# Category: Topology

## 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 – 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,