ICTS

This two-week programme investigates how “refinement’’ appears across enumerative geometry, topological recursion, and gauge and string theories. Over the last twenty years, deep connections have been uncovered among these fields in the unrefined setting, often through structures from integrable systems and representation theory. Classic examples include the Kontsevich–Witten theorem, which links intersection theory on moduli spaces, integrable hierarchies, Virasoro constraints, and topological recursion, as well as comparable correspondences in Gromov–Witten theory, cohomological field theories, and $r$-spin theory. Hurwitz theory provides another rich example, connecting enumerative geometry, topological recursion and integrable systems.

Refinement arises in diverse forms across these subjects. In Donaldson–Thomas theory it corresponds to replacing cohomological with K-theoretic invariants; in refined Gromov–Witten theory and gauge theories it appears through torus actions; in $b$-Hurwitz theory it incorporates contributions from non-orientable surfaces; and in several contexts refinement is reflected through working with $\mathcal{W}$-algebras at arbitrary level or via $\beta$-deformed matrix models. This leads to fundamental questions: how refinements in different settings are related; how refinement manifests in the underlying algebraic and integrable structures; and whether the established unrefined correspondences extend to the refined realm.

The programme aims to introduce young researchers to these interconnected topics and to facilitate collaboration between experts on refinements and on the algebraic structures that connect them. The first week offers mini-courses aimed at young researchers; the second features expert talks and ample discussion time to stimulate further progress on refined structures and their relationships.

Accommodation will be provided for outstation participants at our on campus guest house.

Organised with support from the European Mathematical Society and IMU-CDC.

Eligibility Criteria : The program is primarily intended for researchers working in enumerative geometry and combinatorics, representation theory, and mathematical and theoretical physics. Graduate students and postdoctoral researchers interested in these topics are particularly encouraged to apply.

ICTS is committed to building an environment that is inclusive, non discriminatory and welcoming of diverse individuals. We especially encourage the participation of women and other under-represented groups. 

Program registration start date: December 20th, 2025
Registration deadline for participants: March 31st, 2026