# Category: Algebra

## 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.

## 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:

- 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.