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.

June 1 – 30, 2019 » Homological Algebra, Microlocal Analysis and Symplectic Geometry

Organizers: Emmanuel Giroux (UMI CNRS-CRM & ENS Lyon), Stéphane Guillermou (Grenoble Alpes) 

The purpose of this scientific program will be to present and discuss the recent developments in applications of the microlocal analysis of sheaves to symplectic geometry. We will especially focus on the work of Dmitry Tamarkin, the scholar-in-residence for this program, who will lecture on his microlocal category and its relationships with the Fukaya category. The first week of the program will be devoted to introductory lectures in order to provide young participants with the necessary background. In the subsequent two weeks, Dmitry Tamarkin will present his work in the morning sessions, and more discussions on the contents of his lectures will be scheduled in the afternoon sessions. Finally, a workshop will be organized in the last week of the program.

August 1 – 31, 2019 » Quiver Varieties and Representation Theory

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.

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.