Geometry, Algebra and Combinatorics of Moduli Spaces and Configurations VIII

This is the 8th edition of the graduate school held in Toblach in winter 2017, winter 2018, winter 2019, winter 2020, winter 2023, winter 2024 and winter 2025. There will be three 5 hours minicourses, some talks and ample time for discussion.

Mini-courses

• Ivan Marin, Université de Picardie-Jules Verne, Complex braid groups and their Hecke algebras.
• Christian Stump, Ruhr-Universität Bochum, Algebraic and enumerative invariants of finite graded bounded posets. [show abstract]

  Abstract:
  In this lecture series, I present classical and modern combinatorial invariants of such posets and their unifying nature across algebraic combinatorics. These invariants include classical structures such as the Möbius function, characteristic polynomial, incidence algebra, chain polynomial, ab-index, and (flag) f- and h-vectors. I also present recently defined families of polynomials associated to such a poset that gained a lot of attention in algebraic combinatorics: the Poincaré-extended ab-index, flag coarse Hilbert-Poincaré series, and Chow polynomial. I discuss their properties such as their unimodality, log-concavity, gamma-positivity, and real-rootedness. Throughout the lecture series, I will in particular discuss known results and conjectures for interesting families of examples: boolean lattices, (ordered) set partition lattices, intersection posets of hyperplane arrangements, lattices of flats of matroids, simplicial posets, shellable posets, and posets that admit R-labelings.
  [Hide abstract]

• Mattia Talpo, Università di Pisa, Logarithmic geometry and tropicalization. [show abstract]

  Abstract: Logarithmic geometry is an enhanced version of algebraic geometry, where varieties are equipped with an extra sheaf of monoids, whose sections should be somehow thought of as ``monomial regular functions’’, and which is essentially of a combinatorial nature. Prototypical examples (and building blocks in some sense) are affine toric varieties. This extra structure is often useful when dealing with degenerations of smooth objects, and, correspondingly, compactifications of certain moduli spaces. There are also fruitful connections to tropical geometry, a piecewise-linear, combinatorial version of algebraic geometry, via a tropicalization procedure, leveraging on the close relationship between a toric variety and its fan. I will give an introduction to these topics and their relationship, also mentioning relevant past and current applications.
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Talks:

• Tommaso Faustini, University of Warwick. Flag enumeration for connected split matroids. [show abstract]

  Abstract: Many matroid invariants satisfy a valuative inclusion-exclusion property with respect to subdivisions of the base polytope. However, face counts notoriously do not. This talk focuses on the cd-index, a non-commutative polynomial that compactly encodes the flag f-vector, and its behavior on the base polytopes of connected split matroids. Split matroids are a broad family obtained by "chopping" the hypersimplex with compatible hyperplanes. I will introduce the geometry of these matroids and the basics of the cd-index before presenting a recent formula for their computation. We will see that while the cd-index is not strictly valuative, it is "almost" so: the deviation from valuativeness is controlled entirely by modular pairs of cyclic flats. This structural insight allows us to reduce complex polyhedral calculations to simpler combinatorial countings. [Hide abstract]

• Lorenzo Giordani, Università di Bologna, Ruhr-Universität Bochum. Chow rings for toric arrangements and operads over Feynman categories. [show abstract]

  Abstract: In 1995 De Concini and Procesi introduced wonderful models for subspace arrangements and computed their cohomology (and the complement space's) using combinatorial information only. After their seminal work, a number of variations of the theme appeared, from Chow rings for atomic lattices to wonderful models for toric arrangements. The "set" of all possible Chow rings of simple matroids, which specialize to cohomology rings of wonderful models for hyperplane arrangements, was given an operad-like structure in a work by Coron. There, a Koszul duality property was proven, rediscovering the link between Orlik-Solomon algebras and Chow rings. In this talk I will present work in progress where we endow the "set" of all Chow rings for toric arrangements with a structure of an operad over a suitable Feynman category. In particular, we provide a presentation for it in terms of generators and relations. This is work in progress with Moci, Pagaria, Paolini, and Rossi. [Hide abstract]

• Daniel Holmes, ISTA (Institute of Science and Technology Austria). From Gromov-Witten theory to GKM theory and back. [show abstract]

  Abstract: At the intersection of geometry, combinatorics, and algebra lies a fruitful two-way interaction of Gromov-Witten theory and GKM theory that is established by equivariant localization. In one direction, GKM theory provides a setting where Gromov-Witten invariants are explicitly computable (which we have implemented in joint work with Giosuè Muratore). In the other direction, the axiomatic behavior of Gromov-Witten invariants is a powerful constraint yielding structural results on algebraic and, more generally, Hamiltonian GKM spaces. I will present recent results in both directions. [Hide abstract]

• Lucrezia Beatrice Lorenzi, SISSA. Bonded links and related algebraic structures. [show abstract]

  Abstract: In a recent work, Diamandis, Kauffman, and Lambropoulou introduced bonded links: a modern structure in the realm of knot theory, consisting of classical links endowed with embedded arcs with endpoints on the link, called bonds. Analogously to the classical Alexander theorem and the standard closure of braids, which relate classical links to the braid group (on n strands, B_n), in the suitable topological context their algebraic behaviour is described by the monoid of bonded braids (on n strands, BBM_n). The existence of a map BB_n \to B_n, which sends bonds to Artin pure braid generators A_ij, sheds further light on the nature of bonds. Time permitting, we will dwell in motivation for quantum invariants of said objects. Based on a joint work in progress (in the final stages) with Sofia Lambropoulou. [Hide abstract]

• Lorenzo Perrone, Università di Bologna. Computing Joins in the Weak Order of Coxeter groups of type B. [show abstract]

  Abstract: We present an algorithm for computing the join of two elements in the weak order of Coxeter groups of type B through their combinatorial description. This extends to signed permutations an algorithm of Markowsky that determines the join of two standard permutations. Furthermore, we provide two more combinatorial characterizations of this join. One is based on statistics of signed permutations: inversions, negative entries and negative sum pairs. The other proves a conjecture of Dyer concerning a connection between the weak order and the Bruhat graph. [Hide abstract]

• Anaëlle Pfister, Max Planck Institute for Mathematics in the Sciences. Positive Geometries for Hyperplane and Toric Arrangement. [show abstract]

  Abstract: Positive geometry is a new mathematical topic inspired by observations of geometric structures in particle physics and cosmology. Motivated by the examples of the moduli space of smooth marked curves of genus 0 and of Del Pezzo surfaces, I will explain how hyperplanes and toric arrangements arise as positive geometries. [Hide abstract]

• Tommaso Rossi, Università di Bologna. Homology operations for gravity algebras. [show abstract]

  Abstract: Let B_n be the braid group with n-strands and Z(B_n) its center. The (integral) homology of B_n was computed in the seventies by F. Cohen. In this talk we will see how to compute the homology of H_*(B_n/Z(B_n); F_p) for any n natural number and p prime. The approach will be topological, since the classifying space of B_n/Z(B_n) can be realized as the (homotopy) quotient C_n(R^2)//S^1, where C_n(R^2) is the unordered configuration space of point in the plane. Combining the results of F. Cohen with techniques from equivariant cohomology we can do the computation. This talk is based on https://arxiv.org/abs/2404.10639. [Hide abstract]

• Tommaso Scognamiglio, INDAM, Università di Bologna. Multiplicities for finite reductive groups, Macdonald polynomials and character varieties for Riemann surfaces. [show abstract]

  Abstract: The representation theory of the finite group GLn(Fq) is well understood, thanks to its combinatorial description (given by Green) and its geometric interpretation due to the results of Deligne and Lusztig. Still, we have very little understanding of the multiplicities, i.e. of the decomposition of tensor product of irreducible representations. Hausel, Letellier and Rodriguez-Villegas gave a combinatorial description of the multiplicities in the generic case. This description relates these multiplicities to the cohomology of complex character varieties for GLn(C), i.e. moduli spaces of local systems. The combinatorics appearing in their result is related to Macdonald and Hall-Littlewood polynomials. In a joint work with Emmanuel Letellier, we try to generalize these results to other groups of type A. In particular, we study multiplicities for characteristic functions of character sheaves of SLn(Fq), rather than irreducible characters, and relate them to the cohomology of complex character varieties for PGLn(C). We expect more generally that, for a finite reductive G(Fq), multiplicities should be related to the character varieties for its complex Langlands dual. If time allows, I will point at some combinatorial interpretation of this approach regarding wreath Macdonald polynomials. [Hide abstract]




Schedule




Venue

The school will be held in Dobbiaco (Toblach) at the Kulturzentrum Grand Hotel Centro Culturale, in "La Sala degli Specchi". Kulturzentrum Grand Hotel Centro Culturale is located in front of Toblach train station (see the map).
Participants are expected to arrive on Sunday 31/5 and to leave on Friday 5/6 after lunch.

Registration and support

Participants are required to register by filling in this registration form.
It would be very helpful for the organization if you fill in the form as soon as possible, thanks!

Limited funding will be available to cover local expenses (lodging, breakfast and dinner) of few Master or PhD students.
To apply for financial support you are required to fill in the same form.
The deadline to apply for funding is February 28, 2026.
Again, we encourage to apply as soon as possible in order to help the organization, thanks!

For any question, you can contact the organizers at the following email address: toblachconfigurations@gmail.com.

This workshop is supported by:

MUR, Dipartimento di eccellenza di Matematica, University of Pisa.

Dipartimento di matematica, Università di Bologna.

Instituto Nazionale Di Alta Matematica INDAM.

GNSAGA, INDAM.

University "La Sapienza" of Rome, Progetti di Ateneo Representation Theory and Applications, 2023-2024.

Organizers: Filippo Callegaro, Michele D'Adderio , Luis Ferroni , Marco Franciosi , Roberto Pagaria and Paolo Papi.