Current trends in Calabi-Yau moduli

Strasbourg, June 10-13, 2025

Petite France, painting generated by Dall E

Mini courses:

Tasuki Kinjo: Intrinsic DT theory and Higgs bundles.

Donaldson--Thomas (DT) theory is a counting theory of coherent sheaves on Calabi-Yau threefolds, and more generally, of objects in 3-Calabi--Yau categories. Recently, DT theory and its categorification have been playing an important role in the study of the topology of the moduli space of objects in 3-Calabi--Yau categories as well as in 2-Calabi--Yau categories through a procedure called dimensional reduction. In particular, DT theory has applications to the study of quiver varieties and the moduli space of Higgs bundles on a curve. This mini-course starts with an introduction to a new framework of DT theory intrinsic to the moduli stack, which we call `intrinsic DT theory’. This framework enables us to count non-linear objects such as G-bundles and G-Higgs bundles. Using this theory, we will define the BPS cohomology of the moduli space of G-Higgs bundles on a curve, and propose a formulation of the topological mirror symmetry conjecture for semisimple G in the style of Hausel and Thaddeus. I will also explain that an idea from the non-commutative Calabi--Yau geometry can be used to prove a version of the topological mirror symmetry for the moduli space of GL_n-Higgs bundles known as the χ-independence. This mini-course includes several joint works with Chenjing Bu, Ben Davison, Daniel Halpern-Leistner, Naoki Koseki, Naruki Masuda, Andrés Ibáñez Núñez, and Tudor Pădurariu.

Olivier Schiffmann : Cohomological Hecke operators of punctual modifications on smooth surfaces

Operators of punctual modifications are an old and classical tool in the study of moduli spaces of vector bundles on curves (as well as in the function field version of the Langlands program). Similar operators have been used by Grojnowski, Nakajima (and then many others) in order to describe the cohomology of Hilbert schemes of points on smooth surfaces. The formalism of cohomological Hall algebras (of zero-dimensional sheaves on a given surface S) allows one to study the algebra of all such operators (independently of the particular choice of moduli space of sheaves on S considered). We will explain how to compute these COHAs for a smooth quasi-projective surface (at least when the cohomology of S is pure) in terms of the so-called W_{1+\infty}-algebra associated to H^*(S,Q), and indicate some applications. This is joint work with A. Mellit, A. Minets and E. Vasserot ([2311.13415] Coherent sheaves on surfaces, COHAs and deformed W_{1+\infty}-algebras)

Research talks

Merlin Christ
Ben Davison
Soheyla Feyzbakhsh
Jérémy Guéré
Francesco Sala
Sarunas Kaubrys
Richard Thomas

Schedule
Titles and Abstracts

Registration

Practical information

Location: Salle de Conférence (ground floor), IRMA building, Institut de recherche mathématique avancée (IRMA) .
Address: 7 rue René Descartes, 67084 Strasbourg Cedex.
You could take the tramway Line C, get off at the stop "Universités", and take the road/rue Edmond Labbé.
After 100 meters, without making any turns, the road changes its name into "rue du général Zimmer". Then you can see the 7-floor IRMA building on your left.
For a picture of the building: More information

Coffee breaks: the common room next to the Salle de Conférence.

Organizers

Lie Fu
Mauro Porta