Sara Mehidi (Utrecht): Compactification of covers over nodal degenerations via log geometry
Abstract: When algebraic varieties degenerate, coverings and torsors defined on the smooth locus need not extend across the boundary: ramification and singularities create natural obstructions. Logarithmic geometry provides a framework in which mild ramification and certain singularities behave better, allowing the extension problem to be reformulated in this setting.
Starting with the elementary notion of a torsor and gradually moving to its scheme-theoretic counterpart, I will explain why extension can fail in the classical setting and how logarithmic geometry provides a remedy. This will lead us to logarithmic torsors. For families of nodal curves, these torsors are classified by the logarithmic Jacobian. I will explain how, under suitable hypotheses, this classification and the extension properties of the logarithmic Jacobian yield natural extension results for torsors across nodal degenerations.
Xavier Blot (Amsterdam): From Witten–Kontsevich to integrable observables
Abstract: In the early 90s Witten and Kontsevich discovered that the intersection numbers of the so-called psi-classes on moduli of curves are governed by the KdV hierarchy, providing the first bridge between the enumerative geometry of curves and integrable systems. How far does this extend? Many geometric theories, from Gromov–Witten to r-spin invariants, turn out to be controlled by integrable hierarchies. I will survey this story and present a recent work showing that from the integrability perspective, psi classes are just one choice among many "integrable observables". We recover in this way two famous theories: the Dubrovin—Zhang and double ramification theories while producing a new one coming from quantization.
Tommy Lundemo (Utrecht): Logarithmic cohomology theories in topology and arithmetic
Abstract: I will survey recent work on how ideas from logarithmic geometry have been applied to tackle problems in homotopy theory and arithmetic geometry. Combining results from both sides, we arrive at an application to (the “deformational” part of) the variational Hodge conjecture. This will touch upon joint projects with Binda, Merici, Park, and Østvær.
Giulio Ruzza(Lisbon): Shifted symmetric functions on partitions and quasimodular forms in the quantum KdV hierarchy
Abstract: I will review the classical integrable hierarchy of the KdV equation and its two quantum versions (associated with the two hamiltonian structures of classical KdV), which are prototypical examples of (classical and quantum) integrable systems. Then, I will discuss the natural appearance of shifted symmetric functions on partitions and of quasimodular forms in the spectral problem of the quantum Hamiltonian operators of the KdV hierarchy. The quantization of the second KdV hamiltonian structure is essentially the Virasoro algebra; focusing on the case of Virasoro central charge = -2, I will explain how this case is related to certain monodromy problems in the theory of complex ODEs (a relation predicted by Physicists) and to the combinatorics of a remarkable family of special functions arising from Wronskians of Laguerre polynomials.
Based on joint works with Jan-Willem van Ittersum and with Davide Masoero.
Leonid Ryvkin(Lyon 1): Tepui fibrations and singular vector bundles
Abstract:
As a differential-geometric object, a tepui fibration is a type of singular fiber bundle with smooth base space and smooth fibers, however the fiber dimension might jump when moving from one base point to another. Tepui fibrations naturally turn up in differential geometry, when one is quotienting by a smooth family of symmetries, which degenerates at certain points, e.g. in the context of singular foliations. In this talk I will give an introduction to tepui fibrations and show how they provide a natural way to extend the classical Serre-Swan theorem beyond the setting of projective modules.
Based on joint work with Alfonso Garmendia and David Miyamoto.
https://arxiv.org/pdf/2510.20936
Michael Heins(TU Delft): Convergent twist deformations
We discuss a functorial framework for the convergence of Drinfeld’s Universal Deformation Formula on spaces of analytic or entire vectors.
Algebraically, this type of deformation is based on a Drinfeld twist
F = ∑ₙ₌₀^∞ ℏⁿ · Fₙ ∈ (U(𝔤) ⊗ U(𝔤))[[ℏ]].
It constitutes a distinguished—and, in terms of explicit formulas, rather elusive—formal power series with coefficients in two copies of the universal enveloping algebra U(𝔤) of some Lie algebra 𝔤.
Drinfeld’s principal idea is that the defining axioms of a twist induce formal deformations of any associative algebra 𝒜 on which the Lie algebra 𝔤 acts by derivations. That is, one obtains an associative product on 𝒜[[ℏ]] that deforms the original algebra structure.
By equipping the representation space with a locally convex topology, we overcome the formal character of this deformation: we pass from a formal parameter ℏ to a complex number ℏ ∈ ℂ. This is achieved by matching an equicontinuity condition on the action of the components Fₙ of the twist with the order of analytic vectors of the representation.
Finally, we demonstrate the effectiveness of our machinery by applying it to the explicit non-abelian Drinfeld twists constructed by Giaquinto and Zhang.
This is joint work with Chiara Esposito and Stefan Waldmann.
Francesca Pratali (Utrecht): Operads and trees
In this talk, we will introduce the basic notions of operads and algebras over an operad, and explain how they provide a powerful framework for studying the homotopy theory of seemingly different algebraic structures in a unified way. The fun part of the seminar is that we will use the language of trees, namely certain directed connected graphs whose combinatorics makes operad theory more visual and intuitive. This formalism, introduced by Moerdijk and Weiss, also paves the way to the world of ∞-operads—though that will probably be a story for another seminar!