizers: Jaume Llibre (Barcelona), Dana Schlomiuk (Montréal)

Polynomial vector fields occur in many areas of applied mathematics such as for example in population dynamics, chemistry, electrical circuits, neural networks, shock waves, laser physics, hydrodynamics, etc. They are also important from the theoretical point of view. Three problems about these systems stated more than one hundred years ago are still open. Theoretical developments in this area of research are bound to shed light on these very hard open problems and have an impact on applications. In recent years a number of new significant results were obtained on families of polynomial vector fields. The goal of this workshop is to focus on these new developments, facilitate scientific exchanges and stimulate further activity in this growing area of research.

Some of the points which will be discussed are the following:

  1. The algebraic invariant theory of polynomial differential systems.
  2. Integrability of polynomial differential systems.
  3. Algorithms for effective computations of algebraic and geometric properties of polynomial vector fields.
  4. Hilbert’s 16th problem.
  5. Counting problems on particular solutions of polynomial vector fields.
  6. Singular perturbations problems occurring in planar slow-fast systems.

September 1 – 30, 2019 » Low-Dimensional Topology

Organizers: Steven Boyer (UQAM), Liam Watson (Sherbrooke)

As part of the 50th anniversary program, the CRM will host a thematic month in low-dimensional topology (September 2019). This is an area of research that includes geometric topology in dimensions 3 and 4, knot theory, and geometric group theory (to name a few) while drawing on techniques from symplectic topology and gauge theory towards the resolution of long-standing problems. While many new connections are being established, the field as a whole is at an exciting crossroads; some of the greatest open problems have been resolved—such as the geometrization of 3-manifolds due to Perelman and the positive resolution of the virtual Haken conjecture due to Agol and Wise. These works have opened new vistas of questions and conjectures for further study.

The CRM is delighted that Ciprian Manolescu will serve as distinguished researcher-in-residence for the thematic month. Manolescu’s recent and highly celebrated disproof of the triangulation conjecture [Pin(2)-equivariant Seiberg–Witten Floer homology and the triangulation conjecture, Journal of the American Mathematical Society, 2016] is emblematic of the current activity in low-dimensions described above. Manolescu’s work develops gauge-theoretic tools that resolve low-dimensional problems known to be linked to the existence of triangulations in high dimensions. This has re-invigorated gauge theoretic methods in low-dimensions, as evidenced by a string of exciting work from Manolescu and his students.

The aim of this focused month, which will coalesce around the work of the program’s researcher-in-residence, will be to take stock of current developments in the field and highlight the many exciting new directions present in this area of research. As such, the program will endeavour to include and be accessible to early career researchers in low-dimensional topology and related fields,

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:


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.