April 29 – May 17, 2019 » Faces of Integrability

Organizers: Jacques Hurtubise (McGill), Nicolai Reshetikhin (Berkeley), Lauren K. Williams (Berkeley)

The theory of integrable systems, with its origins in symmetries, has intricate ties to a wide variety of areas of mathematics. Sometimes the ties are straightforward, but in many cases, the links are more complicated, and indeed somewhat difficult to make explicit.  Some of these interfaces, between integrability, geometry, representation theory, and probability theory will be dominating subjects during the conference and satellite activities.  Themes to be covered include the role of cluster algebras and cluster varieties in the description of moduli spaces, the links between integrable systems and representation theory appearing in such areas as quantum groups and quantization of moduli spaces, and the fascinating interfaces of probability theory, combinatorics and integrable systems appearing in several processes linked to statistical mechanical models.

During the first week of activities, April 29 – May 3), introductory lectures for graduate students will take place. It will consist of four four-hour series of lectures:

Gaétan Borot (MPIM)
“Geometric and topological recursion”

Mikhael Gekhtman (Notre Dame)
“Cluster Integrable Systems”

Nicolai Reshetikhin (Berkeley)
“An overview of the construction of integrable systems based on factorizable Poisson Lie groups”

Hugh Thomas (UQAM)
“Introduction to cluster algebras”

A conference will take place during the second week, May 6-10.

Invited Speakers:
Jorgen Andersen (Aarhus), Marco Bertola (Concordia), Alexander Bobenko (Berlin)(*), Alexander Borodin (MIT), Luigi Cantini (Cergy), Filippo Colomo (INSM, Firenze), Sylvie Corteel (Paris-Diderot), Ivan Corwin (Columbia), Rukmini Dey (ICTS- Bangalore), Philippe di Francesco (Illinois)(*), Laszlo Feher (Szeged and Budapest), Vladimir Fock (Strasbourg), Vadim Gorin (MIT)(*), John Harnad (CRM, Concordia), Rinat Kedem (Illinois), Richard Kenyon (Brown), Boris Khesin (Toronto), Alisa Knizel (Columbia), Dmitri Korotkin (Concordia), Osya Mandelshtam (Brown) (*), Marta Mazzocco (Loughborough), Alexander Okounkov (Columbia)(*), Vladimir Rubtsov (Angers)(*), Gus Schrader (Columbia), Vasilisa Schramchenko (Sherbrooke), Alexander Shapiro (Berkeley)(*), Andrey Smirnov (Berkeley)(*), Andrea Sportiello (Paris-Nord), Véronique Terras (Paris-Sud), Taras Skrypnyk (Milan), Jasper Stokman (Amsterdam)(*), Harold Williams (Davis), Pavel Winternitz (CRM, Montréal), Milen Yakimov (Louisiana State)

(*) To be confirmed

During the third week, May 13-17, research discussions and seminars will continue together with follow-up lectures for graduate students.

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

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

The first week (June 3-7) will be devoted to mini-courses meant to introduce the necessary background material:

– Generating Functions, Old and New,
by Sylvain Courte;
– Microlocal Theory of Sheaves,
by Stéphane Guillermou;
– Introduction to 2-Categories,
by André Joyal.

The next two weeks (June 10-14 and June 17-21) will be devoted to series of lectures and discussion sessions on the following recent works:

– Microlocal Category of a Symplectic Manifold,
by Dmitri Tamarkin;
– Wrapped Floer Theory for Liouville Sectors,
by Sheel Ganatra, John Pardon and Vivek Shende;
– Arboreal Skeleta of Weinstein Manifolds,
by David Nadler and Laura Starkston;
– Floer Theory and Quantization of Exact Lagrangians in Cotangent Bundles,
by Claude Viterbo.

A conference on related topics will be held at CRM during the final week of the programme (June 24-28).

More details will be posted on this webpage in the next few weeks.

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.

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.

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

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,