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.
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:
- The algebraic invariant theory of polynomial differential systems.
- Integrability of polynomial differential systems.
- Algorithms for effective computations of algebraic and geometric properties of polynomial vector fields.
- Hilbert’s 16th problem.
- Counting problems on particular solutions of polynomial vector fields.
- Singular perturbations problems occurring in planar slow-fast systems.
Organizers: Joel Kamnitzer (Toronto), Hugh Thomas (UQAM)
The representation theory of quivers (and related preprojective algebras) has been studied by researchers from algebra, while the geometry of quiver varieties has been studied by researchers in geometric representation theory. This activity will bring together members of these two communities to exchange recent progress and to stimulate further research and collaboration. Among other topics, we will discuss quantization of quiver varieties, Coulomb branch constructions using quiver varieties, tilting theory for preprojective algebras, and categorification of cluster algebras.
August 5-9: Focus week on cluster algebras and quiver representations, with a mini-course by Pierre-Guy Plamondon
August 12-16: Workshop on quiver varieties and representation theory
August 19-23: Focus week on quiver varieties, with mini-courses by Michael Finkelberg and Hiraku Nakajima
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,
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.