A one-day workshop on Enumerative Geometry and Geometric Representation Theory

Wednesday Dec 10, 2025, 9:00 AM → 6:00 PM Europe/Rome

Aula F (Polo Fibonacci)

The one-day workshop “Enumerative Geometry and Geometric Representation Theory” will take place at the Department of Mathematics, University of Pisa, on December 10, 2025. The workshop aims to bring together researchers working at the intersection of algebraic geometry, representation theory, and mathematical physics.

In addition to four 1-hour invited talks, a few slots will be available for short contributed presentations. If you are interested in giving a contributed talk, please indicate this in the registration form.

A limited amount of funding is available to support participants. If you wish to request financial support, please do so through the registration procedure.

Deadline to request support and/or apply for a contributed talk: November 14, 2025

Deadline for registration: December 5, 2025

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Speakers:

Andrea Appel (Università degli Studi di Parma)

Andrea Maffei (Università di Pisa)

Yaping Yang (University of Melbourne)

Gufang Zhao (University of Melbourne)

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Contributed Talks:

Tommaso Scognamiglio (Università di Bologna)

Juan Sebastian Numpaque Roa (Universidade do Porto)
 

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Organizers: 

Francesco Sala (Università di Pisa)

Mattia Talpo (Università di Pisa)

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The workshop is supported by the University of Pisa and by Prin 2022 "Geometry of algebraic structures: moduli, invariants, deformations" (2022BTA242).
 

1 Quotients of the double loop group

For a smooth affine algebraic group $G$ over an algebraically closed field, we consider some two-variables generalizations of the affine Grassmannian $G((t))/G[[t]]$, given by quotients of the double loop group $G((x))((y))$. We prove that they are representable by ind-schemes if $G$ is solvable. Given a smooth surface $X$ and a flag of subschemes of $X$, we provide a geometric interpretation of the two-variables Grassmannians, in terms of bundles and trivialisation data defined on appropriate loci in $X$, which depend on the flag. This is a joint work with Valerio Melani and Gabriele Vezzosi.

Speaker: Andrea Maffei (Università di Pisa)

10:00 AM Coffee break

2 Line Operators, cohomological Hall algebras, and affine Grassmannian

Motivated by the study of categories of line operators in 4d N=2 gauge theories, we construct a triangulated monoidal category with a weak braiding starting from a symmetric 3 Calabi-Yau category equipped with additional structures. When the 3 Calabi–Yau category arises from the root datum of an algebraic group, we compare the resulting category with the equivariant derived category of coherent sheaves on the affine Grassmannian.

This talk is based on ongoing joint work with Fujita, Soibelman, and Zhao.

Speaker: Yaping Yang (University of Melbourne)

11:30 AM Break

3 Tensor products of quiver bundles

In this work we introduce a notion of tensor product of (twisted) quiver representations with relations in the category of $\mathcal{O}_X$-modules. As a first application of our notion, we see that tensor products of polystable quiver bundles are polystable and later we use this to both deduce a quiver version of the Segre embedding and to identify distinguished closed subschemes of $\mathsf{GL}(n,\mathbb{C})$-character varieties of free abelian groups.

Speaker: Juan Sebastian Numpaque Roa (Universidade do Porto)

4 Cohomology of singular moduli spaces in the non-abelian Hodge theory of a curve

The geometry of the moduli spaces appearing in the non-abelian Hodge theory of a Riemann surface (i.e. moduli spaces of Higgs bundles and character stacks) is well understood in the smooth case. In the singular case, the situation is more complicated. In this article, we give a conjectural formula for the mixed Poincaré series of character stacks for Riemann surfaces. Furthermore, we verify it under Euler's specialization (i.e., we calculate the E-series). Such a formula was previously known only in the smooth case, thanks to the work of Hausel, Letellier, and Rodriguez-Villegas. Our formula expresses the singular case as a sort of "symmetric power" of the smooth ones. Similar results have appeared in multiple works in related areas. The results of the article provide important evidence in support of the chi-independence property of the coomology of these stacks.

Speaker: Tommaso Scognamiglio (Università di Bologna)

12:40 PM Lunch Break

5 Quantum toroidal algebras, affine Yangians and abelian qKZ equations

In this talk, I will present the construction of abelian meromorphic R-matrices for quantum toroidal algebras, providing a meromorphic braiding on category O representations with respect to the (rational) tensor product induced by Drinfeld coproduct. I will then explain how, under the equivalence of Gautam and Toledano Laredo, these R-matrices compute the monodromy of the abelian qKZ equations for the corresponding affine Yangians. This is based on joint work with S. Gautam.

Speaker: Andrea Appel (Università degli Studi di Parma)

4:00 PM Coffee break

6 Quantum K-Theory of Critical Loci

In this talk, I will define a pullback map from the Grothendieck group of coherent matrix factorizations to that of coherent sheaves on a (-1)-shifted Lagrangian inside the critical locus of a function. This map satisfies natural functoriality under composition of Lagrangian correspondences, along with expected properties such as bivariance and base change. I will explain how this construction arises naturally in the study of quantum K-theory for critical loci, with examples drawn from moduli spaces associated to quivers with potentials. This is based on joint work with Y. Cao and Y. Toda.

Speaker: Gufang Zhao (University of Melbourne)